Probability and Statistics Engineers
Contents:
List of Topics
Introduction to Statistics & Data Analysis
No. of
Weeks
1st
Contact Hours
3
Sample Space, Events, Counting Sample Points , Additive
Rules.
2nd
3
Conditional Probability, Independent Events, Multiplicative
Rules, Bayes Rule.
3rd
3
Concept of Random Variable, Discrete Probability
Distributions, Continuous Probability Distributions.
4th
3
Sampling distribution of the mean.
5th
3
Discrete Uniform Distribution, Binomial Distribution,
Poisson Distribution.
Revision and Test 1
6th
3
7th
3
Continuous Uniform Distribution, Normal Distribution.
8th
3
Exponential Distribution, Random Sampling, Sampling
Distribution of Means, t-Distribution
Introduction to Estimation, Statistical Inference, Estimating
the Mean, Standard Error.
9th
3
10th
3
Estimating the Difference Between Two Means, Estimating a
Proportion, Estimating the Difference between Two
Proportions.
Statistical Hypotheses, Testing a Statistical Hypothesis.
11th
3
12th
3
Revision and Test2
13th
3
One- and Two-Tailed Tests, Tests Concerning a Single
Mean, Tests on a Single Mean when variance is unknown
Tests on Two Means, Test on a Single Proportion, Test on
two Proportions.
.
14th
3
15th
3
Introduction to Statistics & Data Analysis:
1.1 Introduction:
Populations and Samples: The study of statistics revolves around the study of data
sets. This lesson describes two important types of data sets populations and samples.
Population vs Sample
The main difference between a population and sample has to do with how
observations are assigned to the data set.

A population includes each element from the set of observations that can be
made.

A sample consists only of observations drawn from the population.
Depending on the sampling method, a sample can have fewer observations than the
population, the same number of observations, or more observations. More than one
sample can be derived from the same population.
Examples:
• Let X 1 , X 2 , X 3 ,...., X n be the population values (in general, they are
unknown).
• Let x1 , x2 , x3 ,...., xn be the sample values (these values are known) .
• Statistics obtained from the sample are used to estimate (approximate) the
parameters of the population.
Measures of Location (Central Tendency):
• The data (observations) often tend to be concentrated around the center of
the data.
• Some measures of location are: the mean, mode, and median.
• These measures are considered as representatives (or typical values) of the
data. They are designed to give some quantitative measures of where the
center of the data is in the sample.
The Sample mean of the observations ( x ):
If x1 , x2 , x3 ,...., xn are the sample values, then the sample mean is
n
xi
x1  x2  x3  .....  xn 
i 1
x

n
n
Example:
Suppose that the following sample represents the ages (in year) of a sample of 8
students: 20, 23, 19, 24, 25, 24, 26, 21
Then, the sample mean is:
x
20  23  19  24  25  24  26  21 182

 22.75 years
8
8
Measures of Variability (Dispersion or Variation):
• The variation or dispersion in a set of data refers to how spread out the
observations are from each other.
• The variation is small when the observations are close together. There is no
variation if the observations are the same.
• Some measures of dispersion are range, variance, and standard deviation
• These measures are designed to give some quantitative measures of the
variability in the data.
2
The Sample Variance (S ):
Let x1 , x2 , x3 ,...., xn be the observations of the sample. The sample variance is
denoted by S 2 and is defined by:
n
S2 
Note:
(x
i 1
i
 x)2
n 1
(n −1) is called the degrees of freedom (df) associated with the sample
2
variance S
.
The Standard Deviation (S):
The standard deviation is another measure of variation. It is the square root of the
variance, i.e., it is:
n
S  S2 
(x
i 1
i
 x)2
n 1
Example:
Compute the sample variance and standard deviation of the following
observations (ages in year): 20 ,22, 23 , 18 , 17
Solution:
20  22  23  18  17
x
 20
Mean is:
5
n
Variance is:
S2 
S2 
(x
i 1
i
 x)2
n 1
(20  20) 2  (22  20) 2  (23  20) 2  (18  20) 2  (17  20) 2 0  4  9  4  9

 6.5
5 1
4
Standard deviation is: S  6.5 = 2.55
to 3 s.f.