1. Mean
- Raw data: x̄ = Σx/n.
- Frequency table: x̄ = Σfx/Σf.
- Assumed mean: x̄ = A + Σfd/Σf (d = x − A).
- Step deviation: x̄ = A + h × Σfu/Σf (u = (x − A)/h).
Properties: Σ(x − x̄) = 0 always. Adding k to every value adds k to the mean; multiplying by k multiplies the mean by k.
(ASO 2012) x: 1–5, f: 4, 5, 8, p, 2, mean 2.6 → (48 + 4p) = 2.6(19 + p) → p = 1.
(ASO 2021) Mean of first five multiples of 7 = 21.
2. Median
- Raw data: arrange in order. n odd → ((n + 1)/2)th value; n even → average of (n/2)th and (n/2 + 1)th.
- Grouped: Median = l + [(N/2 − cf)/f] × h (l = lower limit of the median class, cf = cumulative frequency before it, f = its frequency, h = class width).
(ASO 2021) N = 100, median class 20–30 (cf 24, f 36) → 20 + (26/36) × 10 = 27.22.
3. Mode
- Raw data: value occurring most often.
- Grouped: Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h (f₁ modal class frequency, f₀ previous, f₂ next).
(ASO 2012) Modal class 20–30 (12, 32, 20) → 20 + 20/32 × 10 = 26.25.
4. Relation
Empirical: Mode = 3 Median − 2 Mean.
Mean 20, median 22 → mode = 66 − 40 = 26.
5. Other facts
- Cumulative frequency (ogive) is used for the median.
- Range = max − min; standard deviation is a measure of dispersion, not central tendency.
- New mean after adding a value: (n x̄ + new)/(n + 1). After a wrong entry: correct total = old total − wrong + right.
(ASO 2021) 30 numbers average 12; first 20 average 11, next 9 average 10 → last = 360 − 220 − 90 = 50.
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
Basic level
Example 1. The geometric mean of 3 and 27 is:
(a) 9 (b) 81 (c) 15 (d) 11
Answer: (a) 9
Solution: GM = √(3×27) = √81 = 9.
Example 2. The median of 37, 18, 32, 52, 27, 30, 15, 49, 55 is:
(a) 35 (b) 32 (c) 30 (d) 37
Answer: (b) 32
Solution: Arrange in order: 15, 18, 27, 30, 32, 37, 49, 52, 55. Middle value = 32.
Example 3. The mean of 58, 68, 72, 79, 66, 73, 18 is:
(a) 63 (b) 68 (c) 60 (d) 62
Answer: (d) 62
Solution: Sum = 434, n = 7. Mean = 62.
Example 4. The median of 6, 72, 29, 59, 78, 50 is:
(a) 59 (b) 54.5 (c) 53.5 (d) 49
Answer: (b) 54.5
Solution: Sorted: 6, 29, 50, 59, 72, 78. Median = 54.5.
Example 5. The mode of 28, 24, 66, 24, 94, 53, 41, 49, 24 is:
(a) 41 (b) 25 (c) 24 (d) 94
Answer: (c) 24
Solution: 24 occurs most often (3 times).
ASO exam level
Example 6. The mode of 8, 17, 18, 17, 8, 10, 5, 17 is:
(a) 10 (b) 8 (c) 12.5 (d) 17
Answer: (d) 17
Solution: Mode = the most frequent value = 17 (it occurs 3 times).
Example 7. The algebraic sum of the deviations of a set of n values from their mean is:
(a) n (b) n − 1 (c) 0 (d) 1
Answer: (c) 0
Solution: Σ(x − mean) = Σx − n × mean = 0 always.
Example 8. The algebraic sum of the deviations of a set of n values from their mean is:
(a) 0 (b) n (c) n − 1 (d) 1
Answer: (a) 0
Solution: Σ(x − mean) = Σx − n × mean = 0 always.
Example 9. Which of the following is NOT a measure of central tendency?
(a) Mean (b) Median (c) Standard deviation (d) Mode
Answer: (c) Standard deviation
Solution: Mean, median and mode describe the centre; standard deviation measures spread.
Example 10. Which of the following is NOT a measure of central tendency?
(a) Median (b) Standard deviation (c) Mode (d) Mean
Answer: (b) Standard deviation
Solution: Mean, median and mode describe the centre; standard deviation measures spread.
Example 11. The mean of 7 numbers is 11. If 39 is included, the new mean is:
(a) 11 (b) 25 (c) 14.5 (d) 15.5
Answer: (c) 14.5
Solution: Total = 77 + 39 = 116. New mean = 116/8 = 14.5.
Example 12. The range of 95, 73, 5, 70, 43, 82, 7, 89 is:
(a) 92 (b) 95 (c) 100 (d) 90
Answer: (d) 90
Solution: Range = 95 − 5 = 90.
Example 13. The range of 73, 92, 19, 66, 40, 16, 12, 98 is:
(a) 110 (b) 86 (c) 98 (d) 88
Answer: (b) 86
Solution: Range = 98 − 12 = 86.
Advanced level
Example 14. For a moderately skewed distribution, mean = 25 and median = 27. The mode is:
(a) 21 (b) 31 (c) 34 (d) 26
Answer: (b) 31
Solution: Mode = 3 Median − 2 Mean = 81 − 50 = 31.
Example 15. For a moderately skewed distribution, mean = 60 and median = 55. The mode is:
(a) 57.5 (b) 70 (c) 48 (d) 45
Answer: (d) 45
Solution: Mode = 3 Median − 2 Mean = 165 − 120 = 45.
Topic-wise practice for Statistics: mean, median, mode (Quant): Grouped data, missing frequency, variance and standard deviation. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Mean of ungrouped and grouped data
- Finding a missing value or frequency
- Median of ungrouped and grouped data
- Mode of a frequency table
- Relation: mode = 3 median − 2 mean
- Variance and standard deviation
- Range, quartiles, GM and HM basics