Why averages work, and which one to trust

You can now calculate every average and predict how it behaves. This layer asks why. Why does sum ÷ count give a fair share? Why must deviations from the mean cancel out? Why is averaging two class averages usually wrong? Why is a cricketer's "average" not the mean of their scores? And when a newspaper says "the average Indian…", which average should it be using?

The arguments here are the kind mathematicians call proofs: reasons that work for every data set, not just the examples we try.

The class marks, salaries and journeys used as examples here are invented; the rainfall figures and Bradman's record are real, and the chapters say where they come from.

Put a ruler on your finger and place identical coins at the positions 3, 4, 5, 6 and 7 cm. Where must your finger go for the ruler to balance? At 5 cm, the mean. Move the coin at 7 to 12 and the balance point shifts right, to 6 cm, which is the new mean (3 + 4 + 5 + 6 + 12 = 30, and 30 ÷ 5 = 6).

This is not a coincidence. The mean is exactly the point where the "pull" of the values on each side cancels out. To see why, measure each value's deviation: how far it is from the mean, with a sign (+ above, − below).

Investigate showed by experiment that adding a constant shifts the mean and multiplying scales it. Now we can prove it using nothing but sum = mean × count.

Section A of Class 7 has 30 students with a mean mark of 62. Section B has 20 students with a mean of 72. What is the mean of all 50 students?

It is tempting to say (62 + 72) ÷ 2 = 67. That is wrong. Section A has more students, so its average should count for more. Go back to totals:

• Section A total = 30 × 62 = 1,860.
• Section B total = 20 × 72 = 1,440.
• All 50 students: (1,860 + 1,440) ÷ 50 = 3,300 ÷ 50 = 66.

The combined mean, 66, is closer to Section A's 62 because Section A is bigger. This is called a weighted mean: each group's mean is weighted by its size.

When data has many different values, a frequency table with one row per value becomes long and bumpy. Here is an invented but realistic set of marks (out of 50) for 40 students in a Class 8 test:

8, 22, 47, 18, 13, 49, 40, 39, 22, 30, 37, 48, 30, 27, 32, 33, 39, 31, 36, 21, 17, 43, 17, 47, 27, 21, 22, 39, 29, 28, 39, 45, 38, 28, 30, 19, 7, 33, 20, 24.

Almost every mark appears only once or twice, so a mark-by-mark table tells us little. Instead we group the marks into class intervals of equal width: 0–10, 10–20, 20–30, 30–40, 40–50.

A double bar graph puts two related data sets on the same axes, with a pair of bars for each category and a legend saying which bar is which. It is the natural way to compare two groups, two years or two places.

India has two monsoons. Mumbai, on the west coast, is soaked by the south-west monsoon from June to September. Chennai, on the east coast, gets its heaviest rain from the north-east monsoon in October to December. A double bar graph makes the contrast leap out.

Nine people sit in a tea stall. Each earns about ₹25,000 a month. Their mean, median and mode income are all ₹25,000. Then a billionaire walks in who earns ₹100 crore a month.

• Total income in the room: 9 × 25,000 + 1,00,00,00,000 = ₹1,00,02,25,000.
• Mean income: ₹1,00,02,25,000 ÷ 10 = ₹10,00,22,500, about ₹10 crore a month each!
• Median income: the 10 values in order are nine 25,000s and one 100 crore; the middle two (5th and 6th) are both 25,000, so the median is ₹25,000.
• Mode: ₹25,000.

The mean now describes nobody in the room. Nine people earn far less, and one earns ten times more. This is why reports on incomes, house prices and wealth use the median: it tells you about a typical person, not about how much money is in the room.

In cricket, a batter's batting average is not the mean runs per innings. It is

batting average = total runs ÷ number of times out.

Innings in which the batter was not out (marked with an asterisk, like 45*) add their runs to the total but do not add to the count of dismissals. The idea: a not-out innings is unfinished, so it would be unfair to treat it as a completed score. The side effect: batters who are often not out, like tail-enders or finishers, can have a batting average higher than any "fair share" of their innings.

The range has the same weakness as the mean: one outlier can wreck it. In the ten values 12, 14, 15, 15, 16, 17, 18, 19, 20, 48, the range is 48 − 12 = 36, but nine of the ten values lie within just 8 of each other.

A sturdier idea, used from Class 9 onwards, is the interquartile range (IQR). Put the data in order and split it into two halves. The median of the lower half is the lower quartile, the median of the upper half is the upper quartile. The IQR is their difference, and it measures the spread of the middle half of the data, ignoring the extremes.

Here the lower half is 12, 14, 15, 15, 16 (median 15) and the upper half is 17, 18, 19, 20, 48 (median 19). IQR = 19 − 15 = 4, a far better description of how spread out most values are.

Collecting data is as old as governments. Rulers needed to know how many people, fields and animals they had, to raise taxes and armies and to plan for famine. The word statistics itself comes from state: it began as "facts about the state".

India has a long record. The Arthashastra, a text on statecraft traditionally linked with Kautilya (Chanakya) more than 2,000 years ago, describes officials who kept records of households, land and livestock. Under Akbar, Raja Todar Mal's revenue system measured land and recorded crop yields, and Abul Fazl's Ain-i-Akbari is packed with tables of data about the empire. Modern India has one of the largest statistical systems in the world, and one of its founders, P. C. Mahalanobis, is remembered on 29 June, his birthday, as National Statistics Day.