Lesson 1 of 11 · 7 min

Why spread matters

NCERT §13.1

The school is picking one opener for the inter-school team. Two batsmen, Arjun and Kabir, have each played 8 innings at the trials and each has scored exactly 400 runs. The coach says their averages are equal, so the choice is a coin toss. Is it?

The story this chapter follows: The school cricket trials

Imagine the school cricket trials. Two openers, Arjun and Kabir, have each played 8 innings; the coach also runs a bowling-machine test and a practice match, and has to pick one opener. All the scores, hits and runs in this chapter are made up for illustration, chosen so the arithmetic comes out clean.

The lesson in notes

In short

A measure of central tendency (mean, median or mode) gives one value around which the data sit. It says nothing about how tightly or loosely the observations are packed around that value.

Mean of n observations: x̄ = (x₁ + x₂ + … + xₙ)/n. Median: arrange the data in order; for odd n it is the ((n + 1)/2)th value, for even n the mean of the (n/2)th and (n/2 + 1)th values.

NCERT's two batsmen have the same mean and the same median, 53, over ten matches, yet one scores anywhere from 0 to 117 while the other stays between 46 and 60. Same centre, very different reliability.

Made-up illustration used through these notes: two openers at a school trial each score 400 runs in 8 innings. Arjun: 42, 48, 50, 52, 55, 45, 50, 58. Kabir: 12, 95, 0, 78, 48, 110, 5, 52. Both have mean 50 and median 50.

Plotted as dots on a number line, Arjun's scores bunch near 50 while Kabir's are strewn from 0 to 110. The centre alone cannot tell the two apart.

Variability is therefore a second feature of data. A single number that describes it is called a measure of dispersion, and this chapter builds such measures for raw (ungrouped) and grouped data.