Chapter 13: Statistics

You already know how to compute the mean, median, and mode , these are measures of central tendency. They tell you where the "centre" of a data set is. But two data sets can have the same mean and look entirely different , one might be tightly bunched, another spread far and wide. This chapter answers: how much do the data vary?

The answer is measures of dispersion. We will study four:

Range , simplest: max minus min.
Mean deviation , average of absolute deviations from a centre (mean or median).
Variance , average of squared deviations from the mean.
Standard deviation , square root of variance.

The standard deviation $\sigma$ is the workhorse of statistics. It has the same units as the data and behaves predictably under shifts and scalings. The variance $\sigma^2$ is more convenient for algebraic manipulation. The coefficient of variation $\sigma / \bar{x}$ (often expressed as a percentage) lets you compare dispersion across data sets with different units or scales.

We will compute these for raw data, frequency-tabulated data, and grouped (continuous frequency) data. For grouped data, we use the short-cut formula and the step-deviation method to keep arithmetic manageable.

For Class XII probability and statistics, the ideas you learn here scale up to random variables, expectation, and population statistics. Mastery here pays dividends.

What's inside

Measures of central tendency: review (mean, median, mode).
Range and mean deviation about mean / median.
Variance and standard deviation for ungrouped data.
Variance and standard deviation for grouped data (short-cut, step-deviation).
Combined mean and standard deviation, properties under shift/scale.
Coefficient of variation; comparing data sets.

Key results / Formula card

Quantity	Formula
Mean (raw)	$\bar{x} = \dfrac{\sum x_i}{n}$
Mean (frequency)	$\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}$
Range	$\max - \min$
Mean deviation about mean	$\text{MD}_{\bar{x}} = \dfrac{\sum
Mean deviation about median	$\text{MD}_M = \dfrac{\sum
Variance (raw)	$\sigma^2 = \dfrac{\sum (x_i - \bar{x})^2}{n}$
Standard deviation	$\sigma = \sqrt{\sigma^2}$
Variance (alt formula)	$\sigma^2 = \dfrac{\sum x_i^2}{n} - \bar{x}^2$
Variance (frequency)	$\sigma^2 = \dfrac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}$
Coefficient of variation	$\text{CV} = \dfrac{\sigma}{\bar{x}} \times 100$
Shift/scale (data $y = a + b x$ )	$\bar{y} = a + b \bar{x}$ , $\sigma_y =

How to read this chapter

For every dataset problem: (1) tabulate, (2) compute $\bar{x}$ , (3) deviations $(x_i - \bar{x})$ or their absolute values / squares, (4) sum and divide. The mechanics are simple; the volume of arithmetic is the challenge. Use the step-deviation method for grouped data to save time.

Chapter 13: Statistics

What's inside

Key results / Formula card

How to read this chapter

Sub-topics

Practice quiz