College Statistics Quiz: Boxplots And Five Number Summary
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Boxplots And Five Number SummaryQuestion 1 of 20

For a particular dataset, the five-number summary is {10, 20, 50, 80, 180}. Which of the following statements is necessarily true?

The distribution is skewed to the right.
There are no outliers in the dataset.
The mean of the dataset is greater than 50.
At least 50% of the data values are greater than or equal to 50.
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College Statistics Quiz

College Statistics Quiz: Boxplots And Five Number Summary

Practice Boxplots And Five Number Summary 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 Boxplots And Five Number Summary, 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

For a particular dataset, the five-number summary is {10, 20, 50, 80, 180}. Which of the following statements is necessarily true?

  1. The distribution is skewed to the right.
  2. There are no outliers in the dataset.
  3. The mean of the dataset is greater than 50.
  4. At least 50% of the data values are greater than or equal to 50. (correct answer)
Explanation: The median (the third number in the summary) is 50. By definition, the median is the value at which 50% of the data is below and 50% is above. Therefore, at least 50% of the data values must be greater than or equal to the median of 50. Choice A is likely true based on the apparent right skew, but not necessarily true. Choice C is also likely due to the probable right-skew, but not guaranteed. Choice B is false; using the 1.5×IQR rule: IQR = 80-20 = 60, upper fence = 80 + 1.5(60) = 170. Since the maximum (180) exceeds 170, it is an outlier.

Question 2

The five-number summary for a dataset of 400 home prices is {$150k, $220k, $280k, $400k, $950k}. Approximately how many homes have prices between $220k and $400k?

  1. 100
  2. 200 (correct answer)
  3. 300
  4. Cannot be determined because the prices are not uniformly distributed.
Explanation: The five-number summary consists of the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. By definition, the first quartile ($220k) is the 25th percentile, and the third quartile ($400k) is the 75th percentile. The data between Q1 and Q3 represents the middle 50% of the observations, regardless of the shape of the distribution. Therefore, the number of homes with prices in this range is approximately 50% of the total, which is 0.50×400=2000.50 \times 400 = 200.

Question 3

A dataset has a first quartile Q1Q_1 of 60 and a third quartile Q3Q_3 of 90. A modified boxplot is created using the 1.5 × IQR rule to identify potential outliers. Which of the following is a possible value for the end of the upper whisker of the boxplot?

  1. 135
  2. 120 (correct answer)
  3. 138
  4. 90
Explanation: First, calculate the interquartile range: IQR=Q3−Q1=90−60=30IQR = Q_3 - Q_1 = 90 - 60 = 30. Next, determine the upper fence for outliers: Q3+1.5×IQR=90+1.5×30=90+45=135Q_3 + 1.5 \times IQR = 90 + 1.5 \times 30 = 90 + 45 = 135. The upper whisker of a modified boxplot extends to the maximum data value that is less than or equal to the upper fence. Therefore, the whisker must end at a data point that is ≤135\leq 135. Choice C (138) is an outlier, so the whisker would not extend to it. Choice A (135) is the fence itself, and while possible, choice B (120) is a more general possible value representing the highest non-outlier data point. Choice D (90) would only be the end of the whisker if all data points above 90 were outliers, which is highly improbable.

Question 4

A dataset has a five-number summary of {100, 120, 135, 160, 200}. If 10 is added to every value in the dataset, what is the new five-number summary?

