Statistics For Business I

Lecture notes on Statistics for business I
CHAPTER ONE
OVERVIEW OF STATISTICS
Definition and classifications of statistics
Definition:
We can define statistics in two ways.
1. Plural sense (lay man definition).
It is an aggregate or collection of numerical facts.
2. Singular sense (formal definition)
Statistics is defined as the science of collecting, organizing, presenting, analyzing
and interpreting numerical data for the purpose of assisting in making a more
effective decision.
Classifications:
Depending on how data can be used statistics is some times divided in to two main areas
or branches.
1. Descriptive Statistics: is concerned with summary calculations, graphs, charts and
tables.
2. Inferential Statistics: is a method used to generalize from a sample to a population.
For example, the average income of all families (the population) in Ethiopia can be
estimated from figures obtained from a few hundred (the sample) families.
 It is important because statistical data usually arises from sample.
 Statistical techniques based on probability theory are required.
Stages in Statistical Investigation
There are five stages or steps in any statistical investigation.

1. Collection of data: the process of measuring, gathering, assembling the raw data up
on which the statistical investigation is to be based.
 Data can be collected in a variety of ways; one of the most common methods
is through the use of survey. Survey can also be done in different methods,
three of the most common methods are:
 Telephone survey
 Mailed questionnaire
 Personal interview.
Exercise: discuss the advantage and disadvantage of the above three methods
with respect to each other.
2. Organization of data: Summarization of data in some meaningful way, e.g table form
3. Presentation of the data: The process of re-organization, classification, compilation,
and summarization of data to present it in a meaningful form.
4. Analysis of data: The process of extracting relevant information from the summarized
data, mainly through the use of elementary mathematical operation.
5. Inference of data: The interpretation and further observation of the various statistical
measures through the analysis of the data by implementing those methods by which
conclusions are formed and inferences made.
 Statistical techniques based on probability theory are required.
Definitions of some terms
a. Statistical Population: It is the collection of all possible observations of a specified

characteristic of interest (possessing certain common property) and being under
study. An example is all of the students in AAU 3101 course in this term.
b. Sample: It is a subset of the population, selected using some sampling technique in
such a way that they represent the population.
c. Sampling: The process or method of sample selection from the population.
d. Sample size: The number of elements or observation to be included in the sample.
e. Census: Complete enumeration or observation of the elements of the population. Or
it is the collection of data from every element in a population
f. Parameter: Characteristic or measure obtained from a population.
g. Statistic: Characteristic or measure obtained from a sample.
h. Variable: It is an item of interest that can take on many different numerical values.
Types of Variables or Data:
1. Qualitative Variables are nonnumeric variables and can't be measured. Examples

include gender, religious affiliation, and state of birth.
2. Quantitative Variables are numerical variables and can be measured. Examples
include balance in checking account, number of children in family. Note that quantitative
variables are either discrete (which can assume only certain values, and there are usually
"gaps" between the values, such as the number of bedrooms in your house) or continuous
(which can assume any value within a specific range, such as the air pressure in a tire.)
Applications, Uses and Limitations of statistics
Applications of statistics:
 In almost all fields of human endeavor.
 Almost all human beings in their daily life are subjected to obtaining numerical
facts e.g. abut price.
 Applicable in some process e.g. invention of certain drugs, extent of
environmental pollution.
 In industries especially in quality control area.
Uses of statistics:
The main function of statistics is to enlarge our knowledge of complex phenomena. The
following are some uses of statistics:
1. It presents facts in a definite and precise form.
2. Data reduction.
3. Measuring the magnitude of variations in data.
4. Furnishes a technique of comparison
5. Estimating unknown population characteristics.
6. Testing and formulating of hypothesis.
7. Studying the relationship between two or more variable.

8. Forecasting future events.
Limitations of statistics
As a science statistics has its own limitations. The following are some of the limitations:
 Deals with only quantitative information.
 Deals with only aggregate of facts and not with individual data items.
 Statistical data are only approximately and not mathematical correct.
 Statistics can be easily misused and therefore should be used be experts.
Scales of measurement
Proper knowledge about the nature and type of data to be dealt with is essential in order
to specify and apply the proper statistical method for their analysis and inferences.
Measurement scale refers to the property of value assigned to the data based on the
properties of order, distance and fixed zero.
In mathematical terms measurement is a functional mapping from the set of objects {O i}

to the set of real numbers {M(Oi)}.
The goal of measurement systems is to structure the rule for assigning numbers to objects
in such a way that the relationship between the objects is preserved in the numbers
assigned to the objects. The different kinds of relationships preserved are called
properties of the measurement system.
Order
The property of order exists when an object that has more of the attribute than another
object, is given a bigger number by the rule system. This relationship must hold for all
objects in the "real world".
The property of ORDER exists

When for all i, j if Oi > Oj, then M(Oi) > M(Oj).
Distance
The property of distance is concerned with the relationship of differences between

objects. If a measurement system possesses the property of distance it means that the unit
of measurement means the same thing throughout the scale of numbers. That is, an inch
is an inch, no matters were it falls - immediately ahead or a mile downs the road.
More precisely, an equal difference between two numbers reflects an equal difference in
the "real world" between the objects that were assigned the numbers. In order to define
the property of distance in the mathematical notation, four objects are required: Oi, Oj, Ok,
and Ol . The difference between objects is represented by the "-" sign; O i - Oj refers to the
actual "real world" difference between object i and object j, while M(O i) - M(Oj) refers to
differences between numbers.
The property of DISTANCE exists, for all i, j, k, l
If Oi-Oj ≥ Ok- Ol then M(Oi)-M(Oj) ≥ M(Ok)-M( Ol ).
Fixed Zero
A measurement system possesses a rational zero (fixed zero) if an object that has none of
the attribute in question is assigned the number zero by the system of rules. The object
does not need to really exist in the "real world", as it is somewhat difficult to visualize a
"man with no height". The requirement for a rational zero is this: if objects with none of
the attribute did exist would they be given the value zero. Defining O 0 as the object with
none of the attribute in question, the definition of a rational zero becomes:
The property of FIXED ZERO exists if M(O0) = 0.
The property of fixed zero is necessary for ratios between numbers to be meaningful.
SCALE TYPES
Measurement is the assignment of numbers to objects or events in a systematic fashion.

Four levels of measurement scales are commonly distinguished: nominal, ordinal,
interval, and ratio and each possessed different properties of measurement systems.
Nominal Scales
Nominal scales are measurement systems that possess none of the three properties stated
above.
 Level of measurement which classifies data into mutually exclusive, all inclusive
categories in which no order or ranking can be imposed on the data.
 No arithmetic and relational operation can be applied.
Examples:
o Political party preference (Republican, Democrat, or Other,)

o Sex (Male or Female.)
o Marital status(married, single, widow, divorce)
o Country code
o Regional differentiation of Ethiopia.
Ordinal Scales
Ordinal Scales are measurement systems that possess the property of order, but not the
property of distance. The property of fixed zero is not important if the property of
distance is not satisfied.
 Level of measurement which classifies data into categories that can be ranked.
Differences between the ranks do not exist.
 Arithmetic operations are not applicable but relational operations are applicable.
 Ordering is the sole property of ordinal scale.
Examples:
o Letter grades (A, B, C, D, F).

o Rating scales (Excellent, Very good, Good, Fair, poor).
o Military status.
Interval Scales
Interval scales are measurement systems that possess the properties of Order and
distance, but not the property of fixed zero.
 Level of measurement which classifies data that can be ranked and differences are
meaningful. However, there is no meaningful zero, so ratios are meaningless.
 All arithmetic operations except division are applicable.
 Relational operations are also possible.
Examples:
o IQ
o Temperature in oF.
Ratio Scales
Ratio scales are measurement systems that possess all three properties: order, distance,
and fixed zero. The added power of a fixed zero allows ratios of numbers to be
meaningfully interpreted; i.e. the ratio of Bekele's height to Martha's height is 1.32,
whereas this is not possible with interval scales.
 Level of measurement which classifies data that can be ranked, differences are
meaningful, and there is a true zero. True ratios exist between the different units
of measure.
 All arithmetic and relational operations are applicable.
Examples:
o Weight
o Height
o Number of students
The following present a list of different attributes and rules for assigning numbers to
objects. Try to classify the different measurement systems into one of the four types of
scales. (Exercise)
1. Your checking account number as a name for your account.

