College Statistics Quiz: Measures Of Spread
20 questions · exam conditions
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Measures Of SpreadQuestion 1 of 20

If the range of a non-empty dataset is zero, which of the following statements must be true?

The mean of the dataset must be zero.
The standard deviation of the dataset must be zero.
The dataset must contain only the number zero.
The interquartile range must be equal to the mean.
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College Statistics Quiz

College Statistics Quiz: Measures Of Spread

Practice Measures Of Spread in College Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Measures Of Spread, giving you a quick way to practice the rules, question types, and explanations that matter most for College Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

If the range of a non-empty dataset is zero, which of the following statements must be true?

  1. The mean of the dataset must be zero.
  2. The standard deviation of the dataset must be zero. (correct answer)
  3. The dataset must contain only the number zero.
  4. The interquartile range must be equal to the mean.
Explanation: The range of a dataset is the difference between the maximum and minimum values. If the range is zero, it means the maximum value is equal to the minimum value. This is only possible if all values in the dataset are identical. When all data points are the same, there is no variability or spread. Therefore, the standard deviation, which measures spread, must be zero. The mean does not have to be zero; for example, the dataset {5, 5, 5} has a range of 0 but a mean of 5. The IQR would also be zero, but it would not necessarily equal the mean.

Question 2

A dataset has a first quartile (Q1) of 60 and a third quartile (Q3) of 90. According to the 1.5 x IQR rule for identifying potential outliers, which of the following data points would be classified as an outlier?

  1. 20
  2. 15
  3. 135
  4. 140 (correct answer)
Explanation: First, calculate the interquartile range (IQR): IQR = Q3 - Q1 = 90 - 60 = 30. Next, determine the fences for outliers. The lower fence is Q1 - 1.5 * IQR = 60 - 1.5 * 30 = 60 - 45 = 15. The upper fence is Q3 + 1.5 * IQR = 90 + 1.5 * 30 = 90 + 45 = 135. Any data point below the lower fence (< 15) or above the upper fence (> 135) is considered an outlier. The value 140 is greater than 135, so it is an outlier. The values 15 and 135 are exactly on the fences and are typically not classified as outliers. The value 20 is between the fences (15 < 20 < 135), so it is not an outlier.

Question 3

A dataset consists of the following 5 numbers: {10, 12, 15, 18, 20}. A sixth number, 70, is added to the dataset. Which of the following measures of spread will experience the largest percentage increase due to the addition of this new number?

  1. Range (correct answer)
  2. Interquartile Range
  3. Standard Deviation
  4. Variance
Explanation: The range is the most sensitive measure of spread to extreme outliers. Let's analyze the change: Original dataset: {10, 12, 15, 18, 20}. Range = 20 - 10 = 10. Q1=11, Q3=19, IQR=8. New dataset: {10, 12, 15, 18, 20, 70}. New Range = 70 - 10 = 60. The percentage increase is ((60-10)/10)*100% = 500%. New Q1=12, New Q3=20, New IQR=8. The percentage increase is 0%. While the standard deviation and variance will also increase, the range is directly defined by the extreme values and thus experiences the most dramatic percentage change when a significant outlier is introduced.

Question 4

The heights of a sample of plants are measured in inches. The variance of these heights is found to be 9 square inches. If the heights are converted to feet by dividing each value by 12, what will be the new standard deviation of the heights, in feet?

  1. 0.0625
  2. 0.25 (correct answer)
  3. 0.75
  4. 3.0
Explanation: This is a multi-step problem. First, find the original standard deviation from the variance. Standard deviation is the square root of variance, so the original standard deviation is 9=3\sqrt{9} = 3 inches. Second, apply the unit conversion. Dividing all data points by a constant (12) also divides their standard deviation by that constant. The new standard deviation in feet is 3 inches12=0.25\frac{3 \text{ inches}}{12} = 0.25 feet. Distractor A is the new variance (9/122=9/144=0.06259/12^2 = 9/144 = 0.0625), not the standard deviation. Distractor C results from incorrectly dividing the variance by 12 (9/12 = 0.75). Distractor D is the original standard deviation in inches.