  1. {100, 120, 145, 170, 210}
  2. {110, 120, 135, 160, 210}
  3. {110, 130, 145, 170, 210} (correct answer)
  4. {100, 130, 145, 160, 200}
Explanation: When you encounter questions about transforming datasets, remember that adding or subtracting a constant to every data point shifts all summary statistics by that same amount. This is because measures of position (like quartiles and extremes) move uniformly when the entire dataset is translated. The original five-number summary is {100, 120, 135, 160, 200}, representing minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum respectively. When 10 is added to every value, each component of the five-number summary increases by exactly 10:
  • Minimum: 100 + 10 = 110
  • Q1: 120 + 10 = 130
  • Median: 135 + 10 = 145
  • Q3: 160 + 10 = 170
  • Maximum: 200 + 10 = 210
This gives us {110, 130, 145, 170, 210}, which is answer choice C. Looking at the wrong answers: Choice A incorrectly keeps the minimum and second quartile unchanged while only transforming some values. Choice B fails to add 10 to most of the middle values, suggesting a misunderstanding of how transformations work uniformly across all data points. Choice D only transforms two values and decreases others, which contradicts the uniform addition operation. Remember this key principle: adding or subtracting a constant shifts all summary statistics by that exact amount, while multiplying or dividing scales them proportionally. This uniform transformation property makes these problems straightforward once you recognize the pattern.

Question 5

The range of a dataset is 50 and the interquartile range (IQR) is 20. An extreme data point is added that increases the maximum value but does not change any other data points. Which of the following is the most likely effect on the range and the IQR?

  1. Both the range and the IQR will increase.
  2. The IQR will increase, but the range will likely remain unchanged.
  3. The range will increase, but the IQR will likely remain unchanged. (correct answer)
  4. Both the range and the IQR will likely remain unchanged.
Explanation: When you encounter questions about how extreme values affect measures of spread, focus on understanding what each measure actually captures. The range measures the total spread from minimum to maximum, while the IQR measures the spread of the middle 50% of data (from Q1 to Q3). Adding an extreme data point that increases the maximum value will directly impact the range since range = maximum - minimum. With a larger maximum and unchanged minimum, the range must increase beyond the original 50. However, the IQR depends only on the 25th and 75th percentiles (Q1 and Q3). When you add a single extreme high value to a dataset, you're typically adding just one data point. This lone addition is unlikely to shift where Q1 and Q3 fall within the ordered dataset, especially if the original dataset was reasonably sized. The middle 50% of values remains essentially the same, so the IQR stays around 20. Looking at each option: A) incorrectly assumes both measures change—but the IQR is resistant to extreme values. B) has this backwards, suggesting the IQR changes while the range doesn't—but extreme values always affect range. D) incorrectly suggests neither measure changes, ignoring that range is sensitive to outliers. Study tip: Remember that measures of spread fall into two categories: sensitive to outliers (range, standard deviation) and resistant to outliers (IQR, median absolute deviation). When extreme values are added, sensitive measures change while resistant measures typically don't.

Question 6

The five-number summary for the weights of a sample of 30 dogs is {8, 25, 40, 55, 90} pounds. If the heaviest dog, weighing 90 pounds, is replaced by a new dog weighing 95 pounds, which of the following summary statistics must change?

  1. The range and the maximum. (correct answer)
  2. The median and the maximum.
  3. The interquartile range and the range.
  4. The median and the interquartile range.
Explanation: When you encounter questions about changes to summary statistics, you need to understand which statistics are sensitive to extreme values and which are resistant to them. Let's examine what happens when we replace the 90-pound dog with a 95-pound dog. The five-number summary consists of minimum (8), first quartile (25), median (40), third quartile (55), and maximum (90). The maximum clearly changes from 90 to 95 pounds since we're replacing the heaviest dog with an even heavier one. The range (maximum - minimum) also must change, going from 82 pounds (90-8) to 87 pounds (95-8). However, the median stays at 40 pounds because we're only changing one extreme value, and the median depends on the middle values of the ordered dataset. Similarly, the interquartile range (third quartile - first quartile = 55-25 = 30) remains unchanged because quartiles, like the median, are resistant to changes in extreme values. Looking at the answer choices: A is correct because both range and maximum change. B is wrong because the median doesn't change when you modify an extreme value. C is wrong because the interquartile range remains unaffected by extreme value changes. D is wrong because neither the median nor interquartile range changes. Study tip: Remember that measures of center (median) and spread (IQR) based on position are resistant to extreme values, while measures involving the actual extreme values (range, maximum, minimum) are sensitive to changes in those extremes.