2. Your checking account balance as a measure of the amount of money you have in
that account.
3. The order in which you were eliminated in a spelling bee as a measure of your
spelling ability.
4. Your score on the first statistics test as a measure of your knowledge of statistics.
5. Your score on an individual intelligence test as a measure of your intelligence.
6. The distance around your forehead measured with a tape measure as a measure of
your intelligence.
7. A response to the statement "Abortion is a woman's right" where "Strongly
Disagree" = 1, "Disagree" = 2, "No Opinion" = 3, "Agree" = 4, and "Strongly
Agree" = 5, as a measure of attitude toward abortion.
8. Times for swimmers to complete a 50-meter race
9. Months of the year Meskerm, Tikimit…
10. Socioeconomic status of a family when classified as low, middle and upper
classes.
11. Blood type of individuals, A, B, AB and O.

12. Pollen counts provided as numbers between 1 and 10 where 1 implies there is
almost no pollen and 10 that it is rampant, but for which the values do not
represent an actual counts of grains of pollen.
13. Regions numbers of Ethiopia (1, 2, 3 etc.)
14. The number of students in a college;
15. the net wages of a group of workers;
16. the height of the men in the same town;
METHODS OF DATA COLLECTION

There are two sources of data:
1. Primary Data
 Data measured or collect by the investigator or the user directly from the
source.
 Two activities involved: planning and measuring.
a) Planning:
 Identify source and elements of the data.
 Decide whether to consider sample or census.
 If sampling is preferred, decide on sample size, selection
method,… etc
 Decide measurement procedure.
 Set up the necessary organizational structure.
b) Measuring: there are different options.
 Focus Group
 Telephone Interview
 Mail Questionnaires
 Door-to-Door Survey
 Mall Intercept
 New Product Registration
 Personal Interview and
 Experiments are some of the sources for collecting the
primary data.
2. Secondary Data
 Data gathered or compiled from published and unpublished sources or
files.
 When our source is secondary data check that:
 The type and objective of the situations.
 The purpose for which the data are collected and
compatible with the present problem.
 The nature and classification of data is appropriate to our
problem.
 There are no biases and misreporting in the published data.
Note: Data which are primary for one may be secondary for the other.
2. METHODS OF DATA PRESENTATION
Having collected and edited the data, the next important step is to organize it. That is to
present it in a readily comprehensible condensed form that aids in order to draw
inferences from it. It is also necessary that the like be separated from the unlike ones.
The presentation of data is broadly classified in to the following two categories:
 Tabular presentation
 Diagrammatic and Graphic presentation.
The process of arranging data in to classes or categories according to similarities

technically is called classification.
Classification is a preliminary and it prepares the ground for proper presentation of data.
Definitions:
 Raw data: recorded information in its original collected form, whether it is counts
or measurements, is referred to as raw data.
 Frequency: is the number of values in a specific class of the distribution.
 Frequency distribution: is the organization of raw data in table form using classes
and frequencies.
There are three basic types of frequency distributions
 Categorical frequency distribution

 Ungrouped frequency distribution
 Grouped frequency distribution
There are specific procedures for constructing each type.
1) Categorical frequency Distribution:
Used for data that can be place in specific categories such as nominal, or ordinal. e.g. marital
status.
Example: a social worker collected the following data on marital status for 25
persons.(M=married, S=single, W=widowed, D=divorced)
M S D W D
S S M M M
W D S M M
W D D S S
S W W D D
Solution:
Since the data are categorical, discrete classes can be used. There are four types of marital
status M, S, D, and W. These types will be used as class for the distribution. We follow
procedure to construct the frequency distribution.
Step 1: Make a table as shown.
Class Tally Frequency Percent
(1) (2) (3) (4)
Step 2: Tally D the data and place the result in

column (2).
W
Step 3: Count the tally and place the result in column (3).
Step 4: Find the percentages of values in each class by using;
Where f= frequency of the class, n=total number of value.
Percentages are not normally a part of frequency distribution but they can be added since
they are used in certain types diagrammatic such as pie charts.
Step 5: Find the total for column (3) and (4).
Combing the entire steps one can construct the following frequency distribution.
Class Tally Frequency Percent
(1) (2) (3) (4)
M ///// 6 20
S //// // 7 28
D //// // 7 28
W //// 5 24
2) Ungrouped frequency Distribution:
-Is a table of all the potential raw score values that could possible occur in the data along
with the number of times each actually occurred.
-Is often constructed for small set or data on discrete variable.
Constructing ungrouped frequency distribution:
 First find the smallest and largest raw score in the collected data.
 Arrange the data in order of magnitude and count the frequency.
 To facilitate counting one may include a column of tallies.
Example:
The following data represent the mark of 20 students.
80 76 90 85 80
70 60 62 70 85
65 60 63 74 75
76 70 70 80 85
Construct a frequency distribution, which is ungrouped.
Solution:
Step 1: Find the range, Range=Max-Min=90-60=30.

Step 2: Make a table as shown
Step 3: Tally the data.
Step 4: Compute the frequency.
Mark Tally Frequency
60 // 2
62 / 1
63 / 1
65 / 1
70 //// 4
74 / 1
75 // 2
76 / 1
80 /// 3
85 /// 3
90 / 1
Each individual value is presented separately, that is why it is named ungrouped

frequency distribution.
3) Grouped frequency Distribution:
-When the range of the data is large, the data must be grouped in to classes that are more than
one unit in width.
Definitions:
 Grouped Frequency Distribution: a frequency distribution when several numbers

are grouped in one class.
 Class limits: Separates one class in a grouped frequency distribution from another.
The limits could actually appear in the data and have gaps between the upper limits of
one class and lower limit of the next.
 Units of measurement (U): the distance between two possible consecutive measures.
It is usually taken as 1, 0.1, 0.01, 0.001, -----.
 Class boundaries: Separates one class in a grouped frequency distribution from

another. The boundaries have one more decimal places than the row data and
therefore do not appear in the data. There is no gap between the upper boundary of
one class and lower boundary of the next class. The lower class boundary is found by
subtracting U/2 from the corresponding lower class limit and the upper class
boundary is found by adding U/2 to the corresponding upper class limit.
 Class width: the difference between the upper and lower class boundaries of any
class. It is also the difference between the lower limits of any two consecutive classes
or the difference between any two consecutive class marks.
 Class mark (Mid points): it is the average of the lower and upper class limits or the
average of upper and lower class boundary.
 Cumulative frequency: is the number of observations less than/more than or equal to

a specific value.
 Cumulative frequency above: it is the total frequency of all values greater than or
equal to the lower class boundary of a given class.
 Cumulative frequency blow: it is the total frequency of all values less than or equal
to the upper class boundary of a given class.
 Cumulative Frequency Distribution (CFD): it is the tabular arrangement of class

interval together with their corresponding cumulative frequencies. It can be more than
or less than type, depending on the type of cumulative frequency used.
 Relative frequency (rf): it is the frequency divided by the total frequency.
 Relative cumulative frequency (rcf): it is the cumulative frequency divided by the

total frequency.
Guidelines for classes
1. There should be between 5 and 20 classes.

