College Statistics Quiz: Shape And Outliers
20 questions · exam conditions
0:00
Shape And OutliersQuestion 1 of 20

A real estate report for a large metropolitan area states that the mean home price is $780,000 and the median home price is $595,000. Which of the following statements is the most reasonable conclusion based on this information?

The distribution of home prices is likely skewed to the right.
The distribution of home prices is likely skewed to the left.
The distribution of home prices is likely symmetric and mound-shaped.
The data must contain an error, as the mean cannot be that much higher than the median.
← Back to quizzes

College Statistics Quiz

College Statistics Quiz: Shape And Outliers

Practice Shape And Outliers 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 Shape And Outliers, 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

A real estate report for a large metropolitan area states that the mean home price is $780,000 and the median home price is $595,000. Which of the following statements is the most reasonable conclusion based on this information?

  1. The distribution of home prices is likely skewed to the right. (correct answer)
  2. The distribution of home prices is likely skewed to the left.
  3. The distribution of home prices is likely symmetric and mound-shaped.
  4. The data must contain an error, as the mean cannot be that much higher than the median.
Explanation: In a skewed distribution, the mean is pulled in the direction of the long tail. When the mean is substantially greater than the median (mean > median), it indicates that there are high-value outliers or a long tail on the right side of the distribution. This pattern is characteristic of a right-skewed distribution, which is common for variables like income and housing prices.

Question 2

A financial analyst is studying the annual salaries of employees at a technology firm. The data is found to be roughly symmetric with a median salary of $95,000. The company then hires a new CEO with an annual salary of $15,000,000. How will the inclusion of the new CEO's salary most likely affect the mean and median salary, and the shape of the distribution?

  1. The mean and median will both increase substantially, and the distribution will remain approximately symmetric.
  2. The mean will increase substantially, the median will remain relatively stable, and the distribution will become right-skewed. (correct answer)
  3. The mean will increase substantially, the median will remain relatively stable, and the distribution will become left-skewed.
  4. The median will increase substantially, the mean will remain relatively stable, and the distribution will become right-skewed.
Explanation: The CEO's salary is an extreme high outlier. The mean is sensitive to outliers and will be pulled upwards significantly by this large value. The median, being a resistant measure of center, will change very little, as it is based on the position of the middle value, not its magnitude. The addition of a high outlier creates a long tail on the right side of the distribution, making it right-skewed.

Question 3

Which of the following conditions, if true for a given dataset, is sufficient to conclude that the distribution is perfectly symmetric?

  1. The mean of the dataset is equal to the median.
  2. The distance from the first quartile to the median is equal to the distance from the median to the third quartile.
  3. For every point above the median, there is a corresponding point an equal distance below the median. (correct answer)
  4. The number of data points below the mean is equal to the number of data points above the mean.
Explanation: This is the definition of symmetry around the median. If every point has a 'mirror image' on the other side of the median, the distribution is perfectly symmetric. The other conditions are necessary for many symmetric distributions but are not sufficient. For example, a distribution can have its mean equal to its median but still be slightly skewed if the tails are not balanced (A). The quartiles can be equidistant from the median, but the whiskers could have different lengths (B). The number of points above and below the mean can be equal in a non-symmetric distribution (D).

Question 4

A researcher collects data on the response times of a cognitive task. Summary statistics for the dataset (n=50) are: Min=5.2, Q1=10.5, Median=12.1, Q3=14.2, Max=25.8. Using the 1.5 × IQR rule for identifying outliers, which of the following statements is true?

  1. There are no outliers in this dataset.
  2. The maximum value of 25.8 is an outlier, but the minimum value of 5.2 is not. (correct answer)
  3. The minimum value of 5.2 is an outlier, but the maximum value of 25.8 is not.
  4. Both the minimum value of 5.2 and the maximum value of 25.8 are outliers.
Explanation: First, calculate the Interquartile Range (IQR): IQR = Q3 - Q1 = 14.2 - 10.5 = 3.7. Next, calculate the lower and upper fences for outliers. Lower fence = Q1 - 1.5 × IQR = 10.5 - 1.5 × 3.7 = 10.5 - 5.55 = 4.95. Upper fence = Q3 + 1.5 × IQR = 14.2 + 1.5 × 3.7 = 14.2 + 5.55 = 19.75. The minimum value, 5.2, is greater than the lower fence of 4.95, so it is not an outlier. The maximum value, 25.8, is greater than the upper fence of 19.75, so it is an outlier.

