LEARNING CONTENT:

 Statistics is the science of conducting studies to collect, organize, summarize,

analyze, and draw conclusions from data.

 A population consists of all subjects (human or otherwise) that are studied.

 A sample is a subset of the population.

 The values that a variable can assume are called data. It can be qualitative
(categorical) or quantitative (numerical and can be ranked). Quantitative data
can be discrete (countable) or continuous (includes decimals).

 Levels of Measurement

Nominal – categorical (names)

Ordinal – nominal, plus can be ranked (order)

Interval – ordinal, plus intervals are consistent

Ratio – interval, plus ratios are consistent, true

zero ORGANIZING DATA:

Data collected in original form is called raw data.

 A frequency distribution is the organization of raw data in table form, using

classes and frequencies. Nominal- or ordinal-level data that can be placed in
categories is organized in categorical frequency distributions.

Example:

Twenty-five army inductees were given a blood test to determine their blood

type. Raw Data: A,B,B,AB,O O,O,B,AB,B B,B,O,A,O

A,O,O,O,AB AB,A,O,B,A

Class Frequency Percent

A 5 20
B 7 28
O 9 36
AB 4 16
Measures of Central Tendency:
Mean:
 The mean is the quotient of the sum of the values and the total number of values.

 The is used for sample mean.

symbolX
X1  X 2  X 3   Xn
X X
 n
n
 For a population, the Greek letter μ (mu) is used for the mean.

X1  X 2  X 3   XN

X
 N
N
EXAMPLE:

The data represent the number of days off per year for a sample of individuals selected
from nine different countries. Find the mean.

20, 26, 40, 36, 23, 42, 35, 24, 30

20  26  40  36  23  42  35  24  30 276
X 9 9  30.7

The mean number of days off is 30.7 years.

Median:
 The median is the midpoint of the data array. The symbol for the median is MD.
It will be one of the data values if there is an odd number of values and will be
the average of two data values if there is an even number of values.

Example 3-4: Hotel Rooms

The number of rooms in the seven hotels in downtown Pittsburgh is 713, 300, 618, 595,
311, 401, and 292. Find the median.
Sort in ascending order.
292, 300, 311, 401, 595, 618, 713
Select the middle
value. MD = 401
The median is 401 rooms.

Mode:
 The mode is the value that occurs most often in a data set. It is sometimes
said to be the most typical case.There may be no mode, one mode
(unimodal), two modes (bimodal), or many modes (multimodal)
Example:

Find the mode of the signing bonuses of eight NFL players for a specific year. The bonuses in
millions of dollars are

18.0, 14.0, 34.5, 10, 11.3, 10, 12.4, 10

You may find it easier to sort first.

10, 10, 10, 11.3, 12.4, 14.0, 18.0,

34.5

Select the value that occurs the

most. The mode is 10 million

dollars.

Measures of Variation: Variance & Standard Deviation

 The variance is the average of the squares of the distance each value is from the
mean.

 The standard deviation is the square root of the variance.

 The standard deviation is a measure of how spread out your

data a The population variance is

2
  X   2
N

 The population standard deviation is

  X
  2
N

Find the variance and standard deviation for the data set for Brand A paint. 10,
60, 50, 30, 40, 20
Months, X 𝜇 X-𝜇 (X − 𝜇)2
10 35 -25 625
60 35 25 625
50 35 15 225
30 35 -5 25
40 35 5 25
20 35 -15 225
1750
 2   X   2

n 1

7
1750 5
 6  17.1 0
 291.7
6

Measures of Position: z-score

 A z-score or standard score for a value is obtained by subtracting the mean
from the value and dividing the result by the standard deviation.

XX
z s

 A z-score represents the number of standard deviations a value is above or below

the mean.

Example 3-29: Test Scores

A student scored 65 on a calculus test that had a mean of 50 and a standard

deviation of 10; she scored 30 on a history test with a mean of 25 and a standard
deviation of 5.
Compare her relative positions on the two tests.

X 65  50
zX    1.5 Calculus
s 10

30  25
X   1.0 History
zX  5
s

She has a higher relative position in the Calculus class.

The Normal Distribution

 Many continuous variables have distributions that are bell-shaped and are called
approximately normally distributed variables.
 The theoretical curve, called the bell curve or the Gaussian distribution, can be
used to study many variables that are not normally distributed but are
approximately normal.
 The shape and position of the normal distribution curve depend on two parameters,
the
mean and the standard deviation.
 Each normally distributed variable has its own normal distribution curve, which
depends on the values of the variable’s mean and standard deviation.

value  mean
z  standard deviation

X
z 

Each month, an American household generates an average of 28 pounds of

newspaper for garbage or recycling. Assume the standard deviation is 2 pounds.
If a household is selected at random, find the probability of its generating
between 27 and 31 pounds per month. Assume the variable is approximately
normally distributed.
Step 1: Draw the normal distribution curve.
Step 2: Find z values corresponding to 27 and 31.
27  28
z  0.5
2
31 28
z  1.5
2

Step 3: Find the area between z = -0.5 and z = 1.5.

Table E gives us an area of 0.9332 – 0.3085 = 0.6247. The probability is 62%.