2. The classes must be mutually exclusive. This means that no data value can fall
into two different classes
3. The classes must be all inclusive or exhaustive. This means that all data values
must be included.
4. The classes must be continuous. There are no gaps in a frequency distribution.
5. The classes must be equal in width. The exception here is the first or last class. It
is possible to have an "below ..." or "... and above" class. This is often used with
ages.
Steps for constructing Grouped frequency Distribution
1. Find the largest and smallest values

2. Compute the Range(R) = Maximum - Minimum
3. Select the number of classes desired, usually between 5 and 20 or use Sturges rule
where k is number of classes desired and n is total number of
observation.
4. Find the class width by dividing the range by the number of classes and rounding
up, not off. .
5. Pick a suitable starting point less than or equal to the minimum value. The starting
point is called the lower limit of the first class. Continue to add the class width to
this lower limit to get the rest of the lower limits.
6. To find the upper limit of the first class, subtract U from the lower limit of the
second class. Then continue to add the class width to this upper limit to find the
rest of the upper limits.
7. Find the boundaries by subtracting U/2 units from the lower limits and adding U/2
units from the upper limits. The boundaries are also half-way between the upper
limit of one class and the lower limit of the next class. !may not be necessary to
find the boundaries.
8. Tally the data.
9. Find the frequencies.
10. Find the cumulative frequencies. Depending on what you're trying to accomplish,
it may not be necessary to find the cumulative frequencies.
11. If necessary, find the relative frequencies and/or relative cumulative frequencies
Example*:
Construct a frequency distribution for the following data.
11 29 6 33 14 31 22 27 19 20
18 17 22 38 23 21 26 34 39 27
Solutions:
Step 1: Find the highest and the lowest value H=39, L=6
Step 2: Find the range; R=H-L=39-6=33
Step 3: Select the number of classes desired using Sturges formula;
=1+3.32log (20) =5.32=6(rounding up)
Step 4: Find the class width; w=R/k=33/6=5.5=6 (rounding up)
Step 5: Select the starting point, let it be the minimum observation.
 6, 12, 18, 24, 30, 36 are the lower class limits.
Step 6: Find the upper class limit; e.g. the first upper class=12-U=12-1=11
 11, 17, 23, 29, 35, 41 are the upper class limits.
So combining step 5 and step 6, one can construct the following classes.
Class limits
6 – 11
12 – 17
18 – 23
24 – 29
30 – 35
36 – 41
Step 7: Find the class boundaries;

E.g. for class 1 Lower class boundary=6-U/2=5.5
Upper class boundary =11+U/2=11.5
 Then continue adding w on both boundaries to obtain the rest boundaries. By

doing so one can obtain the following classes.
Class boundary
5.5 – 11.5
11.5 – 17.5
17.5 – 23.5
23.5 – 29.5
29.5 – 35.5
35.5 – 41.5
Step 8: tally the data.
Step 9: Write the numeric values for the tallies in the frequency column.
Step 10: Find cumulative frequency.
Step 11: Find relative frequency or/and relative cumulative frequency.
The complete frequency distribution follows:
Class Class boundary Class Tally Freq. Cf (less Cf (more rf. rcf (less
limit Mark than than type) than type
type)
6 – 11 5.5 – 11.5 8.5 // 2 2 20 0.10 0.10
12 – 17 11.5 – 17.5 14.5 // 2 4 18 0.10 0.20
18 – 23 17.5 – 23.5 20.5 7 11 16 0.35 0.55

//////
24 – 29 23.5 – 29.5 26.5 //// 4 15 9 0.20 0.75
30 – 35 29.5 – 35.5 32.5 /// 3 18 5 0.15 0.90

36 – 41 35.5 – 41.5 38.5 // 2 20 2 0.10 1.00
Diagrammatic and Graphic presentation of data.
These are techniques for presenting data in visual displays using geometric and pictures.
Importance:
 They have greater attraction.

 They facilitate comparison.
 They are easily understandable.
-Diagrams are appropriate for presenting discrete data.
-The three most commonly used diagrammatic presentation for discrete as well as qualitative
data are:
 Pie charts
 Bar charts
Pie chart
- A pie chart is a circle that is divided in to sections or wedges according to the

percentage of frequencies in each category of the distribution. The angle of the
sector is obtained using:
Example: Draw a suitable diagram to represent the following population in a town.
Men Women Girls Boys
2500 2000 4000 1500
Solutions:
Step 1: Find the percentage.
Step 2: Find the number of degrees for each class.

Step 3: Using a protractor and compass, graph each section and write its name corresponding
percentage.
Class Frequency Percent Degree
Men 2500 25 90
Women 2000 20 72
Girls 4000 40 144
Boys 1500 15 54
Pictogram
-In these diagram, we represent data by means of some picture symbols. We decide
abut a suitable picture to represent a definite number of units in which the variable is
measured.
Example: draw a pictogram to represent the following population of a town.
Year 1989 1990 1991 1992
Population 2000 3000 5000 7000

Bar Charts:
- A set of bars (thick lines or narrow rectangles) representing some magnitude over
time space.
- They are useful for comparing aggregate over time space.
- Bars can be drawn either vertically or horizontally.
- There are different types of bar charts. The most common being :
 Simple bar chart

 Component or sub divided bar chart.
 Multiple bar charts.
Simple Bar Chart
-Are used to display data on one variable.

-They are thick lines (narrow rectangles) having the same breadth. The magnitude of a quantity
is represented by the height /length of the bar.
Example: The following data represent sale by product, 1957- 1959 of a given company for three
products A, B, C.
Product Sales($) Sales($) Sales($)

In 1957 In 1958 In 1959
A 12 14 18
B 24 21 18
C 24 35 54
Solutions:
30
25
Sales in $
20
15
10
5
0
A B C
product
Component Bar chart
-When there is a desire to show how a total (or aggregate) is divided in to its component parts, we
use component bar chart.
-The bars represent total value of a variable with each total broken in to its component parts and
different colours or designs are used for identifications
Example:
Draw a component bar chart to represent the sales by product from 1957 to 1959.
Solutions:100
80
Product C
60
Sales in $
Product B
40
Product A
20
0
1957 1958 1959
Year of production
Multiple Bar charts
- These are used to display data on more than one variable.

- They are used for comparing different variables at the same time.
Example:
Draw a component bar chart to represent the sales by product from 1957 to 1959.
Solutions:
60
50
40 Product A
Sales in $
30 Product B
20 Product C
10
0
1957 1958 1959
Year of production
Graphical Presentation of data
The histogram, frequency polygon and cumulative frequency graph or ogive are most
commonly applied graphical representations for continuous data.
Procedures for constructing statistical graphs:
 Draw and label the X and Y axes.

 Choose a suitable scale for the frequencies or cumulative frequencies and label it on the
Y axes.
 Represent the class boundaries for the histogram or ogive or the mid points for the
frequency polygon on the X axes.
 Plot the points.
 Draw the bars or lines to connect the points.
Histogram
A graph which displays the data by using vertical bars of various height to represent frequencies.
Class boundaries are placed along the horizontal axes. Class marks and class limits are some times
used as quantity on the X axes.
Example: Construct a histogram to represent the previous data (example *).
Frequency Polygon:
- A line graph. The frequency is placed along the vertical axis and classes mid points
are placed along
8 the horizontal axis. It is customer to the next higher and lower
class interval with corresponding frequency of zero, this is to make it a complete
polygon.
Example: Draw a frequency polygon for the above data (example *).
Solutions: 6
4
Va lu e F re q u e n cy
0
2.5 8.5 14.5 20.5 26.5 32.5 38.5 44.5
Class Mid points
Ogive (cumulative frequency polygon)
- A graph showing the cumulative frequency (less than or more than type) plotted
against upper or lower class boundaries respectively. That is class boundaries are
plotted along the horizontal axis and the corresponding cumulative frequencies are
plotted along the vertical axis. The points are joined by a free hand curve.
Example: Draw an ogive curve(less than type) for the above data.(Example *)
3. MEASURES OF CENTERAL TENDENCY