Question 5

A strongly right-skewed distribution of data on household incomes is transformed by taking the natural logarithm of each income value. Which of the following is the most likely effect of this transformation on the shape of the distribution?

  1. The distribution will become more strongly right-skewed.
  2. The distribution will become left-skewed.
  3. The distribution will become more symmetric. (correct answer)
  4. The transformation will not affect the skewness of the distribution.
Explanation: Logarithmic transformations compress the scale of the data. For a right-skewed distribution, the high values in the right tail are very spread out. The log transformation has a larger effect on these large values than on the smaller values, effectively 'pulling in' the long right tail. This process tends to make the distribution more symmetric.

Question 6

A dataset contains an extreme, high outlier. If this outlier is identified and removed from the dataset, which of the following measures is expected to change the most?

  1. The median
  2. The interquartile range (IQR)
  3. The standard deviation (correct answer)
  4. The mode
Explanation: The standard deviation is calculated using the mean and measures the average distance of data points from the mean. Since the mean is heavily influenced by outliers, and the standard deviation calculation involves the squared differences from the mean, it is the most sensitive to extreme values. The median and IQR are resistant to outliers, and the mode is generally unaffected unless the outlier was part of a modal group.

Question 7

For a particular dataset, a data point is found to have a z-score of 2.9. The distribution is known to be unimodal and only slightly skewed. Which of the following is a correct conclusion about this data point?

  1. The point is definitely an outlier by the 1.5 × IQR rule.
  2. The point is unusual, but may or may not be classified as an outlier by the 1.5 × IQR rule. (correct answer)
  3. The point is definitely not an outlier by the 1.5 × IQR rule.
  4. The point cannot be an outlier because a z-score must be greater than 3 to be considered an outlier.
Explanation: A z-score of 2.9 indicates the data point is 2.9 standard deviations above the mean, which is an unusual value in any distribution. However, different rules for identifying outliers can lead to different conclusions. The common 'z-score > 3' rule is just one guideline. The 1.5 × IQR rule identifies outliers based on quartiles. For a perfectly normal distribution, the 1.5 × IQR fences correspond to z-scores of approximately ±2.7. Since this distribution is not specified as normal, we cannot be certain that a z-score of 2.9 falls outside the fences. Thus, the point is unusual, but its classification as an outlier by the IQR method is not guaranteed.

Question 8

Consider the following dataset of observations: 33, 35, 38, 38, 40, 41, 44, 45, 68. Which statement about this dataset is correct?

  1. The value 68 is an outlier, and its presence makes the distribution right-skewed. (correct answer)
  2. The value 33 is an outlier, and its presence makes the distribution left-skewed.
  3. There are no outliers, and the distribution is approximately symmetric.
  4. Both 33 and 68 are outliers according to the 1.5 × IQR rule.
Explanation: First, find the quartiles. With n=9, the median is the 5th value, which is 40. Q1 is the median of the lower half (33, 35, 38, 38), so Q1 = (35+38)/2 = 36.5. Q3 is the median of the upper half (41, 44, 45, 68), so Q3 = (44+45)/2 = 44.5. The IQR = 44.5 - 36.5 = 8. The upper fence is Q3 + 1.5IQR = 44.5 + 1.58 = 44.5 + 12 = 56.5. Since 68 is greater than 56.5, it is an outlier. This high outlier pulls the mean to the right, making the distribution right-skewed.

Question 9

A large dataset is approximately symmetric. The maximum value in the dataset is then erroneously replaced by a number ten times larger than the original maximum. How will this change affect the shape of the distribution and the interquartile range (IQR)?

  1. The distribution will become right-skewed, and the IQR will increase significantly.
  2. The distribution will become right-skewed, and the IQR will likely remain unchanged. (correct answer)
  3. The distribution will remain symmetric, but the range and standard deviation will increase.
  4. The distribution will become left-skewed, and the IQR will likely remain unchanged.
Explanation: Replacing the maximum value with a much larger number creates an extreme high outlier. This will pull the tail of the distribution to the right, making it right-skewed. The interquartile range (IQR = Q3 - Q1) is a resistant measure of spread because it is based on the middle 50% of the data. Since the change was made to the maximum value, it is very unlikely to affect either Q1 or Q3, especially in a large dataset. Therefore, the IQR will likely remain unchanged.

Question 10

Two datasets, A and B, each with 100 data points, have the same mean of 50. The distribution of Dataset A is symmetric and unimodal (bell-shaped). The distribution of Dataset B is symmetric and bimodal, with peaks at 20 and 80. Which dataset is expected to have a larger standard deviation, and why?