Question 7

A climatologist records the daily high temperatures (in degrees Celsius) for a city over a year. The five-number summary for the data is {–10, 5, 12, 20, 35}. The temperatures are then converted to degrees Fahrenheit using the formula F=1.8C+32F = 1.8C + 32. What is the interquartile range (IQR) of the daily high temperatures in degrees Fahrenheit?

  1. 15
  2. 27 (correct answer)
  3. 59
  4. 45
Explanation: The original interquartile range (IQR) in Celsius is Q3−Q1=20−5=15Q_3 - Q_1 = 20 - 5 = 15°C. When data is transformed linearly (y=ax+by = ax + b), measures of spread like the range and IQR are only affected by the multiplicative factor aa, not the additive factor bb. Therefore, the new IQR in Fahrenheit is 1.8×15=271.8 \times 15 = 27. Distractor C incorrectly applies the full transformation to the IQR (1.8×15+32=591.8 \times 15 + 32 = 59). Distractor A is the original IQR in Celsius, failing to apply the transformation. Distractor D is the range of the original data in Celsius (35 - (-10) = 45), a common confusion.

Question 8

A boxplot of employee salaries shows the median is much closer to the first quartile (Q1) than to the third quartile (Q3), and the right whisker is substantially longer than the left whisker. Which statement is the most valid conclusion?

  1. The distribution of salaries is approximately normal.
  2. The mean salary is likely lower than the median salary.
  3. The distribution of salaries is skewed to the right. (correct answer)
  4. There are more employees with salaries between the median and Q3 than between Q1 and the median.
Explanation: A median closer to Q1 and a long right whisker are classic indicators of a right-skewed (or positively skewed) distribution. This means there is a tail of high values pulling the distribution to the right. Choice A is incorrect because a normal distribution would be symmetric. Choice B is incorrect; in a right-skewed distribution, the mean is typically pulled in the direction of the skew, making it greater than the median. Choice D is a common misconception; the number of data points between Q1 and the median is approximately 25%, the same as the number between the median and Q3. The larger distance between the median and Q3 simply means those 25% of salaries are more spread out.

Question 9

A dataset contains 20 positive integers, and its interquartile range is 30. Each value in the dataset is doubled, and then 5 is subtracted. What is the interquartile range of the new, transformed dataset?

  1. 55
  2. 60 (correct answer)
  3. 65
  4. 25
Explanation: Let the original first and third quartiles be Q1Q_1 and Q3Q_3. The original IQR is Q3−Q1=30Q_3 - Q_1 = 30. The transformation applied to each data point xx is y=2x−5y = 2x - 5. The new quartiles will be Q1′=2Q1−5Q'_1 = 2Q_1 - 5 and Q3′=2Q3−5Q'_3 = 2Q_3 - 5. The new IQR is Q3′−Q1′=(2Q3−5)−(2Q1−5)=2Q3−2Q1=2(Q3−Q1)=2×30=60Q'_3 - Q'_1 = (2Q_3 - 5) - (2Q_1 - 5) = 2Q_3 - 2Q_1 = 2(Q_3 - Q_1) = 2 \times 30 = 60. Measures of spread are affected by multiplication/division but not by addition/subtraction. Choice A (55) results from incorrectly applying the entire transformation to the IQR (2×30−52 \times 30 - 5). Choice C (65) results from 2×30+52 \times 30 + 5. Choice D (25) results from incorrectly subtracting 5 from the IQR.

Question 10

A dataset contains 20 positive integers, and its interquartile range is 30. Each value in the dataset is doubled, and then 5 is subtracted. What is the interquartile range of the new, transformed dataset?

