StatisticsIntroduction
"Statistics may be rightly called the science of averages and their estimates." — A.L. Bowley and A.L. Boddington
The inter-school final is on Saturday and Coach Mehra has one batting place left. Two batsmen want it, and on paper they are tied: over their last ten matches, both averaged exactly 53 runs.
The coach does not trust a single number. Before he decides, he wants to see all twenty scores.
Play the season. Each tap plays one match and drops both scores onto the number lines — batsman A on the top line, batsman B on the bottom one. The purple marks show each batsman's mean so far.
Fig. 13.1 (top) and Fig. 13.2 (bottom) — the runs scored by batsmen A and B in their last ten matches.
Both purple marks finish at 53, so the means really are equal. But the blue dots tell two very different stories.
The final needs a steady 50 from the number 3 batsman. Whom should the coach pick?
Batsman B. All ten of his scores lie between 46 and 60, so about 50 runs is what he delivers almost every time. Batsman A swings from a duck to 117. Same average, very different players — and the average on its own could not tell them apart.
Here are the twenty scores as numbers.
Batsman A: 30, 91, 0, 64, 42, 80, 30, 5, 117, 71
Batsman B: 53, 46, 48, 50, 53, 53, 58, 60, 57, 52
Recall that the
Batsman A's ten scores add up to 530, so his mean is
For the
Here is even, so the median is the mean of the
So both
| Batsman A | Batsman B | |
|---|---|---|
| Mean | 53 | 53 |
| Median | 53 | 53 |
Yet the two performances are clearly not the same. Batsman A's scores vary from 0 (the minimum) to 117 (the maximum); batsman B's stay between 46 and 60. In Fig. 13.1 the dots are scattered right across the line, while in Fig. 13.2 they cluster tightly around 53.
A measure of central tendency tells us roughly where the data is centred. It says nothing about how the data is scattered around that centre — and that variability matters just as much. Like the mean or the median, we would like a single number for it. Such a number is called a
Spread is everywhere once you look for it. A factory keeps the spread of its bolt sizes tiny so that every bolt fits its nut; two classes with the same average marks can still have very different weakest and strongest students. Whenever someone asks how reliable or how consistent something is, they are asking about dispersion.
Older Than You Think
The word statistics comes from the Latin status, a political state: for most of history, statistics meant the numbers a ruler kept about the people and the land. Perhaps the first census was held in Egypt around 3050 B.C. In India, about 2000 years ago, the empire of Chandragupta Maurya (324–300 B.C.) ran an efficient system of administrative records, and Kautilya's Arthashastra (around 300 B.C.) describes the recording of births and deaths. Much later, Abul Fazl's Ain-i-Akbari gave a detailed account of the surveys carried out under Akbar.
The mathematics came later. Captain John Graunt of London (1620–1674) studied the records of births and deaths so carefully that he is called the father of vital statistics.
Drag these milestones into the order in which they happened, earliest at the top.
Five thousand years of counting people, and the coach's question is still the one that matters: not just what is typical, but how far things stray from it.