Sample Population Variance Quiz: Sample Vs. Population Variance

1) Which formula represents the population variance (using all values in the population)?

σ² = Σ(x − μ)² ÷ N

S² = Σ(x − x̄)² ÷ (n − 1)

σ = Σ(x − μ) ÷ N

S = Σ(x − x̄) ÷ n

Population variance uses the true population mean μ and divides by N (the population size): σ² = Σ(x − μ)² / N. This averages the squared distances from μ.

Explanation

Population variance uses the true population mean μ and divides by N (the population size): σ² = Σ(x − μ)² / N. This averages the squared distances from μ.

2) Which formula represents the sample variance (based on a subset of the population)?

σ² = Σ(x − μ)² ÷ N

S² = Σ(x − x̄)² ÷ (n − 1)

S² = Σ(x − x̄)² ÷ n

σ² = Σ(x − μ)² ÷ (N − 1)

Sample variance uses the sample mean x̄ and divides by n − 1 to correct bias: s² = Σ(x − x̄)² / (n − 1).

Explanation

Sample variance uses the sample mean x̄ and divides by n − 1 to correct bias: s² = Σ(x − x̄)² / (n − 1).

3) Using n in the denominator for a sample (instead of n − 1) will typically underestimate the population variance.

True

False

With a sample, x̄ is pulled toward the sample values. Dividing by n makes the average squared deviation too small on average. Using n − 1 inflates it slightly to remove that bias.

Explanation

With a sample, x̄ is pulled toward the sample values. Dividing by n makes the average squared deviation too small on average. Using n − 1 inflates it slightly to remove that bias.

4) For data {2, 4, 6, 8}, find the population variance.

4

5

6

6.67

Mean μ = (2+4+6+8)/4 = 20/4 = 5. Deviations: −3, −1, 1, 3. Squares: 9, 1, 1, 9. Sum = 20. Population variance = 20/4 = 5.00.

Explanation

Mean μ = (2+4+6+8)/4 = 20/4 = 5. Deviations: −3, −1, 1, 3. Squares: 9, 1, 1, 9. Sum = 20. Population variance = 20/4 = 5.00.

5) For the same data {2, 4, 6, 8}, find the sample variance.

4

5

6

6.67

Using n − 1: sum of squares = 20 (from Q4). Sample variance = 20/(4−1) = 20/3 = 6.67 (rounded to 2 decimals).

Explanation

Using n − 1: sum of squares = 20 (from Q4). Sample variance = 20/(4−1) = 20/3 = 6.67 (rounded to 2 decimals).

6) The term for the (n − 1) in the sample-variance denominator is __________.

We estimate the sample mean x̄ from the same data, which “uses up” 1 degree of freedom. That’s why we divide by (n − 1) instead of n.

Explanation

We estimate the sample mean x̄ from the same data, which “uses up” 1 degree of freedom. That’s why we divide by (n − 1) instead of n.

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7) Which situation calls for population variance rather than sample variance?

Measuring the heights of every student in a class of 30

Measuring the heights of 10 students chosen from a class of 30

Surveying 50 people from a city of 1,000,000

Testing 12 lightbulbs from a factory’s daily output

If you include every member of the group of interest, you have the population. Use the population formula (divide by N).

Explanation

If you include every member of the group of interest, you have the population. Use the population formula (divide by N).

8) Choose all cases where you should use sample variance (select all that apply).

A survey of 25 randomly chosen shoppers from a mall

All math scores from one 9th‑grade class this semester

Ten randomly chosen apples from a truck with 5,000 apples

Every temperature reading taken this week in your classroom (no missing times)

All measurements from one small box of apples

Samples are subsets: (A) a subset of shoppers, (C) a subset of apples, (E) a subset of a larger shipment. (B) and (D) include all values, so use population variance there.

Explanation

Samples are subsets: (A) a subset of shoppers, (C) a subset of apples, (E) a subset of a larger shipment. (B) and (D) include all values, so use population variance there.

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9) Which statistic is an unbiased estimator of the population variance σ²?

Σ(x − x̄)² ÷ n

Σ(x − μ)² ÷ N

Σ(x − x̄)² ÷ (n − 1)

Σ(x − μ)² ÷ (N − 1)

The sample variance s² = Σ(x − x̄)²/(n − 1) is unbiased for σ². Dividing by n would underestimate σ² on average.

Explanation

The sample variance s² = Σ(x − x̄)²/(n − 1) is unbiased for σ². Dividing by n would underestimate σ² on average.

10) For data {10, 12, 14}, the population variance is __________ (round to 2 decimals).