Introduction
 When we want to make comparison between groups of numbers it is good to have a
single value that is considered to be a good representative of each group. This single value
is called the average of the group. Averages are also called measures of central tendency.
 An average which is representative is called typical average and an average which is
not representative and has only a theoretical value is called a descriptive average. A typical
average should posses the following:
 It should be rigidly defined.
 It should be based on all observation under investigation.
 It should be as little as affected by extreme observations.
 It should be capable of further algebraic treatment.
 It should be as little as affected by fluctuations of sampling.
 It should be ease to calculate and simple to understand.
Objectives:
 To comprehend the data easily.
 To facilitate comparison.
 To make further statistical analysis.
The Summation Notation:

 Let X1, X2 ,X3 …XN be a number of measurements where N is the total number of
observation and Xi is ith observation.
 Very often in statistics an algebraic expression of the form X 1+X2+X3+...+XN is
used in a formula to compute a statistic. It is tedious to write an expression like this
very often, so mathematicians have developed a shorthand notation to represent a
sum of scores, called the summation notation.
 The symbol is a mathematical shorthand for X1+X2+X3+...+XN
The expression is read, "the sum of X sub i from i equals 1 to N." It means "add up all the
numbers."
Example: Suppose the following were scores made on the first homework assignment for
five students in the class: 5, 7, 7, 6, and 8. In this example set of five numbers, where
N=5, the summation could be written:
The "i=1" in the bottom of the summation notation tells where to begin the sequence of
summation. If the expression were written with "i=3", the summation would start with the
third number in the set. For example:
In the example set of numbers, this would give the following result:
The "N" in the upper part of the summation notation tells where to end the sequence of
summation. If there were only three scores then the summation and example would be:
Sometimes if the summation notation is used in an expression and the expression must be
written a number of times, as in a proof, then a shorthand notation for the shorthand
notation is employed. When the summation sign "∑" is used without additional notation,
then "i=1" and "N" are assumed.
For example:
PROPERTIES OF SUMMATION
1. where k is any constant
2. where k is any constant
3. where a and b are any constant
4.
The sum of the product of the two variables could be written:

Example: considering the following data determine
X Y
5 6
7 7
7 8
6 7
8 8
a) e)
b) f)
c) g)
d) h)
Solutions:
a)
b)
c)
d)
e)
f)
g)
h)
Types of measures of central tendency
There are several different measures of central tendency; each has its advantage and
disadvantage.
 The Mean (Arithmetic)
 The Mode
 The Median
The choice of these averages depends up on which best fit the property under discussion.
The Arithmetic Mean

 Is defined as the sum of the magnitude of the items divided by the number of
items.
 The mean of X1, X2 ,X3 …Xn is denoted by A.M ,m or and is given by:
 If X1 occurs f1 times, if X2occurs f2 times, … , if Xn occurs fn times
Then the mean will be , where k is the number of classes and
Example: Obtain the mean of the following number

2, 7, 8, 2, 7, 3, 7
Solution:
Xi fi Xifi
2 2 4
3 1 3
7 3 21
8 1 8
Total 7 36
Arithmetic Mean for Grouped Data
If data are given in the shape of a continuous frequency distribution, then the mean is
obtained as follows:
Xi =the class mark of the ith class and fi = the frequency of the ith
class
Example: calculate the mean for the following age distribution.
Class frequency
6- 10 35
11- 15 23
16- 20 15
21- 25 12
26- 30 9
31- 35 6
Solutions:
 First find the class marks
 Find the product of frequency and class marks
 Find mean using the formula.
Class fi Xi Xifi
6- 10 35 8 280
11- 15 23 13 299
16- 20 15 18 270
21- 25 12 23 276
26- 30 9 28 252
31- 35 6 33 198
Total 100 1575
Exercises:
1. Marks of 75 students are summarized in the following frequency distribution:
Marks No. of students

If 20% of 40-44 7 the students have marks between 55
and 59 45-49 10
i. 50-54 22 Find the missing frequencies f4 and f5.
ii. 55-59 f4 Find the mean.
60-64 f5
65-69 6
70-74 3
Special properties of Arithmetic mean

1. The sum of the deviations of a set of items from their mean is always zero. i.e.
2. If is the mean of observations, if is the mean of observations, … ,

if is the mean of observation, then the mean of all the observation in all
groups often called the combined mean is given by:
Example: In a class there are 30 females and 70 males. If females averaged 60 in an

examination and boys averaged 72, find the mean for the entire class.
Solutions:
3. If a wrong figure has been used

when calculating the mean the
correct mean can be obtained with
out repeating the whole process
using:
Where n is total number of observations.
Example: An average weight of 10 students was calculated to be 65.Latter it was

discovered that one weight was misread as 40 instead of 80 kg. Calculate the correct
average weight.
Solutions:
Merits and Demerits of Arithmetic Mean
Merits:
 It is based on all observation.
 It is suitable for further mathematical treatment.
 It is stable average, i.e. it is not affected by fluctuations of sampling to some extent.
 It is easy to calculate and simple to understand.
Demerits:
 It is affected by extreme observations.
 It can not be used in the case of open end classes.
 It can not be determined by the method of inspection.
 It can not be used when dealing with qualitative characteristics, such as intelligence,
honesty, beauty.
The Mode
- Mode is a value which occurs most frequently in a set of values

- The mode may not exist and even if it does exist, it may not be unique.
- In case of discrete distribution the value having the maximum frequency is the modal
value.
Examples:
1. Find the mode of 5, 3, 5, 8, 9
Mode =5
2. Find the mode of 8, 9, 9, 7, 8, 2, and 5.
It is a bimodal Data: 8 and 9
3. Find the mode of 4, 12, 3, 6, and 7.
No mode for this data.
- The mode of a set of numbers X1, X2, …Xn is usually denoted by .
Mode for Grouped data
If data are given in the shape of continuous frequency distribution, the mode is defined
as:
Where:
Note: The modal class is a class with the highest frequency.
Example: Following is the distribution of the size of certain farms selected at random
from a district. Calculate the mode of the distribution.
Size of farms No. of farms

5-15 8
15-25 12
25-35 17
35-45 29
45-55 31
55-65 5
65-75 3
Solutions:
Merits and Demerits of Mode

Merits:
 It is not affected by extreme observations.
 Easy to calculate and simple to understand.
 It can be calculated for distribution with open end class
Demerits:
 It is not rigidly defined.
 It is not based on all observations
 It is not suitable for further mathematical treatment.
 It is not stable average, i.e. it is affected by fluctuations of sampling to
some extent.
 Often its value is not unique.