Question 5

A dataset of exam scores has a standard deviation of 12 points. The professor decides to curve the exam by multiplying every student's score by 1.1 and then adding 5 points. What will be the new standard deviation of the scores, rounded to the nearest tenth?

  1. 12.0 points
  2. 13.2 points (correct answer)
  3. 18.2 points
  4. 17.0 points
Explanation: Measures of spread, such as the standard deviation, are affected by multiplication or division, but not by addition or subtraction. Adding 5 points to every score shifts the entire distribution but does not change its spread. Multiplying each score by 1.1 will multiply the standard deviation by 1.1. The new standard deviation will be 12×1.1=13.212 \times 1.1 = 13.2.

Question 6

A class is split into two sections for an exam. Section A has 10 students with a mean score of 80 and a score variance of 25. Section B has 20 students with a mean score of 80 and a score variance of 49. If the scores from the two sections are combined into a single dataset of 30 students, what is the variance of the combined scores?

  1. 37.0
  2. 41.0 (correct answer)
  3. 44.3
  4. Cannot be determined from the information given.
Explanation: When two groups with the same mean are combined, the variance of the combined group is the weighted average of the individual variances, where the weights are the group sizes. Let nA,nBn_A, n_B be the sizes and σA2,σB2\sigma_A^2, \sigma_B^2 be the variances. The combined variance σC2\sigma_C^2 is given by σC2=nAσA2+nBσB2nA+nB\sigma_C^2 = \frac{n_A \sigma_A^2 + n_B \sigma_B^2}{n_A + n_B}. Plugging in the values: σC2=(10)(25)+(20)(49)10+20=250+98030=123030=41.0\sigma_C^2 = \frac{(10)(25) + (20)(49)}{10 + 20} = \frac{250 + 980}{30} = \frac{1230}{30} = 41.0.

Question 7

A dataset consists of four positive integers: {1, 2, 3, x}. The sample variance of this dataset is 503\frac{50}{3}. What is the value of x?

  1. 6
  2. 8
  3. 10 (correct answer)
  4. 12
Explanation: Let the mean be yˉ=1+2+3+x4=6+x4\bar{y} = \frac{1+2+3+x}{4} = \frac{6+x}{4}. The formula for sample variance is s2=1n−1∑(yi−yˉ)2s^2 = \frac{1}{n-1} \sum (y_i - \bar{y})^2. So, 503=13[(1−6+x4)2+(2−6+x4)2+(3−6+x4)2+(x−6+x4)2]\frac{50}{3} = \frac{1}{3} \left[ (1 - \frac{6+x}{4})^2 + (2 - \frac{6+x}{4})^2 + (3 - \frac{6+x}{4})^2 + (x - \frac{6+x}{4})^2 \right]. Multiplying both sides by 3 gives 50=∑(yi−yˉ)250 = \sum (y_i - \bar{y})^2. This algebraic path is complex. An alternative is to test the integer options. If x=10, the dataset is {1, 2, 3, 10}. The mean is 1+2+3+104=4\frac{1+2+3+10}{4} = 4. The deviations from the mean are (1-4)=-3, (2-4)=-2, (3-4)=-1, and (10-4)=6. The squared deviations are 9, 4, 1, and 36. The sum of squared deviations is 9+4+1+36=509+4+1+36=50. The sample variance is s2=504−1=503s^2 = \frac{50}{4-1} = \frac{50}{3}, which matches the given information.

Question 8

A researcher calculates the variance for a sample of 10 items and gets a value of s2=20s^2 = 20. She later realizes this sample represents the entire population of interest. What is the true population variance, σ2\sigma^2?

  1. 18.0 (correct answer)
  2. 20.0
  3. 22.2
  4. Cannot be determined without the raw data.
Explanation: The formula for sample variance is s2=∑(xi−xˉ)2n−1s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}, while the formula for population variance is σ2=∑(xi−xˉ)2n\sigma^2 = \frac{\sum(x_i - \bar{x})^2}{n}. The key is the sum of squared deviations, ∑(xi−xˉ)2\sum(x_i - \bar{x})^2. From the sample variance, we can find this sum: ∑(xi−xˉ)2=s2(n−1)=20(10−1)=20(9)=180\sum(x_i - \bar{x})^2 = s^2(n-1) = 20(10-1) = 20(9) = 180. Now, we can calculate the population variance using this sum and dividing by nn: σ2=18010=18.0\sigma^2 = \frac{180}{10} = 18.0.

