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Three averages exist because they answer three different questions.
The mean asks what each share would be if the total were split equally. The median asks which value sits in the middle when everything is lined up. The mode asks which value turns up most often.
They diverge exactly when the data is lopsided, and that is when papers set them. Nine workers earning ₹10,000 and one owner earning ₹5,00,000 give a mean of ₹59,000 — a figure not one of the ten people earns. The median, ₹10,000, describes them honestly.
So the set of formulas is small and the marks are reliable, but there is one discipline that decides whether you get them: the median needs the data sorted first, and most lost marks in this chapter come from candidates who took the middle term of the list as printed. Sort, then count.
1. Find the median of: 12, 7, 15, 9, 20, 5
2. What is the arithmetic mean of the first 15 natural numbers?
3. The mean of 10 observations is 25. Later it is found that one observation of 45 was wrongly recorded as 25. What is the corrected mean?
4. In a class, 20 boys score an average of 60 and 30 girls score an average of 70. What is the average of the whole class?
5. For a distribution the mean is 26 and the median is 27. Using the empirical relation, the mode is:
6. If 5 is added to every observation of a data set, which of these remains unchanged?
7. Find the range of: 23, 8, 41, 16, 30
8. For the data 4, 6, 6, 7, 9, 10, 14, which relation holds?