Mean and representative values

Who batted better?

In one cricket series Shubman scored 0, 17, 21 and 90, and Yashasvi scored 67, 55, 18 and 35. Who did better? Vaishnavi points to Shubman’s 90, the highest score of all. Shreyas adds up the runs: 175 for Yashasvi and 128 for Shubman. Vaishnavi then notices that Yashasvi was steadier, since the gap between his highest and lowest scores is smaller. Each friend looks at the numbers in a different way, and each way says something true.

In the next series Shubman scored 23, 7, 10, 52 and 18, a total of 110, while Yashasvi scored 26, 53, 2 and 15 and missed one match, a total of 96. Now the totals are unfair: Shubman batted in five matches and Yashasvi in only four. What we need is a single number that represents each player’s batting, a representative value, which balances the high scores against the low ones.

The arithmetic mean

The simplest such number is the arithmetic mean, or average:

Mean = sum of all the values ÷ number of values

Shubman’s mean is 110 ÷ 5 = 22 runs a match and Yashasvi’s is 96 ÷ 4 = 24 runs a match, so in this series Yashasvi did better. (The book prints 110 ÷ 5 = 21; the right value is 22, and Yashasvi is still ahead.) Means need not be whole numbers. Two runners whose seven times add up to 122 seconds each have a mean of 122 ÷ 7 ≈ 17.43 seconds, so on average neither is quicker.

The mean as a fair share

Shreyas and four friends collect 3, 8, 10, 5 and 4 guavas, 30 in all. Parag and five friends collect 5, 4, 6, 3, 4 and 8, also 30. If each group shares equally, Shreyas’s group gets 30 ÷ 5 = 6 each and Parag’s gets 30 ÷ 6 = 5 each. Same total, different shares, because the groups differ in size. Picture the bars being levelled: the tall ones give to the short ones until all stand at the mean. In the same way, 2, 7, 9, 4 and 3 hibiscus flowers on five days give a mean of 25 ÷ 5 = 5 flowers a day, the number that would bloom daily if every day were the same.

Indian mathematicians saw the mean in exactly this way. Their names for it include samarajju (Brahmagupta, 628 CE), samikarana, meaning levelling (Mahaviracharya, 850 CE), and samamiti, mean measure (Bhaskaracharya, 1150 CE). The word sama means equal.

Zero or no value?

A player’s scores read 57, 13, 0, 84, —, 51, 27. The 0 is a real innings that earned nothing, so it counts. The dash means the player did not bat, so that match is left out. The mean is (57 + 13 + 0 + 84 + 51 + 27) ÷ 6 = 232 ÷ 6 ≈ 38.67 runs a match.

Sometimes zeros should be left out on purpose. Sita’s mango tree gave 0, 0, 8, 24, 41, 16, 5, 0, 0, 0, 0 and 0 mangoes from January to December. For the typical harvest per month, use only the season, March to July: (8 + 24 + 41 + 16 + 5) ÷ 5 = 94 ÷ 5 = 18.8 mangoes a month. The mean over all 12 months, about 7.8, describes no real month at all.

Averages around us

Averages appear everywhere: average rainfall per day in a month, a scooter’s average mileage, wheat yield per hectare in a state, or the waste a person throws away each day. In Europe in the early 1500s, a land-measuring rod was set at 16 feet by lining up 16 men toe to heel. The rod was then cut into 16 equal parts, each one foot, which made the foot the mean length of sixteen real feet.

The mean also settles the onion question. Over the same twelve months, Yahapur’s prices add up to ₹458 and Wahapur’s to ₹450, so the means are about ₹38.17 and ₹37.50 a kilogram: onions were slightly costlier in Yahapur on average. But one number cannot show how much the prices jump about. For that, see median and outliers, where the median and dot plots come in.

Why it matters

A good representative value lets you compare groups fairly, whether players, classes, towns or farms. Draw the same data as pictures in visualising data. Answers to the textbook questions are in the NCERT solutions for Part 2 Chapter 5; see all of Part 2 Chapter 5 notes.