Question 9

The mean daily high temperature in a city during summer is 85°F with a standard deviation of 5°F. The shape of the temperature distribution is unknown. According to Chebyshev's inequality, at least what percentage of days must have a temperature between 70°F and 100°F?

  1. 75%
  2. 89% (correct answer)
  3. 95%
  4. The percentage cannot be determined.
Explanation: Chebyshev's inequality can be used for any distribution. First, determine how many standard deviations from the mean the interval extends. The interval is from 85−1585 - 15 to 85+1585 + 15. The distance from the mean is 15°F. With a standard deviation of 5°F, this distance represents k=155=3k = \frac{15}{5} = 3 standard deviations. Chebyshev's inequality states that the proportion of data within kk standard deviations of the mean is at least 1−1k21 - \frac{1}{k^2}. For k=3k=3, this is 1−132=1−19=89≈0.8891 - \frac{1}{3^2} = 1 - \frac{1}{9} = \frac{8}{9} \approx 0.889, or 88.9%. The closest answer is 89%.

Question 10

Consider three datasets, each with 100 data points and a range from 0 to 10. Dataset X: A uniform distribution of points spread evenly from 0 to 10. Dataset Y: A U-shaped distribution, with 50 points at the value 0 and 50 points at the value 10. Dataset Z: A single-value distribution, with all 100 points at the value 5. Which of the following correctly orders the datasets by their standard deviation, from smallest to largest?

  1. X < Y < Z
  2. Y < X < Z
  3. Z < X < Y (correct answer)
  4. Z < Y < X
Explanation: Standard deviation measures spread from the mean.
  • Dataset Z has all points identical (at the mean of 5), so there is no spread. Its standard deviation is exactly 0. This is the smallest.
  • Dataset Y has all its points at the maximum possible distance from the mean (which is 5). This represents the maximum possible spread, so it will have the largest standard deviation.
  • Dataset X has its points spread out evenly. Its spread is greater than Z's but less concentrated at the extremes than Y's. Therefore, the correct order of standard deviations from smallest to largest is Z, then X, then Y.

Question 11

Two datasets, Set A and Set B, are compared. Every value in Set B is exactly 10 units greater than the corresponding value in Set A. Which of the following statements correctly describes the relationship between their measures of spread?

  1. The standard deviation of Set B is 10 units greater than that of Set A.
  2. The variance of Set B is 100 units greater than that of Set A.
  3. The range and interquartile range of Set B are equal to those of Set A. (correct answer)
  4. The range of Set B is 10 units greater, but the variance is the same.
Explanation: Adding a constant value (10) to every data point is a linear transformation that shifts the entire distribution. Measures of center (like the mean and median) will increase by 10. However, measures of spread (range, IQR, variance, standard deviation) are unaffected by addition or subtraction because the distances between the data points remain unchanged. Therefore, the range, IQR, variance, and standard deviation of Set B will be identical to those of Set A.

Question 12

For a given dataset, the variance is 64. What is the correct interpretation of the standard deviation?

  1. The standard deviation is 8, which represents the squared spread of the data.
  2. The standard deviation is 64, the same as the variance.
  3. The standard deviation is 4096, which is the variance squared.
  4. The standard deviation is 8, representing a typical deviation from the mean. (correct answer)
Explanation: When you encounter questions about variance and standard deviation, remember that these two measures of spread are mathematically related but have different units and interpretations. Since variance is 64, the standard deviation is 64=8\sqrt{64} = 8. Standard deviation is always the square root of variance. More importantly, standard deviation represents how much individual data points typically deviate from the mean, measured in the same units as the original data. This makes it much more interpretable than variance. Let's examine why each option is correct or incorrect: Option A is wrong because it correctly calculates the standard deviation as 8, but incorrectly describes it as "squared spread." Variance (not standard deviation) involves squared units. Option B is incorrect because it claims standard deviation equals variance. These are never the same value unless variance equals 0 or 1, which are special cases. Option C is wrong because it suggests standard deviation is 642=409664^2 = 4096. This reverses the relationship—standard deviation is the square root of variance, not variance squared. Option D is correct because it properly calculates standard deviation as 8 and accurately describes its meaning: a typical deviation from the mean. Study tip: Remember that standard deviation = variance\sqrt{\text{variance}}, and standard deviation is always more interpretable because it's in the original units of measurement. When you see variance, immediately think about taking its square root to get the standard deviation for easier interpretation of spread.

