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Statistics & Probability

Mathematical Abilities — Paper-I · Official 2026 Coverage

Useful for: SSC CGL

Overview

Tier-II statistics covers dispersion (standard deviation, coefficient of variation) and multi-step probability.

Key points

Variance = mean of squared deviations; SD = √variance.
Coefficient of variation = (SD ÷ mean) × 100%; lower CV means more consistent data.
Without replacement: multiply conditional probabilities.
P(A or B) = P(A) + P(B) − P(A and B).
Adding a constant does not change SD; multiplying by k multiplies SD by |k|.

Explanation

Combined groups: use totals for mean, not average of means.
For "at least one", use 1 − P(none).
Expected value = Σ (value × probability).
Odds in favour a : b → probability a/(a + b).

Step by step

Identify the concept and the formula that links the given values.
Solve step by step, keeping units consistent.
Check the answer against the options and by a quick estimate.

Examples

Data 2, 4, 4, 4, 5, 5, 7, 9: mean 5; variance (9 + 1 + 1 + 1 + 0 + 0 + 4 + 16)/8 = 4 → SD = 2.
Two balls drawn without replacement from 5 red and 3 blue: P(both red) = 5/8 × 4/7 = 5/14.
Mean 50, SD 10 → CV = 20%.

Traps and advanced points

Trap: Changing SD when a constant is added.
Trap: Treating draws without replacement as independent.
Trap: Double-counting in P(A or B).

Quick revision

SD unaffected by shifting.
CV = SD/mean × 100.
Multiply conditional probabilities.

Practice MCQs

For the three observations 98, 100, 102, find the population standard deviation.
1. 2
2. 1.63
3. 2.63
4. 2.67
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Answer: (b) 1.63. Mean is the centre value; SD = √(mean squared deviation).
A bag contains 5 red, 34 blue and 49 green balls. One ball is drawn at random. Probability of drawing a red ball?
1. 49/88
2. 5/83
3. 5/88
4. 34/88
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Answer: (c) 5/88. Probability = favourable outcomes ÷ total equally likely outcomes.
Group A has 13 observations with mean 109; Group B has 22 observations with mean 213. Find the combined mean.
1. 175.37
2. 173.37
3. 174.37
4. 161
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Answer: (c) 174.37. Combined mean = total of both groups ÷ combined count.
A bag contains 8 red, 77 blue and 135 green balls. One ball is drawn at random. Probability of drawing a red ball?
1. 8/220
2. 135/220
3. 8/212
4. 77/220
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Answer: (a) 8/220. Probability = favourable outcomes ÷ total equally likely outcomes.
Group A has 13 observations with mean 85; Group B has 19 observations with mean 165. Find the combined mean.
1. 132.5
2. 125
3. 131.5
4. 133.5
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Answer: (a) 132.5. Combined mean = total of both groups ÷ combined count.
For the three observations 73, 75, 77, find the population standard deviation.
1. 2.63
2. 2.67
3. 1.63
4. 2
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Answer: (c) 1.63. Mean is the centre value; SD = √(mean squared deviation).
A bag contains 9 red, 78 blue and 137 green balls. One ball is drawn at random. Probability of drawing a red ball?
1. 78/224
2. 9/224
3. 137/224
4. 9/215
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Answer: (b) 9/224. Probability = favourable outcomes ÷ total equally likely outcomes.
Group A has 14 observations with mean 86; Group B has 21 observations with mean 167. Find the combined mean.
1. 126.5
2. 135.6
3. 133.6
4. 134.6
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Answer: (d) 134.6. Combined mean = total of both groups ÷ combined count.
For the three observations 73, 76, 79, find the population standard deviation.
1. 6
2. 3.45
3. 3
4. 2.45
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Answer: (d) 2.45. Mean is the centre value; SD = √(mean squared deviation).
A bag contains 10 red, 79 blue and 139 green balls. One ball is drawn at random. Probability of drawing a red ball?
1. 139/228
2. 10/218
3. 79/228
4. 10/228
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Answer: (d) 10/228. Probability = favourable outcomes ÷ total equally likely outcomes.
Group A has 15 observations with mean 87; Group B has 23 observations with mean 169. Find the combined mean.
1. 128
2. 137.63
3. 135.63
4. 136.63
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Answer: (d) 136.63. Combined mean = total of both groups ÷ combined count.
For the three observations 73, 77, 81, find the population standard deviation.
1. 10.67
2. 3.27
3. 4.27
4. 4
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Answer: (b) 3.27. Mean is the centre value; SD = √(mean squared deviation).
A bag contains 11 red, 80 blue and 141 green balls. One ball is drawn at random. Probability of drawing a red ball?
1. 11/232
2. 141/232
3. 11/221
4. 80/232
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Answer: (a) 11/232. Probability = favourable outcomes ÷ total equally likely outcomes.
Group A has 16 observations with mean 88; Group B has 16 observations with mean 171. Find the combined mean.
1. 130.5
2. 128.5
3. 129.5
4. 142.45
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Answer: (c) 129.5. Combined mean = total of both groups ÷ combined count.
For the three observations 73, 78, 83, find the population standard deviation.
1. 4.08
2. 16.67
3. 5.08
4. 5
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Answer: (a) 4.08. Mean is the centre value; SD = √(mean squared deviation).