  1. Dataset A, because its data is more concentrated around the mean.
  2. Dataset B, because more of its data points are located far from the mean. (correct answer)
  3. The standard deviations will be equal because their means and sample sizes are the same.
  4. It cannot be determined without calculating the standard deviation for both datasets.
Explanation: Standard deviation measures the typical distance of data points from the mean. In Dataset A (bell-shaped), most of the data is clustered near the mean of 50. In Dataset B (bimodal with peaks at 20 and 80), most of the data is clustered far from the mean of 50. Since more of the data in Dataset B lies at a greater distance from the mean, its standard deviation will be larger.

Question 11

A dataset consists of the official jersey numbers for the 23 players on a national soccer team roster. These numbers typically range from 1 to 23. Which of the following is the most likely shape of the distribution of these jersey numbers?

  1. Strongly skewed to the right
  2. Strongly skewed to the left
  3. Approximately normal (bell-shaped)
  4. Approximately uniform (correct answer)
Explanation: The jersey numbers will be the integers from 1 to 23, with each number appearing exactly once. A distribution where each value occurs with the same frequency (in this case, once) is a uniform distribution. There is no tendency for the numbers to cluster in the middle (as in a normal distribution) or at one end (as in a skewed distribution). Therefore, the shape will be approximately uniform.

Question 12

A strongly right-skewed distribution of data on household incomes is transformed by taking the natural logarithm of each income value. Which of the following is the most likely effect of this transformation on the shape of the distribution?

  1. The distribution will become more strongly right-skewed.
  2. The distribution will become left-skewed.
  3. The distribution will become more symmetric. (correct answer)
  4. The transformation will not affect the skewness of the distribution.
Explanation: Logarithmic transformations compress the scale of the data. For a right-skewed distribution, the high values in the right tail are very spread out. The log transformation has a larger effect on these large values than on the smaller values, effectively 'pulling in' the long right tail. This process tends to make the distribution more symmetric.

Question 13

A real estate report for a large metropolitan area states that the mean home price is $780,000 and the median home price is $595,000. Which of the following statements is the most reasonable conclusion based on this information?

  1. The distribution of home prices is likely skewed to the right. (correct answer)
  2. The distribution of home prices is likely skewed to the left.
  3. The distribution of home prices is likely symmetric and mound-shaped.
  4. The data must contain an error, as the mean cannot be that much higher than the median.
Explanation: In a skewed distribution, the mean is pulled in the direction of the long tail. When the mean is substantially greater than the median (mean > median), it indicates that there are high-value outliers or a long tail on the right side of the distribution. This pattern is characteristic of a right-skewed distribution, which is common for variables like income and housing prices.

Question 14

For a distribution that is moderately skewed to the left, which of the following best describes the typical relationship among the mean, median, and mode?

  1. The mean, median, and mode are all approximately equal.
  2. Mean < Median < Mode (correct answer)
  3. Mode < Median < Mean
  4. Mean < Mode < Median
Explanation: In a skewed distribution, the mean is pulled in the direction of the long tail. For a left-skewed distribution, the tail is on the left, so the mean is pulled to the left (lower values). The mode is the peak of the distribution, which will be at a higher value. The median lies between the mean and the mode. Therefore, the typical ordering from least to greatest is mean, then median, then mode.

Question 15

Two datasets, X and Y, have the exact same mean, median, and mode. Which of the following statements must be true?

  1. Both datasets must have the same standard deviation.
  2. Both datasets must be unimodal and symmetric.
  3. Both datasets must have the same number of data points.
  4. The two datasets do not necessarily have the same shape. (correct answer)
Explanation: Measures of center (mean, median, mode) do not fully describe a distribution's shape or spread. It is possible to construct two different datasets that share the same mean, median, and mode but have different shapes and standard deviations. For example, Dataset 1: {1, 5, 5, 9} (Mean=5, Median=5, Mode=5) and Dataset 2: {4, 5, 5, 6} (Mean=5, Median=5, Mode=5). They share the same measures of center but have different spreads and shapes. Therefore, one cannot conclude they have the same shape.

Question 16

A dataset consists of the official jersey numbers for the 23 players on a national soccer team roster. These numbers typically range from 1 to 23. Which of the following is the most likely shape of the distribution of these jersey numbers?