Question 13

A dataset of exam scores has a standard deviation of 12 points. The professor decides to curve the exam by multiplying every student's score by 1.1 and then adding 5 points. What will be the new standard deviation of the scores, rounded to the nearest tenth?

  1. 12.0 points
  2. 13.2 points (correct answer)
  3. 18.2 points
  4. 17.0 points
Explanation: Measures of spread, such as the standard deviation, are affected by multiplication or division, but not by addition or subtraction. Adding 5 points to every score shifts the entire distribution but does not change its spread. Multiplying each score by 1.1 will multiply the standard deviation by 1.1. The new standard deviation will be 12×1.1=13.212 \times 1.1 = 13.2.

Question 14

A class is split into two sections for an exam. Section A has 10 students with a mean score of 80 and a score variance of 25. Section B has 20 students with a mean score of 80 and a score variance of 49. If the scores from the two sections are combined into a single dataset of 30 students, what is the variance of the combined scores?

  1. 37.0
  2. 41.0 (correct answer)
  3. 44.3
  4. Cannot be determined from the information given.
Explanation: When two groups with the same mean are combined, the variance of the combined group is the weighted average of the individual variances, where the weights are the group sizes. Let nA,nBn_A, n_B be the sizes and σA2,σB2\sigma_A^2, \sigma_B^2 be the variances. The combined variance σC2\sigma_C^2 is given by σC2=nAσA2+nBσB2nA+nB\sigma_C^2 = \frac{n_A \sigma_A^2 + n_B \sigma_B^2}{n_A + n_B}. Plugging in the values: σC2=(10)(25)+(20)(49)10+20=250+98030=123030=41.0\sigma_C^2 = \frac{(10)(25) + (20)(49)}{10 + 20} = \frac{250 + 980}{30} = \frac{1230}{30} = 41.0.

Question 15

A dataset has a first quartile (Q1) of 60 and a third quartile (Q3) of 90. According to the 1.5 x IQR rule for identifying potential outliers, which of the following data points would be classified as an outlier?

  1. 20
  2. 15
  3. 135
  4. 140 (correct answer)
Explanation: First, calculate the interquartile range (IQR): IQR = Q3 - Q1 = 90 - 60 = 30. Next, determine the fences for outliers. The lower fence is Q1 - 1.5 * IQR = 60 - 1.5 * 30 = 60 - 45 = 15. The upper fence is Q3 + 1.5 * IQR = 90 + 1.5 * 30 = 90 + 45 = 135. Any data point below the lower fence (< 15) or above the upper fence (> 135) is considered an outlier. The value 140 is greater than 135, so it is an outlier. The values 15 and 135 are exactly on the fences and are typically not classified as outliers. The value 20 is between the fences (15 < 20 < 135), so it is not an outlier.

Question 16

The mean daily high temperature in a city during summer is 85°F with a standard deviation of 5°F. The shape of the temperature distribution is unknown. According to Chebyshev's inequality, at least what percentage of days must have a temperature between 70°F and 100°F?

  1. 75%
  2. 89% (correct answer)
  3. 95%
  4. The percentage cannot be determined.
Explanation: Chebyshev's inequality can be used for any distribution. First, determine how many standard deviations from the mean the interval extends. The interval is from 85−1585 - 15 to 85+1585 + 15. The distance from the mean is 15°F. With a standard deviation of 5°F, this distance represents k=155=3k = \frac{15}{5} = 3 standard deviations. Chebyshev's inequality states that the proportion of data within kk standard deviations of the mean is at least 1−1k21 - \frac{1}{k^2}. For k=3k=3, this is 1−132=1−19=89≈0.8891 - \frac{1}{3^2} = 1 - \frac{1}{9} = \frac{8}{9} \approx 0.889, or 88.9%. The closest answer is 89%.