  1. 55
  2. 60 (correct answer)
  3. 65
  4. 25
Explanation: Let the original first and third quartiles be Q1Q_1 and Q3Q_3. The original IQR is Q3−Q1=30Q_3 - Q_1 = 30. The transformation applied to each data point xx is y=2x−5y = 2x - 5. The new quartiles will be Q1′=2Q1−5Q'_1 = 2Q_1 - 5 and Q3′=2Q3−5Q'_3 = 2Q_3 - 5. The new IQR is Q3′−Q1′=(2Q3−5)−(2Q1−5)=2Q3−2Q1=2(Q3−Q1)=2×30=60Q'_3 - Q'_1 = (2Q_3 - 5) - (2Q_1 - 5) = 2Q_3 - 2Q_1 = 2(Q_3 - Q_1) = 2 \times 30 = 60. Measures of spread are affected by multiplication/division but not by addition/subtraction. Choice A (55) results from incorrectly applying the entire transformation to the IQR (2×30−52 \times 30 - 5). Choice C (65) results from 2×30+52 \times 30 + 5. Choice D (25) results from incorrectly subtracting 5 from the IQR.

Question 11

A climatologist records the daily high temperatures (in degrees Celsius) for a city over a year. The five-number summary for the data is {–10, 5, 12, 20, 35}. The temperatures are then converted to degrees Fahrenheit using the formula F=1.8C+32F = 1.8C + 32. What is the interquartile range (IQR) of the daily high temperatures in degrees Fahrenheit?

  1. 15
  2. 27 (correct answer)
  3. 59
  4. 45
Explanation: The original interquartile range (IQR) in Celsius is Q3−Q1=20−5=15Q_3 - Q_1 = 20 - 5 = 15°C. When data is transformed linearly (y=ax+by = ax + b), measures of spread like the range and IQR are only affected by the multiplicative factor aa, not the additive factor bb. Therefore, the new IQR in Fahrenheit is 1.8×15=271.8 \times 15 = 27. Distractor C incorrectly applies the full transformation to the IQR (1.8×15+32=591.8 \times 15 + 32 = 59). Distractor A is the original IQR in Celsius, failing to apply the transformation. Distractor D is the range of the original data in Celsius (35 - (-10) = 45), a common confusion.

Question 12

The range of a dataset is 50 and the interquartile range (IQR) is 20. An extreme data point is added that increases the maximum value but does not change any other data points. Which of the following is the most likely effect on the range and the IQR?

  1. Both the range and the IQR will increase.
  2. The IQR will increase, but the range will likely remain unchanged.
  3. The range will increase, but the IQR will likely remain unchanged. (correct answer)
  4. Both the range and the IQR will likely remain unchanged.
Explanation: When you encounter questions about how extreme values affect measures of spread, focus on understanding what each measure actually captures. The range measures the total spread from minimum to maximum, while the IQR measures the spread of the middle 50% of data (from Q1 to Q3). Adding an extreme data point that increases the maximum value will directly impact the range since range = maximum - minimum. With a larger maximum and unchanged minimum, the range must increase beyond the original 50. However, the IQR depends only on the 25th and 75th percentiles (Q1 and Q3). When you add a single extreme high value to a dataset, you're typically adding just one data point. This lone addition is unlikely to shift where Q1 and Q3 fall within the ordered dataset, especially if the original dataset was reasonably sized. The middle 50% of values remains essentially the same, so the IQR stays around 20. Looking at each option: A) incorrectly assumes both measures change—but the IQR is resistant to extreme values. B) has this backwards, suggesting the IQR changes while the range doesn't—but extreme values always affect range. D) incorrectly suggests neither measure changes, ignoring that range is sensitive to outliers. Study tip: Remember that measures of spread fall into two categories: sensitive to outliers (range, standard deviation) and resistant to outliers (IQR, median absolute deviation). When extreme values are added, sensitive measures change while resistant measures typically don't.

Question 13

A dataset has a first quartile Q1Q_1 of 60 and a third quartile Q3Q_3 of 90. A modified boxplot is created using the 1.5 × IQR rule to identify potential outliers. Which of the following is a possible value for the end of the upper whisker of the boxplot?

