Standard Deviation and Variance

  • 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 (N-1). So, sample variance is: \[\begin{aligned} \sigma^2 &= \dfrac{1}{N-1} \left[(X_1 - \mu)^2 + (X_2 - \mu)^2 + .... + (X_N - \mu)^2 \right]\\ &= \dfrac{1}{N-1} \Sigma (X_i - \mu)^2\end{aligned}\]

Solved Examples

Solved Example:

8-4-01

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:

Solution:

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 (n-1) 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:

8-4-02

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)

Correct Answer: A

Solved Example:

8-4-03

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:

Correct Answer: C

Solved Example:

8-4-04

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)

Correct Answer: C