Question 17

Consider three datasets, each with 100 data points and a range from 0 to 10. Dataset X: A uniform distribution of points spread evenly from 0 to 10. Dataset Y: A U-shaped distribution, with 50 points at the value 0 and 50 points at the value 10. Dataset Z: A single-value distribution, with all 100 points at the value 5. Which of the following correctly orders the datasets by their standard deviation, from smallest to largest?

  1. X < Y < Z
  2. Y < X < Z
  3. Z < X < Y (correct answer)
  4. Z < Y < X
Explanation: Standard deviation measures spread from the mean.
  • Dataset Z has all points identical (at the mean of 5), so there is no spread. Its standard deviation is exactly 0. This is the smallest.
  • Dataset Y has all its points at the maximum possible distance from the mean (which is 5). This represents the maximum possible spread, so it will have the largest standard deviation.
  • Dataset X has its points spread out evenly. Its spread is greater than Z's but less concentrated at the extremes than Y's. Therefore, the correct order of standard deviations from smallest to largest is Z, then X, then Y.

Question 18

Two datasets, Set A and Set B, are compared. Every value in Set B is exactly 10 units greater than the corresponding value in Set A. Which of the following statements correctly describes the relationship between their measures of spread?

  1. The standard deviation of Set B is 10 units greater than that of Set A.
  2. The variance of Set B is 100 units greater than that of Set A.
  3. The range and interquartile range of Set B are equal to those of Set A. (correct answer)
  4. The range of Set B is 10 units greater, but the variance is the same.
Explanation: Adding a constant value (10) to every data point is a linear transformation that shifts the entire distribution. Measures of center (like the mean and median) will increase by 10. However, measures of spread (range, IQR, variance, standard deviation) are unaffected by addition or subtraction because the distances between the data points remain unchanged. Therefore, the range, IQR, variance, and standard deviation of Set B will be identical to those of Set A.

Question 19

The heights of a sample of plants are measured in inches. The variance of these heights is found to be 9 square inches. If the heights are converted to feet by dividing each value by 12, what will be the new standard deviation of the heights, in feet?

  1. 0.0625
  2. 0.25 (correct answer)
  3. 0.75
  4. 3.0
Explanation: This is a multi-step problem. First, find the original standard deviation from the variance. Standard deviation is the square root of variance, so the original standard deviation is 9=3\sqrt{9} = 3 inches. Second, apply the unit conversion. Dividing all data points by a constant (12) also divides their standard deviation by that constant. The new standard deviation in feet is 3 inches12=0.25\frac{3 \text{ inches}}{12} = 0.25 feet. Distractor A is the new variance (9/122=9/144=0.06259/12^2 = 9/144 = 0.0625), not the standard deviation. Distractor C results from incorrectly dividing the variance by 12 (9/12 = 0.75). Distractor D is the original standard deviation in inches.

Question 20

A dataset consists of the following 5 numbers: {10, 12, 15, 18, 20}. A sixth number, 70, is added to the dataset. Which of the following measures of spread will experience the largest percentage increase due to the addition of this new number?

  1. Range (correct answer)
  2. Interquartile Range
  3. Standard Deviation
  4. Variance
Explanation: The range is the most sensitive measure of spread to extreme outliers. Let's analyze the change: Original dataset: {10, 12, 15, 18, 20}. Range = 20 - 10 = 10. Q1=11, Q3=19, IQR=8. New dataset: {10, 12, 15, 18, 20, 70}. New Range = 70 - 10 = 60. The percentage increase is ((60-10)/10)*100% = 500%. New Q1=12, New Q3=20, New IQR=8. The percentage increase is 0%. While the standard deviation and variance will also increase, the range is directly defined by the extreme values and thus experiences the most dramatic percentage change when a significant outlier is introduced.