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Measures of Central Tendency & Dispersion

Paper-II Statistics (Post-specific) · Statistics Syllabus

Useful for: SSC CGL

Overview

Averages describe the centre of data and measures of dispersion describe its spread; grouped-data formulas are central to Paper-II.

Key points

For positive data, AM ≥ GM ≥ HM; for two numbers, GM² = AM × HM.
Grouped median = L + ((N/2 − cf)/f) × h.
Grouped mode = L + ((f₁ − f₀)/(2f₁ − f₀ − f₂)) × h.
Quartile deviation = (Q₃ − Q₁)/2; coefficient of QD = (Q₃ − Q₁)/(Q₃ + Q₁).
Coefficient of variation = (SD/mean) × 100%.

Explanation

Mean deviation is least when taken about the median.
SD is affected by change of scale but not by change of origin.
Combined SD needs group means, SDs and sizes.
Lower CV means greater consistency.

Step by step

Identify the measure or test asked and its formula.
Organise the data (table, totals) before computing.
Interpret the result (sign, size, significance).

Examples

For 4 and 9: AM = 6.5, GM = 6, HM = 2 × 4 × 9/13 = 72/13 ≈ 5.54; check GM² = 36 = 6.5 × 72/13.
Q₁ = 20, Q₃ = 40 → QD = 10; coefficient = 20/60 = 0.33.
Series A: mean 50, SD 5 (CV 10%); series B: mean 40, SD 6 (CV 15%) → A is more consistent.

Traps and advanced points

Trap: Using N instead of N/2 in the median formula.
Trap: Thinking SD changes when a constant is added.
Trap: Comparing SDs instead of CVs for consistency.

Quick revision

AM ≥ GM ≥ HM.
Median and mode grouped formulas.
CV for consistency.

Practice MCQs

The mean of 4, 8, 6, 10, 12 is
1. 40
2. 6
3. 10
4. 8
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Answer: (d) 8. Sum 40 ÷ 5.
The median of 3, 9, 5, 7, 11, 1 is
1. 7
2. 5
3. 6
4. 9
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Answer: (c) 6. Sorted 1, 3, 5, 7, 9, 11; median = (5 + 7) ÷ 2.
The mode of 2, 3, 3, 5, 7, 3, 5 is
1. 3
2. 5
3. 2
4. 7
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Answer: (a) 3. 3 occurs three times.
For a moderately skewed distribution, the empirical relation is
1. Mode = Mean × Median
2. Mean = 3 Median − 2 Mode
3. Median = Mean + Mode
4. Mode = 3 Median − 2 Mean
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Answer: (d) Mode = 3 Median − 2 Mean. Karl Pearson's empirical formula.
The range of 15, 22, 9, 30, 18 is
1. 9
2. 30
3. 21
4. 15
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Answer: (c) 21. 30 − 9.
Variance is the
1. square root of the standard deviation
2. square of the standard deviation
3. mean of the data
4. range divided by 2
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Answer: (b) square of the standard deviation. SD is the positive square root of variance.

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Notes and questions are written by Kundali IQ for practice. They are not official papers; always check the latest official notification for the syllabus and pattern.