Mean μ = (10+12+14)/3 = 36/3 = 12. Deviations: −2, 0, 2. Squares: 4, 0, 4. Sum = 8. Population variance = 8/3 = 2.67.

Explanation

Mean μ = (10+12+14)/3 = 36/3 = 12. Deviations: −2, 0, 2. Squares: 4, 0, 4. Sum = 8. Population variance = 8/3 = 2.67.

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11) For data {10, 12, 14}, the sample variance is __________ (round to 2 decimals).

Sum of squared deviations is 8 (from Q10). Sample variance uses n − 1 = 2. So s² = 8/2 = 4.00.

Explanation

Sum of squared deviations is 8 (from Q10). Sample variance uses n − 1 = 2. So s² = 8/2 = 4.00.

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12) For the same data set, which value is usually larger: population variance or sample variance?

Population variance is larger

Sample variance is larger

They are always equal

There is no consistent pattern

Sample variance divides by n − 1, which is smaller than n, so the quotient is slightly larger on the same sum of squares.

Explanation

Sample variance divides by n − 1, which is smaller than n, so the quotient is slightly larger on the same sum of squares.

13) For data {1, 1, 1, 7}, find the population variance.

5.25

6.75

8

9

Mean μ = (1+1+1+7)/4 = 10/4 = 2.5. Deviations: −1.5, −1.5, −1.5, 4.5. Squares: 2.25, 2.25, 2.25, 20.25. Sum = 27. Population variance = 27/4 = 6.75.

Explanation

Mean μ = (1+1+1+7)/4 = 10/4 = 2.5. Deviations: −1.5, −1.5, −1.5, 4.5. Squares: 2.25, 2.25, 2.25, 20.25. Sum = 27. Population variance = 27/4 = 6.75.

14) For data {1, 1, 1, 7}, find the sample variance.

5.25

6.75

8

9

Use n − 1: sum of squares = 27 (from Q13). Sample variance = 27/(4−1) = 27/3 = 9.00.

Explanation

Use n − 1: sum of squares = 27 (from Q13). Sample variance = 27/(4−1) = 27/3 = 9.00.

15) If you have the entire group’s data (not a subset), you should divide by N when computing variance.

True

False

Population variance uses all values and divides by N. There is no need to correct for bias with n − 1 when you already have the full population.

Explanation

Population variance uses all values and divides by N. There is no need to correct for bias with n − 1 when you already have the full population.

16) Match the scenario to the correct variance type. Select all that should use population variance.

All 28 quiz scores from Ms. Lee’s class (every student)

12 randomly selected quiz scores from Ms. Lee’s class

Every bottle filled on Line A during one shift (no missing bottles)

20 randomly selected bottles from a day’s production

All readings from every sensor in one machine

A, C, and E include all items in the group under study, so use population variance. B and D are subsets, so they require sample variance.

Explanation

A, C, and E include all items in the group under study, so use population variance. B and D are subsets, so they require sample variance.

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17) A school has 240 ninth‑graders. You randomly survey 30 of them about study time and compute variance. Which denominator should you use?

Divide by 240

Divide by 30

Divide by 29

Divide by 239

You have a sample of n = 30 from a larger population. Use n − 1 = 29 for sample variance.

Explanation

You have a sample of n = 30 from a larger population. Use n − 1 = 29 for sample variance.

18) A teacher computes variance of test scores for all 25 students in a class using (n − 1). What is the issue?

They should divide by N = 25 because the class is the population

They should divide by n = 24

They should divide by n = 26

No issue; (n − 1) is always correct

The data include the entire class (the population). Population variance divides by N (here, 25), not n − 1.

Explanation

The data include the entire class (the population). Population variance divides by N (here, 25), not n − 1.

19) Using n instead of (n − 1) for sample variance makes the result too __________ on average.

Dividing by n gives a smaller average squared deviation than the true σ² when using a sample. Using n − 1 corrects this downward bias.

Explanation

Dividing by n gives a smaller average squared deviation than the true σ² when using a sample. Using n − 1 corrects this downward bias.

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20) Why do we divide by (n − 1) for sample variance?

To avoid negative deviations

Because we estimate the mean from the data and must correct the bias

To make calculations faster

To avoid fractions

The sample mean x̄ is computed from the same data, which reduces variability in deviations. Dividing by (n − 1) (degrees of freedom) corrects this bias, making s² an unbiased estimator of σ².

Explanation

The sample mean x̄ is computed from the same data, which reduces variability in deviations. Dividing by (n − 1) (degrees of freedom) corrects this bias, making s² an unbiased estimator of σ².

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Sample Population Variance Quiz: Sample vs. Population Variance