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Elementary Statistics

Mathematics · Algebra, Geometry & Data

Useful for: SSC CGL, Railway RRB, CTET, OTET, OSSTET, UGC NET, OSSC RHT — Regular High School Teacher, UPSC NDA/NA

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Overview

Mean, median and mode summarise data; ungrouped data questions need careful ordering and counting.

Key points

Mean = sum ÷ number of observations.
Median: middle value of ordered data; for even n, the average of the two middle values.
Mode: the most frequent value; data can be bimodal.
Empirical relation (moderately skewed data): Mode ≈ 3 Median − 2 Mean.
Range = maximum − minimum.

Explanation

Adding a constant to every value adds it to the mean, median and mode; the range is unchanged.
Multiplying every value by k multiplies mean, median, mode and range by k.
Combined mean of two groups = (n1x̄1 + n2x̄2)/(n1 + n2).
Always sort the data before finding the median.

Step by step

Sort the data.
Compute the measure asked.
Check with a quick estimate.

Examples

Data 2, 3, 3, 5, 7: mean = 20/5 = 4, median = 3, mode = 3.
Group A: 20 students, mean 40; group B: 30 students, mean 50 → combined mean = (800 + 1,500)/50 = 46.
Mean 30 and median 28 → mode ≈ 3 × 28 − 2 × 30 = 24.

Traps and advanced points

Trap: Finding the median without sorting.
Trap: Averaging the two group means without weights.
Trap: Thinking a range changes when a constant is added.

Quick revision

Sort before median.
Combined mean is weighted.
Mode ≈ 3 Median − 2 Mean.

Practice MCQs

From the perspective of definition and scope, which one of the following statements about Statistics is correct?
1. Statistics has no exceptions or limitations in exam analysis.
2. Statistics in CDS is elementary-mathematics level and rewards speed with conceptual accuracy.
3. Do not skip unit or sign checks in Statistics.
4. Statistics is unrelated to institutions, processes or real-world applications.
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Answer: (b) Statistics in CDS is elementary-mathematics level and rewards speed with conceptual accuracy.. Statistics in CDS is elementary-mathematics level and rewards speed with conceptual accuracy.
Consider the following statements on Statistics, focusing on definition and scope: 1. Statistics in CDS is elementary-mathematics level and rewards speed with conceptual accuracy. 2. Check domains, signs and units before finalizing the option. Which of the statements given above is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
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Answer: (c) Both 1 and 2. Both statements are correct.
Which option best connects the definition and scope dimension of Statistics with its broader exam relevance?
1. Statistics can be understood without any mechanism, limitation or exception.
2. Write the relevant Statistics relation before calculation. Also, Check signs, units and approximation before marking.
3. Do not skip unit or sign checks in Statistics.
4. Every statement about Statistics is absolute and context-free.
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Answer: (b) Write the relevant Statistics relation before calculation. Also, Check signs, units and approximation before marking.. Write the relevant Statistics relation before calculation. Also, Check signs, units and approximation before marking.
A bag has 3 red and 7 blue balls. One ball is drawn at random. Probability of red?
1. 1/10
2. 3/7
3. 3/10
4. 7/10
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Answer: (c) 3/10. Favourable outcomes=3; total=10; probability=3/10.
A bag has 5 red and 12 blue balls. One ball is drawn at random. Probability of red?
1. 12/17
2. 5/17
3. 1/17
4. 5/12
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Answer: (b) 5/17. Favourable outcomes=5; total=17; probability=5/17.
A bag has 4 red and 14 blue balls. One ball is drawn at random. Probability of red?
1. 4/18
2. 4/14
3. 1/18
4. 14/18
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Answer: (a) 4/18. Favourable outcomes=4; total=18; probability=4/18.
A bag has 8 red and 12 blue balls. One ball is drawn at random. Probability of red?
1. 12/20
2. 1/20
3. 8/12
4. 8/20
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Answer: (d) 8/20. Favourable outcomes=8; total=20; probability=8/20.
A bag has 4 red and 15 blue balls. One ball is drawn at random. Probability of red?
1. 4/15
2. 15/19
3. 4/19
4. 1/19
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Answer: (c) 4/19. Favourable outcomes=4; total=19; probability=4/19.
A bag has 5 red and 8 blue balls. One ball is drawn at random. Probability of red?
1. 5/13
2. 1/13
3. 8/13
4. 5/8
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Answer: (a) 5/13. Favourable outcomes=5; total=13; probability=5/13.
A bag has 4 red and 12 blue balls. One ball is drawn at random. Probability of red?
1. 1/16
2. 4/12
3. 12/16
4. 4/16
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Answer: (d) 4/16. Favourable outcomes=4; total=16; probability=4/16.
A bag has 2 red and 12 blue balls. One ball is drawn at random. Probability of red?
1. 2/14
2. 1/14
3. 12/14
4. 2/12
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Answer: (a) 2/14. Favourable outcomes=2; total=14; probability=2/14.
Consider the following statements on Statistics, focusing on definition and scope: 1. Do not apply a memorized Statistics formula without checking its conditions. 2. Statistics questions reward a formula-first approach and accurate algebraic manipulation. Which of the statements given above is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
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Answer: (b) 2 only. Statement 2 is correct; Statement 1 is a common misconception.
A bag has 2 red and 6 blue balls. One ball is drawn at random. Probability of red?
1. 6/8
2. 2/6
3. 2/8
4. 1/8
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Answer: (c) 2/8. Favourable outcomes=2; total=8; probability=2/8.
A bag has 3 red and 14 blue balls. One ball is drawn at random. Probability of red?
1. 3/17
2. 1/17
3. 14/17
4. 3/14
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Answer: (a) 3/17. Favourable outcomes=3; total=17; probability=3/17.
A bag has 3 red and 10 blue balls. One ball is drawn at random. Probability of red?
1. 3/13
2. 3/10
3. 1/13
4. 10/13
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Answer: (a) 3/13. Favourable outcomes=3; total=13; probability=3/13.