  1. Strongly skewed to the right
  2. Strongly skewed to the left
  3. Approximately normal (bell-shaped)
  4. Approximately uniform (correct answer)
Explanation: The jersey numbers will be the integers from 1 to 23, with each number appearing exactly once. A distribution where each value occurs with the same frequency (in this case, once) is a uniform distribution. There is no tendency for the numbers to cluster in the middle (as in a normal distribution) or at one end (as in a skewed distribution). Therefore, the shape will be approximately uniform.

Question 17

A large dataset is approximately symmetric. The maximum value in the dataset is then erroneously replaced by a number ten times larger than the original maximum. How will this change affect the shape of the distribution and the interquartile range (IQR)?

  1. The distribution will become right-skewed, and the IQR will increase significantly.
  2. The distribution will become right-skewed, and the IQR will likely remain unchanged. (correct answer)
  3. The distribution will remain symmetric, but the range and standard deviation will increase.
  4. The distribution will become left-skewed, and the IQR will likely remain unchanged.
Explanation: Replacing the maximum value with a much larger number creates an extreme high outlier. This will pull the tail of the distribution to the right, making it right-skewed. The interquartile range (IQR = Q3 - Q1) is a resistant measure of spread because it is based on the middle 50% of the data. Since the change was made to the maximum value, it is very unlikely to affect either Q1 or Q3, especially in a large dataset. Therefore, the IQR will likely remain unchanged.

Question 18

A financial analyst is studying the annual salaries of employees at a technology firm. The data is found to be roughly symmetric with a median salary of $95,000. The company then hires a new CEO with an annual salary of $15,000,000. How will the inclusion of the new CEO's salary most likely affect the mean and median salary, and the shape of the distribution?

  1. The mean and median will both increase substantially, and the distribution will remain approximately symmetric.
  2. The mean will increase substantially, the median will remain relatively stable, and the distribution will become right-skewed. (correct answer)
  3. The mean will increase substantially, the median will remain relatively stable, and the distribution will become left-skewed.
  4. The median will increase substantially, the mean will remain relatively stable, and the distribution will become right-skewed.
Explanation: The CEO's salary is an extreme high outlier. The mean is sensitive to outliers and will be pulled upwards significantly by this large value. The median, being a resistant measure of center, will change very little, as it is based on the position of the middle value, not its magnitude. The addition of a high outlier creates a long tail on the right side of the distribution, making it right-skewed.

Question 19

A five-number summary for a set of exam scores is {20, 50, 60, 65, 100}. Based on the 1.5 × IQR rule, what can be concluded about outliers in this dataset?

  1. There are no outliers.
  2. There is at least one high outlier, but there are no low outliers.
  3. There is at least one low outlier, but there are no high outliers.
  4. There is at least one high outlier and at least one low outlier. (correct answer)
Explanation: First, calculate the IQR: IQR = Q3 - Q1 = 65 - 50 = 15. Next, calculate the fences. The lower fence is Q1 - 1.5 × IQR = 50 - 1.5 × 15 = 50 - 22.5 = 27.5. The upper fence is Q3 + 1.5 × IQR = 65 + 1.5 × 15 = 65 + 22.5 = 87.5. The minimum value is 20, which is less than the lower fence of 27.5, so there is at least one low outlier. The maximum value is 100, which is greater than the upper fence of 87.5, so there is at least one high outlier.

Question 20

A researcher collects data on the response times of a cognitive task. Summary statistics for the dataset (n=50) are: Min=5.2, Q1=10.5, Median=12.1, Q3=14.2, Max=25.8. Using the 1.5 × IQR rule for identifying outliers, which of the following statements is true?

  1. There are no outliers in this dataset.
  2. The maximum value of 25.8 is an outlier, but the minimum value of 5.2 is not. (correct answer)
  3. The minimum value of 5.2 is an outlier, but the maximum value of 25.8 is not.
  4. Both the minimum value of 5.2 and the maximum value of 25.8 are outliers.
Explanation: First, calculate the Interquartile Range (IQR): IQR = Q3 - Q1 = 14.2 - 10.5 = 3.7. Next, calculate the lower and upper fences for outliers. Lower fence = Q1 - 1.5 × IQR = 10.5 - 1.5 × 3.7 = 10.5 - 5.55 = 4.95. Upper fence = Q3 + 1.5 × IQR = 14.2 + 1.5 × 3.7 = 14.2 + 5.55 = 19.75. The minimum value, 5.2, is greater than the lower fence of 4.95, so it is not an outlier. The maximum value, 25.8, is greater than the upper fence of 19.75, so it is an outlier.