Q.Find variance and S.D. for the following set of numbers. 65, 77, 81, 98, 100, 80, 129

Standard deviation is defined as the positive square root of variance.

By definition, standard deviation $\sigma = \sqrt{\text{variance}}$. Given variance $= 144$:

$\sigma = \sqrt{144} = 12$

Standard deviation is always taken as the non-negative root (a spread measure cannot be negative), which rules out option (b) $-12$.

The square of the standard deviation is called the variance, option (c).

Standard deviation ($\sigma$) measures the spread of data about the mean. By definition, variance ($\sigma^2$) is exactly the square of the standard deviation: $\text{Variance} = \sigma^2 = (\text{Standard Deviation})^2$.

So among the given options, the square of standard deviation is the Variance, not mean-deviation, range, or mean.

Standard deviation $=\sqrt{\text{Variance}}$; here $\sqrt{225}=15$.

By definition, variance is the square of the standard deviation, so standard deviation $=\sqrt{\text{variance}}$.

Given variance $=225$: standard deviation $=\sqrt{225}=15$.

Variance is defined as the square of the standard deviation.

By definition, variance $= (\text{standard deviation})^2$.

Given standard deviation $= 0.012$:

$\text{Variance} = (0.012)^2 = 0.012 \times 0.012 = 0.000144$

Given standard deviation $=1.6$, the variance is $2.56$.

By definition, variance $=(\text{standard deviation})^2$. Here standard deviation $=1.6$, so variance $=1.6\times1.6=2.56$.

This difference is called the range of the series, the simplest measure of dispersion.

Step 1. Range is defined as: Range $=$ Maximum value $-$ Minimum value.

Step 2. It measures how spread out the data is, from the smallest to the largest observation.