Understanding Variance as a Measure of Spread

1) What does variance measure in a data set?

The middle value of the data

The most frequently occurring value

How spread out the data values are from the mean

The difference between the maximum and minimum values

Variance measures how far data points are from the mean. It shows the average squared distance of each value from the mean.

Explanation

Variance measures how far data points are from the mean. It shows the average squared distance of each value from the mean.

2) Which of the following best describes the first step in calculating variance?

Square all the data values

Subtract each value from the median

Add all the deviations together

Find the mean of the data set

The first step is to calculate the mean (average), because variance measures how values deviate from that mean.

Explanation

The first step is to calculate the mean (average), because variance measures how values deviate from that mean.

3) After finding the mean, what is the next step in calculating variance?

Square the mean

Find the deviation of each value from the mean

Add all values together

Divide by the number of data points

After finding the mean, subtract the mean from each data value to find how far each value deviates from it.

Explanation

After finding the mean, subtract the mean from each data value to find how far each value deviates from it.

4) Why do we square the deviations when calculating variance?

To make the calculations easier

To find the median

To eliminate negative values and emphasize larger deviations

To reduce the size of the numbers

Deviations can be negative, so squaring them makes all values positive and gives more weight to larger differences from the mean.

Explanation

Deviations can be negative, so squaring them makes all values positive and gives more weight to larger differences from the mean.

5) What is the final step in calculating variance?

Multiply all squared deviations together

Find the square root of the sum

Subtract the mean from each squared deviation

Find the average of the squared deviations

After squaring and summing the deviations, divide by the number of data points (n) or by (n - 1) for a sample to get the variance.

Explanation

After squaring and summing the deviations, divide by the number of data points (n) or by (n - 1) for a sample to get the variance.

6) If a data set has a variance of 0, what does this tell you about the data?

The data has a wide spread

The mean is 0

All values in the data set are the same

The data set has no mean

A variance of zero means every data point is equal to the mean — no variability at all.

Explanation

A variance of zero means every data point is equal to the mean — no variability at all.

7) Data Set A has a variance of 15, and Data Set B has a variance of 45. Which statement is true?

Data Set B is more spread out than Data Set A

Data Set A is more spread out than Data Set B

Both data sets have the same spread

Data Set A has a larger range than Data Set B

A larger variance value means greater spread. Since 45 is larger than 15, Data Set B is more variable.

Explanation

A larger variance value means greater spread. Since 45 is larger than 15, Data Set B is more variable.

8) Two classes took the same test. Class 1 has a variance of 25, and Class 2 has a variance of 9. What can you conclude?

Class 1's scores are more spread out than Class 2's scores

Class 2's scores are more spread out than Class 1's scores

Both classes performed equally

Class 2 has a higher mean score

Variance shows how much the scores differ from the mean. Class 1’s higher variance (25) means scores varied more.

Explanation

Variance shows how much the scores differ from the mean. Class 1’s higher variance (25) means scores varied more.

9) Which data set would you expect to have a larger variance?

Heights of students in the same grade level

Heights of people ranging from toddlers to adults

Test scores where everyone got 85%

Daily temperatures in a city during one week in summer

Including toddlers and adults gives a much wider range of heights, so the variance will be higher.

Explanation

Including toddlers and adults gives a much wider range of heights, so the variance will be higher.

10) If you add the same constant value to every number in a data set, what happens to the variance?

The variance increases

The variance decreases

The variance becomes zero

The variance stays the same

Adding or subtracting a constant shifts all values equally and doesn’t affect their spread. Variance remains the same.

Explanation

Adding or subtracting a constant shifts all values equally and doesn’t affect their spread. Variance remains the same.

11) What is the mean number of points scored per player for Team Alpha?

12

13

14

15

(12 + 14 + 13 + 15 + 11) / 5 = 65 / 5 = 13.

Explanation

(12 + 14 + 13 + 15 + 11) / 5 = 65 / 5 = 13.

12) What is the mean number of points scored per player for Team Beta?

13

10

12

15

(5 + 20 + 8 + 22 + 10) / 5 = 65 / 5 = 13.

Explanation

(5 + 20 + 8 + 22 + 10) / 5 = 65 / 5 = 13.

13) Which team's scores appear to be more spread out from the mean?

Team Alpha

Team Beta

Both teams have equal spread

Cannot be determined

Team Alpha’s scores are close to the mean (11–15), while Team Beta’s scores vary widely (5–22). Team Beta’s scores are more spread out.

Explanation

Team Alpha’s scores are close to the mean (11–15), while Team Beta’s scores vary widely (5–22). Team Beta’s scores are more spread out.

14) Based on the variance concept, which team had more consistent scoring among its players?

Team Beta

Both teams were equally consistent

Neither team was consistent

Team Alpha

Smaller variance means more consistent performance. Team Alpha’s scores are much closer together than Team Beta’s.

Explanation

Smaller variance means more consistent performance. Team Alpha’s scores are much closer together than Team Beta’s.

15) A data set has values: 10, 10, 10, 10, 10. What is the variance?

0

5

10

50

All values are identical, so there’s no difference from the mean. The variance is zero.

Explanation

All values are identical, so there’s no difference from the mean. The variance is zero.

16) Which of the following would NOT affect the variance of a data set?

Removing the highest value

Doubling every value

Adding 5 to every value

Replacing a middle value with an extreme value

Adding or subtracting a constant shifts the data but doesn’t change how spread out it is. Variance stays the same.

Explanation

Adding or subtracting a constant shifts the data but doesn’t change how spread out it is. Variance stays the same.

17) Two stores tracked customer wait times. Store A has a variance of 4 minutes squared, and Store B has a variance of 16 minutes squared. Which store has more predictable wait times?

Store B

Both stores are equally predictable

Cannot be determined without the mean

Store A

Smaller variance means less spread and more consistent wait times. Store A (variance = 4) is more predictable.

Explanation

Smaller variance means less spread and more consistent wait times. Store A (variance = 4) is more predictable.

18) In the formula for variance, what does 'n' typically represent?

The number of data values

The mean of the data

The sum of all deviations

The largest value in the data set

In the population variance formula, n represents the total number of data points in the set.

Explanation

In the population variance formula, n represents the total number of data points in the set.

19) A teacher wants to compare the consistency of test scores between two classes. Which measure should the teacher compare?

The means of both classes

The medians of both classes

The variances of both classes

The modes of both classes

Variance measures the degree of spread or consistency. Comparing variances reveals which class’s scores are steadier.

Explanation

Variance measures the degree of spread or consistency. Comparing variances reveals which class’s scores are steadier.

20) Data Set X: 2, 4, 6, 8, 10 (mean = 6) Data Set Y: 1, 5, 6, 7, 11 (mean = 6) Both data sets have the same mean. Which statement about their variances is most likely true?

Data Set X has a larger variance

Data Set Y has a larger variance

Both have the same variance

Data Set X has zero variance

For Data Set X, deviations from mean 6 are -4, -2, 0, +2, +4 (squares = 16, 4, 0, 4, 16 → total = 40). For Data Set Y, deviations are -5, -1, 0, +1, +5 (squares = 25, 1, 0, 1, 25 → total = 52). Because 52 is greater than 40, Data Set Y has the larger variance.

Explanation

For Data Set X, deviations from mean 6 are -4, -2, 0, +2, +4 (squares = 16, 4, 0, 4, 16 → total = 40). For Data Set Y, deviations are -5, -1, 0, +1, +5 (squares = 25, 1, 0, 1, 25 → total = 52). Because 52 is greater than 40, Data Set Y has the larger variance.