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SSC CPO Standard Deviation & Variance

Study Material — 14 PYQs (2020–2020) · Concept Notes · Shortcuts

SSC CPO Standard Deviation & Variance is a frequently tested subtopic — 14 previous year questions from 2020–2020 papers are included below with concept notes, key rules and shortcut tricks.

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Previous Year Questions

SSC CPO Standard Deviation & Variance — Past Exam Questions

14 questions from actual SSC CPO papers · all shown free · click option to reveal solution

Exam Q 12020Previous Year Pattern

The variance of the dataset {2, 4, 6, 8, 10} is:

Exam Q 22020Previous Year Pattern

If the standard deviation of a dataset is 5, what is the variance?

Exam Q 32020Previous Year Pattern

The mean of five numbers is 12 and their variance is 4. If each number is increased by 3, what will be the new variance?

Exam Q 42020Previous Year Pattern

The standard deviation of the dataset {5, 5, 5, 5, 5} is:

Exam Q 52020Previous Year Pattern

If each value in a dataset is multiplied by 2, how does the variance change?

Exam Q 62020Previous Year Pattern

The standard deviation of five numbers is 4. If the sum of the numbers is 50, what is the sum of their squares?

Exam Q 72020Previous Year Pattern

The variance of a set of 8 observations is 64. The sum of the observations is 160. What is the sum of the squares of the deviations from the mean?

Exam Q 82020Previous Year Pattern

The variance of a dataset is 36. If each observation is multiplied by 5, what will be the new variance?

Exam Q 92020Previous Year Pattern

A dataset has mean 25 and standard deviation 5. If 10 is subtracted from each observation, what will be the new standard deviation?

Exam Q 102020Previous Year Pattern

Three datasets X, Y, and Z each have 20 observations with the same mean. The standard deviations are: X = 2, Y = 4, Z = 6. If all observations in each dataset are divided by 2, which dataset will have the smallest standard deviation after the transformation?

Exam Q 112020Previous Year Pattern

A sample of 50 observations has mean 60 and variance 100. If 10 new observations, each with value 70, are added to the sample, what is the new variance (rounded to nearest integer)?

Exam Q 122020Previous Year Pattern

The variance of a dataset is 144. If each observation is multiplied by 5 and then 3 is subtracted from each result, what is the new variance?

Exam Q 132020Previous Year Pattern

Two datasets have the same mean of 50. Dataset A has 8 observations with standard deviation 6, and Dataset B has 12 observations with standard deviation 4. If the datasets are combined, what is the variance of the combined dataset (rounded to nearest integer)?

Exam Q 142020Previous Year Pattern

Three groups have the following statistics: Group 1 (n₁=10, mean=30, variance=25), Group 2 (n₂=15, mean=35, variance=16), Group 3 (n₃=25, mean=32, variance=9). What is the overall variance when all three groups are combined?

Concept Notes

Standard Deviation & Variance— Rules & Concept

Core ConceptRead this first — the foundation of the topic

Standard Deviation and Variance are measures that tell us how spread out data is from the average. Think of them as measuring 'how different' the numbers are from each other. CORE CONCEPT:

Imagine you have marks of 5 students: 10, 20, 30, 40, 50. All are spread out widely. Now imagine: 28, 29, 30, 31, 32. These are clustered tightly around 30. Both have the same average (30), but the spread is different. Variance and Standard Deviation measure this spread. Variance (σ²) = Average of squared differences from the mean

Standard Deviation (σ) = Square root of Variance KEY RULES:

1. Standard Deviation is always non-negative (≥0) 2. If all numbers are identical, SD = 0

3. Larger SD means more scattered data; smaller SD means data is clustered 4. Standard Deviation is preferred over Variance because it's in the same units as original data

Formula BlockMemorise — at least one formula appears in every paper
Variance = Σ(x - mean)² / n
Standard Deviation = √Variance
For quick calculation: Variance = (Σx² / n) - (mean)²
Exam PatternsWhat examiners ask — read before attempting PYQs

SSC CGL typically asks: - Calculate SD or Variance from raw data - Compare spread of two datasets - Identify which dataset has more variation - Effect on SD when all values are multiplied or added by a constant SHORTCUT/TRICK: When values are multiplied by k: New SD = k × Original SD When values are added by c: New SD = Original SD (unchanged) This saves calculation time significantly.

Worked ExampleSolve this step-by-step before moving on
1
Step 1

Find mean = (2+4+6+8+10)/5 = 30/5 = 6

2
Step 2

Find differences from mean: (2-6)=-4, (4-6)=-2, (6-6)=0, (8-6)=2, (10-6)=4

3
Step 3

Square differences: 16, 4, 0, 4, 16

4
Step 4

Variance = (16+4+0+4+16)/5 = 40/5 = 8

5
Step 5

Standard Deviation = √8 = 2.83 (approximately)

Exam TrapsCommon mistakes students make — avoid these

Students forget to divide by n after summing squared differences. They calculate Σ(x-mean)² but stop there—this is NOT variance. You MUST divide by n.

Also, many confuse which measure to use; remember SD is more commonly reported in exams because it's interpretable.

Key Points to Remember

  • Variance measures average squared distance of data points from the mean; Standard Deviation is its square root.
  • Formula: Variance = Σ(x - mean)² / n; SD = √Variance
  • If all data values are identical, Standard Deviation = 0 (no spread).
  • When multiplying all values by k: New SD = k × Original SD; when adding c: SD remains unchanged.
  • Standard Deviation is preferred in exams because it's expressed in the same units as original data, making it more interpretable.
  • Larger SD indicates data is spread out; smaller SD indicates data is clustered around the mean.

Exam-Specific Tips

  • Standard Deviation formula: σ = √[Σ(x - μ)² / n], where μ is the arithmetic mean and n is number of observations.
  • Variance is the square of Standard Deviation: σ² = Variance.
  • Alternative variance formula for quick calculation: Variance = (Σx² / n) - (mean)².
  • Property: If each observation is multiplied by constant k, new SD = k × original SD (linear transformation rule).
  • Property: If constant c is added to each observation, SD remains unchanged (addition does not affect spread).
  • Standard Deviation of any dataset is always a non-negative value (σ ≥ 0).
  • Coefficient of Variation (CV) = (SD / Mean) × 100, used to compare variability across different datasets with different means.
  • For grouped data, class midpoints are used in calculations, and frequency weights are applied in variance formula.

60-Second Revision — Standard Deviation & Variance

  • Formula: Variance = Σ(x-mean)²/n; SD = √Variance. Use quick formula: Variance = (Σx²/n) - (mean)² to save time.
  • Multiply by k → SD multiplies by k; Add constant c → SD unchanged. Use this trick for transformation questions.
  • SD=0 only when all values are identical. Higher SD = more scattered; Lower SD = clustered around mean.
  • Trap: Don't forget to divide by n after summing squared differences—this is the most common error in calculations.
  • Remember: SD is preferred over Variance in SSC exams because it's in original units and easier to interpret.
  • For two datasets with same mean: compare SDs to determine which is more variable/consistent.
  • In exam, if asked 'which dataset varies more?'—calculate or compare SD values; higher SD = more variation.
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