Note: being the point of maximum density, mode is especially useful in finding the most
popular size in studies relating to marketing, trade, business, and industry. It is the
appropriate average to be used to find the ideal size.
The Median
- In a distribution, median is the value of the variable which divides it in to two equal halves.
- In an ordered series of data median is an observation lying exactly in the middle of the series.
It is the middle most value in the sense that the number of values less than the median is equal
to the number of values greater than it.
-If X1, X2, …Xn be the observations, then the numbers arranged in ascending order will be
X[1], X[2], …X[n], where X[i] is ith smallest value.
X[1]< X[2]< …<X[n]
-Median is denoted by .
Median for ungrouped data
Example: Find the median of the following numbers.

a) 6, 5, 2, 8, 9, 4.
b) 2, 1, 3, 5, 8.
Solutions:
a) First order the data: 2, 4, 5, 6, 8, 9
Here n=6
b) Order the data :1, 2, 3, 5, 8

Here n=5
Median for grouped data
If data are given in the shape of continuous frequency distribution, the median is defined
as:
Remark:
The median class is the class with the smallest cumulative frequency (less than type) greater
than or equal to .
Example: Find the median of the following distribution.
Class Frequency
40-44 7
45-49 10
50-54 22
55-59 15
60-64 12
65-69 6
70-74 3
Solutions:
 First find the less than cumulative frequency.
 Identify the median class.
 Find median using formula.
Class Frequency Cumu.Freq(less

than type)
40-44 7 7
45-49 10 17
50-54 22 39
55-59 15 54
60-64 12 66
65-69 6 72
70-74 3 75
Merits and Demerits of Median
Merits:
 Median is a positional average and hence not influenced by extreme observations.
 Can be calculated in the case of open end intervals.
 Median can be located even if the data are incomplete.
Demerits:
 It is not a good representative of data if the number of items is small.
 It is not amenable to further algebraic treatment.
It is susceptible to sampling fluctuations.
4. Measures of Dispersion (Variation)
Introduction and objectives of measuring Variation

-The scatter or spread of items of a distribution is known as dispersion or variation. In
other words the degree to which numerical data tend to spread about an average value is
called dispersion or variation of the data.
-Measures of dispersions are statistical measures which provide ways of measuring the
extent in which data are dispersed or spread out.
Objectives of measuring Variation:
 To judge the reliability of measures of central tendency

 To control variability itself.
 To compare two or more groups of numbers in terms of their variability.
 To make further statistical analysis.
Absolute and Relative Measures of Dispersion
The measures of dispersion which are expressed in terms of the original unit of a series
are termed as absolute measures. Such measures are not suitable for comparing the
variability of two distributions which are expressed in different units of measurement and
different average size. Relative measures of dispersions are a ratio or percentage of a
measure of absolute dispersion to an appropriate measure of central tendency and are thus
pure numbers independent of the units of measurement. For comparing the variability of
two distributions (even if they are measured in the same unit), we compute the relative
measure of dispersion instead of absolute measures of dispersion.
Types of Measures of Dispersion
Various measures of dispersions are in use. The most commonly used measures of
dispersions are:
1) Range and relative range
2) Standard deviation and coefficient of variation.
3) Coefficient of determination and Z-scores
The Range (R)
The range is the largest score minus the smallest score. It is a quick and dirty measure of
variability, although when a test is given back to students they very often wish to know
the range of scores. Because the range is greatly affected by extreme scores, it may give a
distorted picture of the scores. The following two distributions have the same range, 13,
yet appear to differ greatly in the amount of variability.
Distribution 1: 32 35 36 36 37 38 40 42 42 43 43 45
Distribution 2: 32 32 33 33 33 34 34 34 34 34 35 45
For this reason, among others, the range is not the most important measure of variability.
Range for grouped data:
If data are given in the shape of continuous frequency distribution, the range is computed
as:
This is some times expressed as:
Merits and Demerits of range
Merits:
 It is rigidly defined.
 It is easy to calculate and simple to understand.
Demerits:
 It is not based on all observation.
 It is highly affected by extreme observations.
 It is affected by fluctuation in sampling.
 It is not liable to further algebraic treatment.
 It can not be computed in the case of open end distribution.
 It is very sensitive to the size of the sample.
Relative Range (RR)
It is also some times called coefficient of range and given by:
Example:
1. Find the relative range of the above two distribution. (Exercise!)

2. If the range and relative range of a series are 4 and 0.25 respectively. Then what is the
value of: a) Smallest observation b) Largest observation
Solution: (2)
Coefficient of Variation (C.V)
 Is defined as the ratio of standard deviation to the mean usually expressed as percents.
 The distribution having less C.V is said to be less variable or more consistent.
Example: An analysis of the monthly wages paid (in Birr) to workers in two firms A and
B belonging to the same industry gives the following results
Value Firm A Firm B

Mean wage 52.5 47.5
Median wage 50.5 45.5
Variance 100 121
In which firm A or B is there greater variability in individual wages?
Solutions:
Calculate coefficient of variation for both firms.
Since C.VA < C.VB, in firm B there is greater variability in individual wages.
Exercise: A meteorologist interested in the consistency of temperatures in three cities during

a given week collected the following data. The temperatures for the five days of the week in
the three cities were
City 1 25 24 23 26 17
City2 22 21 24 22 20
City3 32 27 35 24 28
Which city have the most consistent temperature, based on these data?
1. Two groups of people were trained to perform a certain task and tested to find out
which group is faster to learn the task. For the two groups the following information
was given:
Value Group one Group two
Mean 10.4 min 11.9 min
Stan.dev. 1.2 min 1.3 min
Relatively speaking:
a) Which group is more consistent in its performance
Solutions:
a) Use coefficient of variation.
Since C.V2 < C.V1, group 2 is more consistent.
CHAPTER TWO
PROBABILITY AND PROBABILITY DISTRIBUTION
2.1. ELEMENTARY PROBABILITY

Introduction
 Probability theory is the foundation upon which the logic of inference is built.
 It helps us to cope up with uncertainty.
 In general, probability is the chance of an outcome of an experiment. It is the
measure of how likely an outcome is to occur.
Definitions of some probability terms
1. Experiment: Any process of observation or measurement or any process which
generates well defined outcome.
2. Probability Experiment: It is an experiment that can be repeated any number of times under
similar conditions and it is possible to enumerate the total number of outcomes with out
predicting an individual out come. It is also called random experiment.
Example: If a fair die is rolled once it is possible to list all the possible outcomes
i.e.1, 2, 3, 4, 5, 6 but it is not possible to predict which outcome will occur.
3. Outcome :The result of a single trial of a random experiment

4. Sample Space: Set of all possible outcomes of a probability experiment
5. Event: It is a subset of sample space. It is a statement about one or more outcomes of a
random experiment .They are denoted by capital letters.
Example: Considering the above experiment let A be the event of odd numbers, B be the event
of even numbers, and C be the event of number 8.
Remark:
If S (sample space) has n members then there are exactly 2 n subsets or
events.
6. Equally Likely Events: Events which have the same chance of occurring.
7. Complement of an Event: the complement of an event A means non- occurrence of
A and is denoted by contains those points of the sample space which don’t
belong to A.
8. Elementary Event: an event having only a single element or sample point.
9. Mutually Exclusive Events: Two events which cannot happen at the same time.
10. Independent Events: Two events are independent if the occurrence of one does not
affect the probability of the other occurring.
11. Dependent Events: Two events are dependent if the first event affects the outcome or
occurrence of the second event in a way the probability is changed.
Example: .What is the sample space for the following experiment
a) Toss a die one time.

b) Toss a coin two times.
c) A light bulb is manufactured. It is tested for its life length by time.
Solution
a) S={1,2,3,4,5,6}
b) S={(HH),(HT),(TH),(TT)}
c) S={t /t≥0}
 Sample space can be
 Countable ( finite or infinite)

 Uncountable.
Counting Rules
In order to calculate probabilities, we have to know
 The number of elements of an event
 The number of elements of the sample space.
That is in order to judge what is probable, we have to know what is possible.
 In order to determine the number of outcomes, one can use several rules of counting.
- The addition rule
- The multiplication rule
- Permutation rule
- Combination rule
 To list the outcomes of the sequence of events, a useful device called tree diagram is
used.
Example: A student goes to the nearest snack to have a breakfast. He can take tea, coffee, or
milk with bread, cake and sandwitch. How many possibilities does he have?
Solutions:
Tea
Bread
Cake
Sandwich
Coeffee
Bread
Cake
Sandwitch
Milk
Bread
Cake
Sandwitch
 There are nine possibilities.
The Multiplication Rule:
If a choice consists of k steps of which the first can be made in n 1 ways, the second can be
made in n2 ways…, the kth can be made in nk ways, then the whole choice can be made in
Example: The digits 0, 1, 2, 3, and 4 are to be used in 4 digit identification card. How many
different cards are possible if
a) Repetitions are permitted.
b) Repetitions are not permitted.
Solutions
a)
1st digit 2nd digit 3rd digit 4th digit
5 5 5 5
There are four steps
1. Selecting the 1st digit, this can be made in 5 ways.
2. Selecting the 2nd digit, this can be made in 5 ways.
3. Selecting the 3rd digit, this can be made in 5 ways.

