Measures of Central Tendency (Mean, Median, Mode) — Practice

Question 1

The monthly incomes (₹000) of 8 workers are: 12, 15, 18, 22, 10, 14, 20, 25. Calculate the arithmetic mean.

Accepted Answer

X̄ = ΣX / N. ΣX = 12 + 15 + 18 + 22 + 10 + 14 + 20 + 25 = 136. N = 8. X̄ = 136 / 8 = ₹17,000. Interpretation: The average monthly income of the 8 workers is ₹17,000.

Question 2

Find the median for the following frequency distribution: Marks: 0–20: 5, 20–40: 15, 40–60: 20, 60–80: 8, 80–100: 2. N = 50.

Accepted Answer

Step 1: Build cumulative frequency table. 0–20: f = 5, CF = 5. 20–40: f = 15, CF = 20. 40–60: f = 20, CF = 40. 60–80: f = 8, CF = 48. 80–100: f = 2, CF = 50. Step 2: N/2 = 50/2 = 25. Median class = class where CF first exceeds 25 = 40–60 (CF jumps from 20 to 40). Step 3: L = 40, N/2 = 25, CF (before median class) = 20, f = 20, h = 20. Median = 40 + [(25 − 20) / 20] × 20 = 40 + [5/20] × 20 = 40 + 5 = 45 marks.

Question 3

For a frequency distribution, Mean = 54 and Median = 52. (a) Find Mode using the empirical relationship. (b) Calculate the arithmetic mean for the following discrete series: X: 10, 20, 30, 40, 50; f: 4, 7, 12, 8, 4. (c) State when to use each measure of central tendency.

Accepted Answer

(a) Empirical relationship: Mode = 3 Median − 2 Mean = 3(52) − 2(54) = 156 − 108 = 48. (b) Arithmetic mean for discrete series: ΣfX = (10×4) + (20×7) + (30×12) + (40×8) + (50×4) = 40 + 140 + 360 + 320 + 200 = 1060. Σf = 4 + 7 + 12 + 8 + 4 = 35. X̄ = 1060 / 35 = 30.28. (c) When to use each: Mean: When data is symmetric with no extreme outliers; when further mathematical analysis is needed; example: average marks of students in a class test where scores are normally distributed. Median: When data is skewed (income, house prices, wealth); when there are extreme outliers; example: median household income is more representative than mean income because a few very rich households distort the mean. Mode: When you want the most typical/common value; for qualitative/nominal data; example: the modal occupation in India is farming; the most common shoe size stocked by a shopkeeper.

Practice — Measures of Central Tendency (Mean, Median, Mode)

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