Q.Define the following terms: Median, Standard Deviation and variance.

This is a plain theory question asking for definitions of three measures of dispersion (and central tendency).

Median

The median is the middle value of a dataset when the observations are arranged in ascending or descending order. If the number of observations $n$ is odd, the median is the $\left(\frac{n+1}{2}\right)^{\text{th}}$ term. If $n$ is even, the median is the average of the $\left(\frac{n}{2}\right)^{\text{th}}$ and $\left(\frac{n}{2}+1\right)^{\text{th}}$ terms.

The median divides the dataset into two equal halves and is a measure of central tendency that is robust to outliers — extreme values do not distort it the way they distort the mean.

Variance

Variance measures the average squared deviation of each observation from the mean. It quantifies how spread out the data points are around the mean.

For a population of $N$ values $x_1, x_2, \ldots, x_N$ with mean $\mu$, the variance $\sigma^2$ is

For a sample of $n$ values with mean $\bar{x}$, the sample variance $s^2$ is

The denominator $n-1$ (Bessel's correction) is used in the sample formula to obtain an unbiased estimate of the population variance. Variance is always non-negative, and a variance of zero means all values are identical.

Standard Deviation

Standard deviation is the square root of the variance. It measures the average distance of each observation from the mean, in the same units as the original data. …

These three statistical measures help us understand data in complementary ways. The median tells us where the center of the data lies when arranged in order, while variance and standard deviation quantify how spread out the values are around the mean. Together, they give a complete picture of a dataset's shape and variability.

Median

The median is the middle value in a dataset when all observations are arranged in ascending (or descending) order. It divides the dataset into two equal halves: 50% of the values lie below the median, and 50% lie above it.

For a dataset with $n$ observations:

• If $n$ is odd, the median is the value at position $\frac{n+1}{2}$.
• If $n$ is even, the median is the average of the values at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$.

Example: For the dataset [3, 7, 9, 15, 21], the median is $9$ (the middle value). For [3, 7, 9, 15], the median is $\frac{7+9}{2} = 8$.

The median is particularly useful when dealing with skewed distributions or outliers, because unlike the mean, it is not affected by extreme values. If a dataset contains a few very large or very small numbers, the median remains a robust measure of central tendency.

Variance

Variance measures how far each value in a dataset is from the mean, on average. It quantifies the spread or dispersion of the data. A small variance indicates that the data points are clustered closely around the mean, while a large variance indicates they are spread out.

where $x_i$ represents each data value, $\bar{x}$ is the mean of the dataset, and $n$ is the number of observations.

For a sample (rather than the entire population), we divide by $n-1$ instead of $n$ to get an unbiased estimate:

Example: For the dataset [2, 4, 6, 8, 10], the mean is $\bar{x} = 6$. The variance is:

Standard Deviation …