Measures of Central Tendencies and Dispersions
Mean
Learning Objectives:

Calculate Mean for grouped and ungrouped data.
For a set of samples consisting individual numbers, \(\bar{x} = \dfrac{\Sigma x_i}{N}\)
or for a set of samples having frequencies, \(\bar{x} = \dfrac{\Sigma w_i x_i}{\Sigma w_i}\)
Solved Example: 8101
For the data set, what is the sample mean? $5,\ 5,\ 10,\ 15,\ 20$
A. 10
B. 11
C. 12.5
D. 13
\[\mathrm{Mean} = \dfrac{\Sigma X_i}{N} = \dfrac{5+ 5 + 10 + 15 + 20}{5} = \dfrac{55}{5} = 11\]
Correct Answer: B
Solved Example: 8102
In FEsuccess.com office, there are 18 employees with various designations. A recent survey about their salaries is tabulated as follows:
Annual Salary Range  Number of Employees 

\$0 to less than \$20000  5 
\$20000 to less than \$40000  7 
\$40000 to less than \$60000  4 
\$60000 to less than \$80000  2 
What is the mean salary of people working in the office of FEsuccess.com?
A. \$20018
B. \$27881
C. \$29672
D. \$33333
\[\Sigma f_i x_i = \$600000\] \[\mathrm{Mean} = \dfrac{\Sigma f_i x_i}{N} = \dfrac{600000}{18} = \$33333.3\]
Correct Answer: D
Median
Learning Objectives:

Calculate Median for grouped and ungrouped data.
Median = Middle value after arranging the data in ascending or descending order.

In case of odd no. of observations it is the middle value.

In case of even no. of observations, it is the average of two middle values.
Solved Example: 8201
For the data set, what is the sample median? $5,\ 5,\ 10,\ 15,\ 20$
A. 10
B. 11
C. 12.5
D. 13
Sample is the middle value, when the data is arranged in an ascending or descending order.
Here, the given data is already arranged in the ascending order.
Middle value = 10
Correct Answer: A
Solved Example: 8202
In FEsuccess.com office, there are 18 employees with various designations. A recent survey about their salaries is tabulated as follows:
Annual Salary Range  Number of Employees 

\$0 to less than \$20000  5 
\$20000 to less than \$40000  7 
\$40000 to less than \$60000  4 
\$60000 to less than \$80000  2 
What is the median salary of people working in the office of FEsuccess.com?
A. \$22558
B. \$25041
C. \$29832
D. \$31429
By looking at the cumulative frequency column, median class is \$20000 to less than \$40000, because 9$^{\mathrm{th}}$ observation is in the second row. please refer (means 6 to 12)
\begin{align*}
\mathrm{Median} &= l+ \left(\dfrac{\dfrac{N}{2}  F}{f}\right) \times h\\ &= \$20000 + \left(\dfrac{\dfrac{18}{2}  5}{7}\right) \times \$20000 \\&= \$31429
\end{align*}
where,
l = Lower limit of the median class = \$20000
f = Frequency of the median class = 7
F = Cumulative freq. of the class preceding the median class = 5
N = Total number of observations = 18
h = Width of the median class = \$20000
Correct Answer: D
Mode
Learning Objectives:

Calculate Mode for grouped and ungrouped data.
Mode = Most frequently occurring value
Solved Example: 8301
For the data set, what is the sample mode? \[5,\ 5,\ 10,\ 15,\ 20\]
A. 10
B. 11
C. 5
D. 13
Mode is the most frequently occurring value. Here 5 is the occurring twice, more than occurrence of any other value.
Correct Answer: C
Solved Example: 8302
If x is the mean of data 3, x, 2 and 4, then the mode is________: (GATE ME 2019 Shift II)
A. 2
B. 3
C. 4
D. The data set does not have a mode.
Correct Answer: B
Standard Deviation and Variance
Learning Objectives:

Calculate Standard Deviation and Variance.
The variance of the population is the arithmetic mean of the squared deviations from the population mean. If \(\mu\) is the arithmetic mean of a discrete population of size N, the population variance is defined by: \[\begin{aligned} \sigma^2 &= \dfrac{1}{N} \left[(X_1  \mu)^2 + (X_2  \mu)^2 + .... + (X_N  \mu)^2 \right]\\ &= \dfrac{1}{N} \Sigma (X_i  \mu)^2\end{aligned}\]
If your data set is a sample from a population, the variance of that sample is calculated by dividing the sum of the squared deviations by the number of data points minus one (N1). So, sample variance is: \[\begin{aligned} \sigma^2 &= \dfrac{1}{N1} \left[(X_1  \mu)^2 + (X_2  \mu)^2 + .... + (X_N  \mu)^2 \right]\\ &= \dfrac{1}{N1} \Sigma (X_i  \mu)^2\end{aligned}\]
Solved Example: 8401
The standard deviation of the following sample: \[65,\ 59,\ 55,\ 62,\ 54,\ 85\] of the test results of 6 students taken from a population of 100 students is:
A. 10.554
B. 11.396
C. 12.512
D. 13.082
Step 1: Calculate the mean of the sample.
Count, N= 6, Sum, $\Sigma$ x= 380, Mean, $\bar{x}$=63.3
Step 2: Calculate the deviations from the mean of each observation.
65  63.3 = 1.7,
59  63.3 =  4.3,
55  63.3 =  8.3
62  63.3 =  1.3,
54  63.3 =  9.3
85  63.3 = 21.7
Step 3: Take the square of these deviations and calculate the average of them.
Since the question clearly mentions that it is a sample, we will divide by (n1) rather than n.
Variance, \begin{align*} s^2 &= \dfrac{1.7^2 + ( 4.3)^2 + ( 8.3)^2 +( 1.3)^2 + ( 9.3)^2 +(21.7)^2}{5}\\ &= \dfrac{649.34}{5}\\ &= 129.87 \end{align*}Step 4: Take the square root of sample variance, that's the standard deviation, $s$: $\sqrt{129.87}$ = 11.396
Correct Answer: B
Solved Example: 8402
A machine produces 0, 1 or 2 defective pieces in a day with an associated probability of $\dfrac{1}{6}$, $\dfrac{2}{3}$ and $\dfrac{1}{6}$ respectively. The mean value and the variance of the number of defective pieces produced by the machine in a day, respectively, are: (GATE ME 2014 Shift III)
A. 1 and $\dfrac{1}{3}$
B. $\dfrac{1}{3}$ and 1
C. 1 and $\dfrac{4}{3}$
D. $\dfrac{1}{3}$ and $\dfrac{4}{3}$
Correct Answer: A
Solved Example: 8403
If the difference between the expectation of the square of a random variable (E[X$^2$]) and the square of the expectation of the random variable (E[X])$^2$ is denoted by R, then:
A. R = 0
B. R < 0
C. R ≥ 0
D. R > 0
Correct Answer: C
Solved Example: 8404
A person decides to toss a fair coin repeatedly until he gets a head. He will make at most 3 tosses. Let the random variable Y denote the number of heads. The value of var {Y}, where var{.} denotes the variance, equals. (GATE EE 2017 Shift II)
A. $\dfrac{7}{8}$
B. $\dfrac{49}{64}$
C. $\dfrac{7}{64}$
D. $\dfrac{105}{64}$
Correct Answer: C