One number for a whole group

Shubman and Yashasvi score in a cricket series: Shubman 0, 17, 21, 90 and Yashasvi 67, 55, 18, 35. Who performed better?

The class disagrees, and every argument has something in it. Vaishnavi says Shubman, because he scored the highest in a single match — 90. Shreyas says Yashasvi, because his TOTAL is 175 against 128. Vaishnavi then notices that Yashasvi is more CONSISTENT: the gap between his highest and lowest score is smaller.

Now a second series. Shubman: 23, 7, 10, 52, 18. Yashasvi: 26, 53, 2, and 15 — he missed one match. Vaishnavi says Shubman is better, with a total of 110 against 96.

Shreyas objects, and he is right: Yashasvi made 96 runs in FOUR matches while Shubman made 110 in FIVE. Comparing totals is unfair when the groups are different sizes.

So can a SINGLE number represent a whole group of numbers? One way is to balance out the highs and the lows: add up all the runs and divide by the number of matches. That value is the AVERAGE, or arithmetic mean (in Hindi, माध्य). It is the same 'average' you hear about on cricket commentary: a batter's batting average is the runs scored divided by the number of times out.

Shubman's average is 110 ÷ 5 = 22 runs per match. Yashasvi's is 96 ÷ 4 = 24. In this series Yashasvi's average is higher. MEAN = (sum of all the values) ÷ (number of values).

The mean can also be understood as a fair share. Shreyas and 4 friends collect 3, 8, 10, 5 and 4 guavas — 30 in all, so sharing equally gives 30 ÷ 5 = 6 each. Parag and 5 friends collect 5, 4, 6, 3, 4 and 8 — also 30, but shared among 6 gives 5 each. Same total, different group size, different share. Ancient Indian scholars saw it this way too: Brahmagupta called it samarajju and Bhāskarācārya samamiti, 'mean measure', from sama meaning equal.