LEARNING CONTENT:

During this time of technology and advanced communication, enormous amount

of information are passed from one point to another using different media. Thus, there is
a need to secure these information using codes or other methods, one of these is called
cryptography. But before going to the specific topics of these codes and cryptography,
we will first be aware that most of these rely on the mathematical concept called
modular arithmetic.

Modulo n

Two integers a and b are said to be congruent modulo n, where n is a natural

number, if
𝑎−𝑏
is an integer. In this case, we write a≡b mod n. The number n is called the modulus.
𝑛
The statement a≡b mod n is called congruence.
Furthermore, if a ≡ b mod n and a and b are whole numbers, then a and b have the same
remainder when divided by n or in ordinary terms, when we divide a with n, the remainder is
b.

Example: Determine whether the congruence is true.

a. 29 ≡ 8 mod 3. This is true or congruent 29−8

because = 7 and 7 is a whole number.
29 8 3
Furthermore, and have the same remainder which is 2.
3 3
15−4
b. 15 ≡ 4 mod 6. This is false or not congruent because is not a whole
6
number.

One simple application of this is in terms of determining the time using the 12-hour clock,
like:

a) What is 7 hours after 8 o’clock?

This can be done by counting 7 hours from 8 o’clock which is 3 o’clock. But using
the
8+7 15
above formula, this can be computed = ≡ 3 mod 12.This means that if we
by 12 12

divided 15 with 2, we get 3 as the remainder and this what we would want to get.

b) But what if the problem is; What 70 hours after 7 o’clock? So it will take time to
count 70 hours from 8 o’clock. But what we can do is to add 70 hours to 7
o’clock and that
77
this will be 77. So using the formula, ≡ 5 mod 12. This means that it will be 5
12
o’clock.

Another important application of this concept is in Universal Product Code or UPC or

bar codes found in the different items like grocery and almost all items sold in the
stores. They are very helpful to keep track of inventory. This means that its purpose is
to make it easy to identify product features such as the brand name, item, size, etc.

The UPC is a 12-digit number that satisfi es a congruence equation that is similar to
the one for ISBNs. The last digit is the check digit. If we label the 12 digits of the UPC
as d1, d2, ... , d12, we can write a formula for the UPC check digit d12.

The formula for determining d12 is:

d12 = (3𝑑1 + 𝑑2 + 3𝑑3 + 𝑑4 + 3𝑑5 + 𝑑6 + 3𝑑7 + 𝑑8 + 3𝑑9 +𝑑10 + 3𝑑11) mod 10 and based on
this formula, d12 should be 0, If not 0, then the UPC is not valid.

This can also be applied in determining whether a credit card is valid or not is
calculated by calculating the last digit using the following steps:

=Beginning with the next-to-last digit (the last digit is the check digit) and
reading from right to left, double every other digit. If a digit becomes a two-digit
number after being doubled, treat the number as two individual digits. Now find the
sum of the new list of digits;
the final sum must be equal to sum mod 10. The last digit must be equal to d16 = 10-sum
mod
10. (Example, if the sum is 34 then this is 4 mod 10, thus, the last digit is 10-4 =6.)

d16 = (𝑑15 +2 𝑑14 + 𝑑13 +2𝑑12 + 𝑑11 + 2𝑑10 + 𝑑9 + 2𝑑8 + 𝑑7 + 2𝑑6 + 𝑑5+2𝑑4 + 𝑑3 + 2𝑑2 + 𝑑1) mod
10

Example: Check if this is a valid Credit Card Number; 5234 8213 3410 1298.

d16 = (9 +2x2 + 1 +2x0 +1 + 2x4 + 3 + 2x3 + 1 + 2x2 + 8 +2x4 + 3 + 2x2 + 5) mod

10 = 65 mod
10 since this 5 mod 10, the credit card number is not valid.

COMMUNICATION SECURITY:

• Cryptography: process of making and using codes to secure transmission of

information

• Encryption: converting original message into a form unreadable by

unauthorized individuals (vs decryption)

• Cryptanalysis: process of obtaining original message from encrypted message

without knowing algorithms

Cryptology: science of encryption; combines cryptography and cryptanalysis

Classical Cryptosystems

• Shift Ciphers (Caesar) y= (x+k )mod 26

• Affine Ciphers y=ax+b (mod 26)

• Vigenere Ciphers codes=(02,14,03,04,18)

Shift Ciphers

Each letter in the plaintext is replaced by a letter some fixed number of positions
down the alphabet
 Each letter in the plaintext is replaced by a letter some fixed number of positions
down the alphabet

EXAMPLE:

a) Encrypt “Pray for Marawi” using a shift cypher k

= 11, that is, apply the formula

c = (p + 11) mod 26

Where c is the ciphertext and p is the plaintext.

plain P R A Y F O R M A R A W I
p 15 17 0 24 6 14 17 12 0 17 0 22 8
C 0 2 11 9 17 25 18 23 11 2 11 7 19
ciphr A C L J R Z S X L C L H T

b) Decrypt: GUNBLOFYM using k=20 (backward)

p = (c- 20) mod 26

ciphr G U N B L O F Y M
C 6 20 13 1 11 14 5 24 12
p 12 0 19 7 17 20 11 4 18
plain M A T H R U L E S