4. Selecting the 4th digit, this can be made in 5 ways.
b)
1st digit 2nd digit 3rd digit 4th digit

5 4 3 2
There are four steps

5. Selecting the 1st digit, this can be made in 5 ways.
6. Selecting the 2nd digit, this can be made in 4 ways.
7. Selecting the 3rd digit, this can be made in 3 ways.
8. Selecting the 4th digit, this can be made in 2 ways.
Permutation
An arrangement of n objects in a specified order is called permutation of the objects.

Permutation Rules:
1. The number of permutations of n distinct objects taken all together is n!
Where
2. The arrangement of n objects in a specified order using r objects at a time is called
the permutation of n objects taken r objects at a time. It is written as and
the formula is
3. The number of permutations of n objects in which k1 are alike k2 are alike ---- etc
is
Example:
1. Suppose we have a letters A,B, C, D
a) How many permutations are there taking all the four?
b) How many permutations are there two letters at a time?
2. How many different permutations can be made from the letters in the word
“CORRECTION”?
Solutions:
1.
a)
b)
2.
Exercises:
1. Six different statistics books, seven different physics books, and 3 different
Economics books are arranged on a shelf. How many different arrangements are
possible if;
i. The books in each particular subject must all stand together
ii. Only the statistics books must stand together
2. If the permutation of the word WHITE is selected at random, how many of the
permutations
i. Begins with a consonant?
ii. Ends with a vowel?
iii. Has a consonant and vowels alternating?
Combination
A selection of objects with out regard to order is called combination.

Example: Given the letters A, B, C, and D list the permutation and combination for selecting
two letters.
Solutions:
Permutation Combination
AB BA CA DA AB BC
AC BC CB DB AC BD
AD BD CD DC AD DC
Note that in permutation AB is different from BA. But in combination AB is the same as BA.
Combination Rule
The number of combinations of r objects selected from n objects is denoted by

and is given by the formula:
Examples:
1. In how many ways a committee of 5 people be chosen out of 9 people?
Solutions:
2. Among 15 clocks there are two defectives .In how many ways can an inspector chose
three of the clocks for inspection so that:
a) There is no restriction.
b) None of the defective clock is included.
c) Only one of the defective clocks is included.
d) Two of the defective clock is included.
Solutions:
a) If there is no restriction select three clocks from 15 clocks and this can be
done in :
b) None of the defective clocks is included.

This is equivalent to zero defective and three non defective, which can be done
in:
c) Only one of the defective clocks is included.

This is equivalent to one defective and two non defective, which can be done in:
d) Two of the defective clock is included.

This is equivalent to two defective and one non defective, which can be done in:
Exercises:
1. Out of 5 Mathematician and 7 Statistician a committee consisting of 2
Mathematician and 3 Statistician is to be formed. In how many ways this can
be done if
a) There is no restriction
b) One particular Statistician should be included
c) Two particular Mathematicians can not be included on the committee.
2. If 3 books are picked at random from a shelf containing 5 novels, 3 books of
poems, and a dictionary, in how many ways this can be don if
a) There is no restriction.
b) The dictionary is selected?
c) 2 novels and 1 book of poems are selected?
Approaches to measuring Probability

There are four different conceptual approaches to the study of probability theory. These
are:
 The classical approach.
 The frequentist approach.
 The axiomatic approach.
 The subjective approach.
The classical approach
This approach is used when:

- All outcomes are equally likely.
- Total number of outcome is finite, say N.
Definition: If a random experiment with N equally likely outcomes is conducted and out
of these NA outcomes are favourable to the event A, then the probability that event A
occur denoted is defined as:
Examples:
1. A fair die is tossed once. What is the probability of getting
a) Number 4?
b) An odd number?
c) An even number?
d) Number 8?
Solutions:
First identify the sample space, say S
a) Let A be the event of number 4
b) Let A be the event of odd numbers
c) Let A be the event of even numbers
d) Let A be the event of number 8

Ø
2. A box of 80 candles consists of 30 defective and 50 non defective candles. If 10

of this candles are selected at random, what is the probability
a) All will be defective.
b) 6 will be non defective
c) All will be non defective
Solutions:
a) Let A be the event that all will be defective.

b) Let A be the event that 6 will be non defective.
c) Let A be the event that all will be non defective.
Exercises:
1. What is the probability that a waitress will refuse to serve alcoholic beverages to
only three minors if she randomly checks the I.D’s of five students from among
ten students of which four are not of legal age?
2. If 3 books are picked at random from a shelf containing 5 novels, 3 books of
poems, and a dictionary, what is the probability that
a) The dictionary is selected?
b) 2 novels and 1 book of poems are selected?
Short coming of the classical approach:

This approach is not applicable when:
- The total number of outcomes is infinite.
- Outcomes are not equally likely.
The Frequentist Approach
This is based on the relative frequencies of outcomes belonging to an event.
Definition: The probability of an event A is the proportion of outcomes favourable to A in

the long run when the experiment is repeated under same condition.
Example: If records show that 60 out of 100,000 bulbs produced are defective. What is the
probability of a newly produced bulb to be defective?
Solution:
Let A be the event that the newly produced bulb is defective.
Axiomatic Approach:
Let E be a random experiment and S be a sample space associated with E. With each event
A a real number called the probability of A satisfies the following properties called axioms of
probability or postulates of probability.
1.
2.
3. If A and B are mutually exclusive events, the probability that one or the other occur
equals the sum of the two probabilities. i. e.
4.
5.
6. P(ø) =0, ø is the impossible event.
Remark: Venn-diagrams can be used to solve probability problems.
AUB AnB A
In general
2 Conditional Probability, Independence, and Bayes’ Theorem

2.2 Conditional Probability and Independence

2.12.1 Conditional probability
Let a and B be two events in a probability space (F, S, P) and P(A) > 0. The conditional
probability of A given that B has occurred denoted by P(A/B) is given by:
Example: A survey of 1000 student drivers in a high school of whom 600 have attended a
driving course was conducted over a one year period to find out how many of them were
involved in at least one accident in which they were at fault. The result is summarized as
follows:
Had accident No accident Total

Attended Course 30 570 600
Didn’t attend course 70 330 400
Total 100 900 1000
i) What is the probability that a student was involved in any accident?
ii) What is the probability that a student who took the course was involved in at
least one accident?
iii) What is the probability that a student who took the course was involved in no
accident?
Solution: Let A be the event that a student was involved in at least one accident, and T be
the event that a student attended/took the driving course.
a) Required is P(A).
b) Required is P(A/T).
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Note that the event T has already occurred and our sample space is to be restricted
to the first row only, i.e., the restricted sample space we have to consider now
consists of the 600 students who attended the course. Therefore N=600 and n = 30
giving us the same result as above.
c) P(A’/T) is required.
Example: A teacher gives two tests of which 25% of the students passed both and 42% of
them passed the first test. What percent of the students who passed the first test also
passed the second?
Solution: Let Ti represent the event that a student passes test i. Requited is
(why?)
Example: Suppose that the probability that it is Friday and a student is absent is 0.03.
What is the probability that a student is absent given that today is Friday?
Solution: Let A = student is absent and B = today is Friday, required is therefore P(A/B).
(Why?)
Now rewriting the definition of conditional probability we get

