Notes 6

MEASURES OF CENTRAL LOCATION
 Any measure indicating the center of the data.

 It is a single figure which is representative of the general level of magnitudes or values of
the items in a set of data.
Uses of Measures of Central Location
1. It provides a “summary” of a data.
2. It provides a “common denominator” for comparing two groups of data.
3. It provides a measure of “typical size”.
4. It gives a mathematical concept to the relationship between different groups.
Mean (Arithmetic Mean)
Two Main Types of Mean
1. Simple Arithmetic Mean

∑
Population mean = µ =
∑
Sample mean = ̅ =
2. Weighted Arithmetic Mean or Weighted Mean

∑
̅ =
∑
 The mean reflects the magnitude of every observation, since every observation contributes to
the value of the mean.
 It is easily affected by the presence of extreme values, and hence not a good measure of central
tendency when extreme observations occur.
Example:
1. The number of goals scored by a college lacrosse team for a given season are:
4, 9, 0, 1, 3, 24, 12, 3, 30, 12, 7, 13, 18, 4, 5, 15
Compute the arithmetic mean of the sample.
Solution:
1. Determine the sample size n (pr population size, N).
n = 16
2. Sum all the observations, ∑
∑ = x1 + x 2 + . . . + x n
= 4 + 9 + 0 + 1 + 3 + 24 + 12 + 3 + 30 + 12 + 7 + 13 + 18 + 4 + 5 + 15
= 160
3. Solve for the mean, ̅ .
∑
̅ =
= 160
16
= 10
2. The score of the student in his Stat 115 course are as follows:
Assignments 87
Quizzes 90
Midterm 88
Final 84
If the following weights are applied, Assignments, 0.15; Quizzes, 0.25; Midterm, 0.30; and Final,
0.30; find the weighted mean of the student.
Solution:
1. Construct the following table
wi xi wixi
0.15 87 13.05
0.25 90 22.5
0.30 88 26.4
0.30 84 25.2
∑ =1 ∑ = 87.15
2. Solve for the weighted mean, ̅ .

∑
̅ =
∑
= 87.15
1
= 87.15
Median
 Is a single value which divides an array of observations into two equal parts such that half of
the observations fall below it and half of the observations fall above.
 The observations must in an increasing order (array)
 There are two cases that should be determined first before computing for median.
CASE 1 : n is odd
̃ = the value of the middle observation in the array.
̃=
CASE 2 : n is even
̃ = the average of the two middle values in the array.
̃=
 It is a positional value and hence is not affected by the presence of extreme value unlike
mean.
 The median is not amenable to further computations
Example:
The number of goals scored by a college lacrosse team for a given season are:
4, 9, 0, 1, 3, 24, 12, 3, 30, 12, 7, 13, 18, 4, 5, 15
Compute the median of the sample.
Solution:
1. Arrange the observations in an increasing order.
0, 1, 3, 3, 4, 4, 5, 7, 9, 12, 12, 13, 15, 18, 24, 30
2. Determine n if odd or ever.
n = 16, which is even
3. Determine the sample median, ̃.
̃=
=
=8
Mode
 Mode is the value which occurs most frequently in the given data set.
Mode = ̂ = value(s) which occurs most frequently.
Example:
1. Raw data: 4 5 3 3 3 6 4 3 3 3 ̂ = 3 (unimodal)
2. Raw data: 4 3 5 4 3 7 3 4 ̂ = 3, 4 (bimodal)
3. Raw data: 4 5 3 4 5 3 6 ̂ = 3, 4, 5 (multimodal)
CIVIL STATUS NO. OF EMPLOYEES

SINGLE 23
MARRIED 70
WIDOW/WIDOWER 16
̂ = married
4. The number of goals scored by a college lacrosse team for a given season are:
4, 9, 3, 1, 3, 24, 12, 3, 30, 12, 7, 13, 18, 4, 5, 15
Compute the mode of the sample.

̂=3
Summary of the Measures of Central Location
Mean Median Mode
Description It is the sum of all The middle score for The most frequent score in our
values/scores in the data a set of data that has data set.
set divided by the total been arranged in
number of values/scores. order of magnitude
Formula when n is odd

The value/score or category with
the highest frequency.
when n is even
Type of Data Discrete and Continuous Discrete and Discrete, Continuous, Categorical
Continuous
Characteristics 1. Not often one of the 1. less affected by 1. It represents the highest bar
actual values/scores outliers and in a bar chart or histogram.
in the data set. skewed data
2. It is the value that
produces the lowest
amount of error from
all other values in the 2. It is not unique, so it leaves us
data set. with problems when we have
3. It includes every two or more values that share
value in your data set the highest frequency.
as part of the
calculation
4. The sum of the
deviations of each
value from the mean
is always zero 3. It will not provide us with a
very good measure of central
tendency when the most
common mark is far away
from the rest of the data in
the data set.
When to use 1. When there 1. When the mean 1. When mean and median is
is(are) no outliers is not applicable not applicable.
in the data set. because of 2. When you have categorical
2. When the data outliers and data.
set is not skewed data.
skewed.
Shapes of Curves (Skewness)
Shapes of Curves (Kurtosis)

Skewness and Measure of Central Location

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MEASURES OF CENTRAL LOCATION

 Any measure indicating the center of the data.

Mean (Arithmetic Mean)

Two Main Types of Mean

1. Simple Arithmetic Mean

2. Weighted Arithmetic Mean or Weighted Mean

2. Solve for the weighted mean, ̅ .

1. Raw data: 4 5 3 3 3 6 4 3 3 3 ̂ = 3 (unimodal)

2. Raw data: 4 3 5 4 3 7 3 4 ̂ = 3, 4 (bimodal)

3. Raw data: 4 5 3 4 5 3 6 ̂ = 3, 4, 5 (multimodal)

CIVIL STATUS NO. OF EMPLOYEES

4, 9, 3, 1, 3, 24, 12, 3, 30, 12, 7, 13, 18, 4, 5, 15

Compute the mode of the sample.

Formula when n is odd

Shapes of Curves (Kurtosis)

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