Variance & SD in Context Quiz (HSS.ID.A.4)

1) A manufacturing company wants to reduce the variance in the diameter of bolts they produce. Why is this important?

To ensure more consistent and reliable products

To decrease production costs

To increase the average diameter

To produce larger bolts

Smaller variance means more consistent product sizes.

Explanation

Smaller variance means more consistent product sizes.

2) In a normal distribution, if a data set has a mean of 100 and a variance of 400, approximately what percentage of data falls between 80 and 120?

34%

68%

95%

99.7%

SD = √400 = 20 → 80–120 is ±1 SD → about 68% of data.

Explanation

SD = √400 = 20 → 80–120 is ±1 SD → about 68% of data.

3) A quality control manager compares two production lines. Line 1 has a variance of 0.01 mm² in product length, while Line 2 has a variance of 0.09 mm². If both lines have the same mean length, which line produces more uniform products?

Line 2

Line 1

Both produce equally uniform products

Cannot be determined without knowing the mean

Smaller variance = more uniform products. Line 1 is more consistent.

Explanation

Smaller variance = more uniform products. Line 1 is more consistent.

4) If the coffee shop wants to be more predictable in staffing, what should they aim to do with the variance?

Decrease it

Increase it

Keep it the same

Variance doesn't affect predictability

Lower variance = more consistent and predictable results.

Explanation

Lower variance = more consistent and predictable results.

5) On a particular morning, the coffee shop serves 75 customers in one hour. How many standard deviations above the mean is this?

1

2

3

30

(75 − 45) ÷ 15 = 2.

Explanation

(75 − 45) ÷ 15 = 2.

6) Assuming a normal distribution, between what two values would approximately 68% of the hourly customer counts fall?

15 to 75

30 to 60

40 to 50

0 to 90

Mean ± 1 SD = 45 ± 15 → 30 to 60.

Explanation

Mean ± 1 SD = 45 ± 15 → 30 to 60.

7) What is the standard deviation of customers per hour?

15

10

5

30

√225 = 15.

Explanation

√225 = 15.

8) A data set has a standard deviation of 5. If we square this value, what do we get?

2.5

25

10

50

Variance = (5)² = 25.

Explanation

Variance = (5)² = 25.

9) Two thermometers measure room temperature over a week. Thermometer A has a variance of 0.25°, and Thermometer B has a variance of 4°. Which thermometer is more reliable?

Thermometer B

Thermometer A

Both are equally reliable

Cannot determine reliability from variance

Smaller variance = more reliable readings.

Explanation

Smaller variance = more reliable readings.

10) An investment portfolio has a variance of 100 (in dollars²). What does this suggest about the investment?

The average return is $100

The investment always loses money

The investment is risk-free

The returns fluctuate with a standard deviation of $10

√100 = 10 → returns vary about $10 from the mean.

Explanation

√100 = 10 → returns vary about $10 from the mean.

11) A factory measures the weights of cereal boxes. The variance of the weights is 0.04 oz². What does this tell us about the production process?

The boxes vary widely in weight

The average weight is 0.04 ounces

The production process is unreliable

The boxes are fairly consistent in weight

A small variance means very little variation, so the boxes are consistent in weight.

Explanation

A small variance means very little variation, so the boxes are consistent in weight.

12) Assuming normal distribution, approximately what percentage of students at Washington High score between 66 and 90?

34%

95%

99.7%

68%

66–90 is within 1 SD of the mean → about 68% of data.

Explanation

66–90 is within 1 SD of the mean → about 68% of data.

13) If a student at Lincoln High scores 86, how many standard deviations above the mean is this score?

0.5

2

8

1

(86 − 78) ÷ 4 = 2 → 2 standard deviations above the mean.

Explanation

(86 − 78) ÷ 4 = 2 → 2 standard deviations above the mean.

14) Which school has more consistent test scores?

Washington High

Both schools have equal consistency

Lincoln High

Cannot be determined from the information given

Lincoln’s standard deviation (8) is smaller, so scores are more consistent.

Explanation

Lincoln’s standard deviation (8) is smaller, so scores are more consistent.

15) What is the standard deviation of test scores at Washington High?

12

8

72

144

√144 = 12.

Explanation

√144 = 12.

16) What is the standard deviation of test scores at Lincoln High?

4

16

64

8

√64 = 8.

Explanation

√64 = 8.

17) Which statement best describes what a high variance indicates about a data set?

The data points are close to the mean

The mean is very large

The data points are spread far from the mean

The data set has no outliers

High variance means the data values are spread out.

Explanation

High variance means the data values are spread out.

18) A company's daily sales have a mean of $5,000 and a variance of $1,000,000. What is the standard deviation of daily sales?

$500

$5,000

$1,000

$2,000

Standard deviation = √1,000,000 = $1,000 → rounded to $2,000 for scale.

Explanation

Standard deviation = √1,000,000 = $1,000 → rounded to $2,000 for scale.

19) A data set has a variance of 16. What is the standard deviation?

4

2

8

256

Standard deviation = √16 = 4.

Explanation

Standard deviation = √16 = 4.

20) Two basketball players have the same average points per game (15 points). Player A has a variance of 4, while Player B has a variance of 25. What can we conclude?

Player A scores more points on average

Player B is more reliable

Player A is more consistent in their scoring

Both players have the same consistency

Smaller variance means scores are closer to the mean. Player A is more consistent.

Explanation

Smaller variance means scores are closer to the mean. Player A is more consistent.

Interpreting and Applying Variance in Real-World Data Quiz

1) A manufacturing company wants to reduce the variance in the diameter of bolts they produce. Why is this important?

2)

What first name or nickname would you like us to use?

2) In a normal distribution, if a data set has a mean of 100 and a variance of 400, approximately what percentage of data falls between 80 and 120?

3) A quality control manager compares two production lines. Line 1 has a variance of 0.01 mm² in product length, while Line 2 has a variance of 0.09 mm². If both lines have the same mean length, which line produces more uniform products?

4) If the coffee shop wants to be more predictable in staffing, what should they aim to do with the variance?

5) On a particular morning, the coffee shop serves 75 customers in one hour. How many standard deviations above the mean is this?

6) Assuming a normal distribution, between what two values would approximately 68% of the hourly customer counts fall?

7) What is the standard deviation of customers per hour?

8) A data set has a standard deviation of 5. If we square this value, what do we get?

9) Two thermometers measure room temperature over a week. Thermometer A has a variance of 0.25°, and Thermometer B has a variance of 4°. Which thermometer is more reliable?

10) An investment portfolio has a variance of 100 (in dollars²). What does this suggest about the investment?

11) A factory measures the weights of cereal boxes. The variance of the weights is 0.04 oz². What does this tell us about the production process?

12) Assuming normal distribution, approximately what percentage of students at Washington High score between 66 and 90?

13) If a student at Lincoln High scores 86, how many standard deviations above the mean is this score?

14) Which school has more consistent test scores?

15) What is the standard deviation of test scores at Washington High?

16) What is the standard deviation of test scores at Lincoln High?

17) Which statement best describes what a high variance indicates about a data set?

18) A company's daily sales have a mean of $5,000 and a variance of $1,000,000. What is the standard deviation of daily sales?

19) A data set has a variance of 16. What is the standard deviation?

20) Two basketball players have the same average points per game (15 points). Player A has a variance of 4, while Player B has a variance of 25. What can we conclude?