. This is sometimes referred to as the general multiplication
rule of probability. Extending this rule to three events A, B and C, we have
1.1.1 Independent Events
A set of events is said to be independent if the occurrence of any one of them doesn’t
depend on the occurrence or non-occurrence of the others. Accordingly, two events A
and B are independent if P(B/A) = P(B). Equivalently, .
Conversely, if , then A and B are independent events.
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Example: What is the probability of randomly drawing two aces in succession from an
ordinary deck of 52 playing cards, if the selection procedure is (a) without replacement
and (b) with replacement?
Solution: Let Ai = the ith card drawn is an ace, i= 1,2. Required is .
a) Without replacement:
, .
Therefore
b) With replacement:
, . Therefore the required
probability becomes . NB: the two events happen to be independent.
1.2 Bayes’ Theorem
The general multiplication rule is useful in solving many problems in which the ultimate
outcome of an experiment depends on outcomes of various intermediate stages. For
instance suppose that a company is receives stabilizers from two different suppliers with
60% of them coming from the first supplier. Suppose stabilizers from supplier 1and 10%
of those from supplier 2 are respectively defective. We would like to know the
probability that any one of the stabilizers received by the company is defective. Let A = a
stabilizer is defective and Bi = it comes from supplier i, i = 1, 2. Accordingly, P(B 1) = 0.6,
P(B2) = 0.4, P(A/B1) = 0.2 and P(A/B2) = 0.1. So the question is to find P(A).The events
associated with this problem are depicted as in the following diagram.
Page 48 of 63
B2
B1
S A
A  B1
A  B
2
From the diagram it can be clearly seen that event A is the union of and
which are disjoint. This implies that . Hence it follows:
As shown in the tree diagram below there are two ways of reaching A, via B1 and via B2.
B1 A
B2 A
Theorem:
If B1,B2,…,Bk are partitions of S and P(Bi) > 0 for i = 1, 2, …, k, then for ny event A in S
such that P(A) ,
Example: For the yearly CBTP session, the CBE office of Jimma University rents cars.
Suppose for the current session the office rents 50% of the cars from company one, 30%
from company two and the remaining from company three. If 10%, 20% and 40% of the
Page 49 of 63
cars from company one, two and three respectively need check up before service, what is
the probability that a car delivered to the CBE will need check up.
Solution: Let A be the event that a car needs check up and B be the event that a car comes
from company i, where i = 1,2, and 3. We have that P(B 1) = 0.5, P(B2) = 0.3, P(B3) = 0.2,
P(A/B1) = 0.1, P(A/B2) = 0.2, P(A/B3) = 0.4. Now the required probability, i.e., P(A) can
be calculated as:
Now suppose in the preceding example we are interested in determining the probability
that a particular car is actually from company two given that it needs check up. Such a
probability can be calculated using the following procedure:
The method used to solve the above-mentioned probability (i.e., P(B 2/A)) can easily be
generalized to yield the following formula called Bayes’ Theorem.
Theorem:
If B1,B2,…,Bk are partitions of S and P(Bi) > 0 for i = 1, 2, …, k, then for any event A in
S, where r = 1, 2, …, k.
NB: The numerator in the above formula is the probability of reaching A via the rth
branch, while the denominator is the sum of the probabilities of reaching A via the k
branches. Bayes’ theorem provides a formula for finding the probability that the ‘effect’
A was caused by the event Br.
2. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
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Definition: A random variable is a numerical description of the outcomes of the experiment or
a numerical valued function defined on sample space, usually denoted by capital letters.
Example: If X is a random variable, then it is a function from the elements of the sample space
to the set of real numbers. i.e. X is a function X: S  R
A random variable takes a possible outcome and assigns a number to it.
Example: Flip a coin three times, let X be the number of heads in three tosses.
X = {0, 1, 2, 3, 4, 5}
X assumes a specific number of values with some probabilities.
Random variables are of two types:
1. Discrete random variable: are variables which can assume only a specific number of
values. They have values that can be counted
Examples:
 Toss coin n times and count the number of heads.
 Number of children in a family.
 Number of car accidents per week.
 Number of defective items in a given company.
 Number of bacteria per two cubic centimeter of water.
2. Continuous random variable: are variables that can assume all values between any two
give values.
Examples:
 Height of students at certain college.
 Mark of a student.
 Life time of light bulbs.
 Length of time required to complete a given training.
Probability Distribution
Definition: a probability distribution consists of value that a random variable can assume and
the corresponding probabilities of the values.
Example: Consider the experiment of tossing a coin three times. Let X is the number of
heads. Construct the probability distribution of X.
Solution:
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 First identify the possible value that X can assume.
 Calculate the probability of each possible distinct value of X and express X in the
form of frequency distribution.
0 1 2 3
 Probability distribution is denoted by P for discrete and by f for continuous random

variable.
Properties of Probability Distribution:
1.
2.
Note:
1. If X is a continuous random variable then
2. Probability of a fixed value of a continuous random variable is zero.
3. If X is discrete random variable then
4. Probability means area for continuous random variable.
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Introduction to expectation
Definition:
1. Let a discrete random variable X assume the values X1, X2, ….,Xn with the
probabilities P(X1), P(X2), ….,P(Xn) respectively. Then the expected value of X,
denoted as E(X) is defined as:
2. Let X be a continuous random variable assuming the values in the interval (a, b) such
that ,then
Examples:
1. What is the expected value of a random variable X obtained by tossing a coin
three times where X is the number of heads?
Solution:
First construct the probability distribution of X
0 1 2 3
2. Suppose a charity organization is mailing printed return-address stickers to over

one million homes in Ethiopia. Each recipient is asked to donate either $1, $2, $5,
$10, $15, or $20. Based on past experience, the amount a person donates is
believed to follow the following probability distribution:
$1 $2 $5 $10 $15 $20
0.1 0.2 0.3 0.2 0.15 0.05
What is expected that an average donor to contribute?

Solution:
$1 $2 $5 $10 $15 $20 Total
Page 53 of 63
0.1 0.2 0.3 0.2 0.15 0.05 1
0.1 0.4 1.5 2 2.25 1 7.25
Mean and Variance of a random variable

Let X is given random variable.
1. The expected value of X is its mean
2. The variance of X is given by:
Where:
Examples:
1. Find the mean and the variance of a random variable X in example 2 above.
Solution:
$1 $2 $5 $10 $15 $20 Total
0.1 0.2 0.3 0.2 0.15 0.05 1
0.1 0.4 1.5 2 2.25 1 7.25
0.1 0.8 7.5 20 33.75 20 82.15
Exercise: Two dice are rolled. Let X is a random variable denoting the sum of the numbers
on the two dice.
i) Give the probability distribution of X
ii) Compute the expected value of X and its variance
 There are some general rules for mathematical expectation.
Let X and Y are random variables and k is a constant.
RULE 1:
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RULE 2:
RULE 3:
RULE 4:
RULE 5:
Common Discrete Probability Distributions
1. Binomial Distribution
A binomial experiment is a probability experiment that satisfies the following four
requirements called assumptions of a binomial distribution.
1. The experiment consists of n identical trials.
2. Each trial has only one of the two possible mutually exclusive outcomes, success or
a failure.
3. The probability of each outcome does not change from trial to trial, and
4. The trials are independent, thus we must sample with replacement.
Examples of binomial experiments

 Tossing a coin 20 times to see how many tails occur.
 Asking 200 people if they watch BBC news.
 Registering a newly produced product as defective or non defective.
 Asking 100 people if they favor the ruling party.
 Rolling a die to see if a 5 appears.
Definition: The outcomes of the binomial experiment and the corresponding probabilities of these
outcomes are called Binomial Distribution.
Then the probability of getting successes in trials becomes:
And this is some times written as:

When using the binomial formula to solve problems, we have to identify three things:
 The number of trials ( )
 The probability of a success on any one trial ( ) and
 The number of successes desired ( ).
Examples:
1. What is the probability of getting three heads by tossing a fair con four times?
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Solution: Let X be the number of heads in tossing a fair coin four times
2. Suppose that an examination consists of six true and false questions, and assume that a student
has no knowledge of the subject matter. The probability that the student will guess the correct
answer to the first question is 30%. Likewise, the probability of guessing each of the remaining
questions correctly is also 30%.
a) What is the probability of getting more than three correct answers?
b) What is the probability of getting at least two correct answers?
c) What is the probability of getting at most three correct answers?
d) What is the probability of getting less than five correct answers?
Solution: Let X = the number of correct answers that the student gets.
a)
Thus, we may conclude that if 30% of the exam questions are answered by guessing, the
probability is 0.071 (or 7.1%) that more than four of the questions are answered correctly
by the student.
b)
c)
Page 56 of 63
d)
Exercises:
a. Suppose that 4% of all TVs made by A&B Company in 2000 are defective. If eight of
these TVs are randomly selected from across the country and tested, what is the
probability that exactly three of them are defective? Assume that each TV is made
independently of the others.
b. An allergist claims that 45% of the patients she tests are allergic to some type of weed.
What is the probability that
I. Exactly 3 of her next 4 patients are allergic to weeds?
II. None of her next 4 patients are allergic to weeds?
c. Explain why the following experiments are not Binomial
I. Rolling a die until a 6 appears.
II. Asking 20 people how old they are.
III. Drawing 5 cards from a deck for a poker hand.
Remark: If X is a binomial random variable with parameters n and p then
,
2. Poisson Distribution
A random variable X is said to have a Poisson distribution if its probability distribution is
given by:
The Poisson distribution depends only on the average number of occurrences per unit
time of space.
The Poisson distribution is used as a distribution of rare events, such as: Arrivals,
Accidents, Number of misprints, Hereditary, Natural disasters like earth quake, etc.
The process that gives rise to such events is called Poisson process.
Example: If 1.6 accidents can be expected an intersection on any given day, what is the
probability that there will be 3 accidents on any given day?
Solution: Let X =the number of accidents,
Page 57 of 63
Exercise: On the average, five smokers pass a certain street corners every ten minutes,
what is the probability that during a given 10 minutes the number of smokers passing will
be
a. 6 or fewer
b. 7 or more
c. Exactly 8…….
If X is a Poisson random variable with parameter then
,
Note: The Poisson probability distribution provides a close approximation to the binomial
probability distribution when n is large and p is quite small or quite large with .
Usually we use this approximation if . In other words, if and [or

], then we may use Poisson distribution as an approximation to binomial distribution.
Example: Find the binomial probability P(X=3) by using the Poisson distribution if
and . Solution:
Common Continuous Probability Distributions

1. Normal Distribution
A random variable X is said to have a normal distribution if its probability density function is
Properties of Normal Distribution:
Page 58 of 63
1. It is bell shaped and is symmetrical about its mean and it is mesokurtic. The maximum
ordinate is at and is given by
2. It is asymptotic to the axis, i.e., it extends indefinitely in either direction from the
mean.
3. It is a continuous distribution.
4. It is a family of curves, i.e., every unique pair of mean and standard deviation defines a
different normal distribution. Thus, the normal distribution is completely described by
two parameters: mean and standard deviation.
5. Total area under the curve sums to 1, i.e., the area of the distribution on each side of the
mean is 0.5.
6. It is unimodal, i.e., values mound up only in the center of the curve.

7.
8. The probability that a random variable will have a value between any two points is
equal to the area under the curve between those points.
Note: To facilitate the use of normal distribution, the following distribution known as the
standard normal distribution was derived by using the transformation
Properties of the Standard Normal Distribution:

- Same as a normal distribution, but also mean is zero, variance is one, standard
Deviation is one
- Areas under the standard normal distribution curve have been tabulated in various
ways. The most common ones are the areas between
- Given normal distributed random variable X with mean
Note:
Page 59 of 63
Examples:
1. Find the area under the standard normal distribution which lies
a) Between
Solution:
b) Between
Solution:
c) To the right of
Solution:
d) To the left of
Solution:
e) Between
Solution:
f) Between
Solution:
2. Find the value of Z if

a) The normal curve area between 0 and z(positive)
is 0.4726
Page 60 of 63
Solution
b) The area to the left of z is 0.9868

Solution
3. A random variable X has a normal distribution with mean 80 and standard deviation
4.8. What is the probability that it will take a value
a) Less than 87.2
b) Greater than 76.4
c) Between 81.2 and 86.0
Solution
a)
b)
c)
Page 61 of 63
4. A normal distribution has mean 62.4.Find its standard deviation if 20.0% of the area
under the normal curve lies to the right of 72.9
Solution
5. A random variable has a normal distribution with .Find its mean if the
probability that the random variable will assume a value less than 52.5 is 0.6915.
Solution
Exercise: Of a large group of men, 5% are less than 60 inches in height and 40% are
between 60 & 65 inches. Assuming a normal distribution, find the mean and standard
deviation of heights.
Page 62 of 63
Page 63 of 63

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Lecture notes on Statistics for business I

Stages in Statistical Investigation

There are five stages or steps in any statistical investigation.

 Statistical techniques based on probability theory are required.

Definitions of some terms

a. Statistical Population: It is the collection of all possible observations of a specified

Types of Variables or Data:

1. Qualitative Variables are nonnumeric variables and can't be measured. Examples

Applications, Uses and Limitations of statistics

7. Studying the relationship between two or more variable.

In mathematical terms measurement is a functional mapping from the set of objects {O i}

The property of ORDER exists

When for all i, j if Oi > Oj, then M(Oi) > M(Oj).

The property of distance is concerned with the relationship of differences between

The property of DISTANCE exists, for all i, j, k, l

If Oi-Oj ≥ Ok- Ol then M(Oi)-M(Oj) ≥ M(Ok)-M( Ol ).

The property of FIXED ZERO exists if M(O0) = 0.

Measurement is the assignment of numbers to objects or events in a systematic fashion.

 No arithmetic and relational operation can be applied.

o Political party preference (Republican, Democrat, or Other,)

o Marital status(married, single, widow, divorce)

o Regional differentiation of Ethiopia.

 Ordering is the sole property of ordinal scale.

o Letter grades (A, B, C, D, F).

1. Your checking account number as a name for your account.

11. Blood type of individuals, A, B, AB and O.

METHODS OF DATA COLLECTION

2. METHODS OF DATA PRESENTATION

The presentation of data is broadly classified in to the following two categories:

The process of arranging data in to classes or categories according to similarities

There are three basic types of frequency distributions

 Categorical frequency distribution

 Grouped frequency distribution

There are specific procedures for constructing each type.

1) Categorical frequency Distribution:

Step 1: Make a table as shown.

Class Tally Frequency Percent

(1) (2) (3) (4)

Step 2: Tally D the data and place the result in

Step 4: Find the percentages of values in each class by using;

Where f= frequency of the class, n=total number of value.

Step 5: Find the total for column (3) and (4).

Class Tally Frequency Percent

(1) (2) (3) (4)

-Is often constructed for small set or data on discrete variable.

Constructing ungrouped frequency distribution:

 To facilitate counting one may include a column of tallies.

The following data represent the mark of 20 students.

Construct a frequency distribution, which is ungrouped.

Step 1: Find the range, Range=Max-Min=90-60=30.

Each individual value is presented separately, that is why it is named ungrouped

3) Grouped frequency Distribution:

 Grouped Frequency Distribution: a frequency distribution when several numbers

 Class boundaries: Separates one class in a grouped frequency distribution from

 Cumulative frequency: is the number of observations less than/more than or equal to

 Cumulative Frequency Distribution (CFD): it is the tabular arrangement of class

 Relative frequency (rf): it is the frequency divided by the total frequency.

 Relative cumulative frequency (rcf): it is the cumulative frequency divided by the