  1. 135
  2. 120 (correct answer)
  3. 138
  4. 90
Explanation: First, calculate the interquartile range: IQR=Q3−Q1=90−60=30IQR = Q_3 - Q_1 = 90 - 60 = 30. Next, determine the upper fence for outliers: Q3+1.5×IQR=90+1.5×30=90+45=135Q_3 + 1.5 \times IQR = 90 + 1.5 \times 30 = 90 + 45 = 135. The upper whisker of a modified boxplot extends to the maximum data value that is less than or equal to the upper fence. Therefore, the whisker must end at a data point that is ≤135\leq 135. Choice C (138) is an outlier, so the whisker would not extend to it. Choice A (135) is the fence itself, and while possible, choice B (120) is a more general possible value representing the highest non-outlier data point. Choice D (90) would only be the end of the whisker if all data points above 90 were outliers, which is highly improbable.

Question 14

A boxplot of employee salaries shows the median is much closer to the first quartile (Q1) than to the third quartile (Q3), and the right whisker is substantially longer than the left whisker. Which statement is the most valid conclusion?

  1. The distribution of salaries is approximately normal.
  2. The mean salary is likely lower than the median salary.
  3. The distribution of salaries is skewed to the right. (correct answer)
  4. There are more employees with salaries between the median and Q3 than between Q1 and the median.
Explanation: A median closer to Q1 and a long right whisker are classic indicators of a right-skewed (or positively skewed) distribution. This means there is a tail of high values pulling the distribution to the right. Choice A is incorrect because a normal distribution would be symmetric. Choice B is incorrect; in a right-skewed distribution, the mean is typically pulled in the direction of the skew, making it greater than the median. Choice D is a common misconception; the number of data points between Q1 and the median is approximately 25%, the same as the number between the median and Q3. The larger distance between the median and Q3 simply means those 25% of salaries are more spread out.

Question 15

The five-number summary for the weights of a sample of 30 dogs is {8, 25, 40, 55, 90} pounds. If the heaviest dog, weighing 90 pounds, is replaced by a new dog weighing 95 pounds, which of the following summary statistics must change?

  1. The range and the maximum. (correct answer)
  2. The median and the maximum.
  3. The interquartile range and the range.
  4. The median and the interquartile range.
Explanation: When you encounter questions about changes to summary statistics, you need to understand which statistics are sensitive to extreme values and which are resistant to them. Let's examine what happens when we replace the 90-pound dog with a 95-pound dog. The five-number summary consists of minimum (8), first quartile (25), median (40), third quartile (55), and maximum (90). The maximum clearly changes from 90 to 95 pounds since we're replacing the heaviest dog with an even heavier one. The range (maximum - minimum) also must change, going from 82 pounds (90-8) to 87 pounds (95-8). However, the median stays at 40 pounds because we're only changing one extreme value, and the median depends on the middle values of the ordered dataset. Similarly, the interquartile range (third quartile - first quartile = 55-25 = 30) remains unchanged because quartiles, like the median, are resistant to changes in extreme values. Looking at the answer choices: A is correct because both range and maximum change. B is wrong because the median doesn't change when you modify an extreme value. C is wrong because the interquartile range remains unaffected by extreme value changes. D is wrong because neither the median nor interquartile range changes. Study tip: Remember that measures of center (median) and spread (IQR) based on position are resistant to extreme values, while measures involving the actual extreme values (range, maximum, minimum) are sensitive to changes in those extremes.

Question 16

For a particular dataset, the five-number summary is {10, 20, 50, 80, 180}. Which of the following statements is necessarily true?

  1. The distribution is skewed to the right.
  2. There are no outliers in the dataset.
  3. The mean of the dataset is greater than 50.
  4. At least 50% of the data values are greater than or equal to 50. (correct answer)
Explanation: The median (the third number in the summary) is 50. By definition, the median is the value at which 50% of the data is below and 50% is above. Therefore, at least 50% of the data values must be greater than or equal to the median of 50. Choice A is likely true based on the apparent right skew, but not necessarily true. Choice C is also likely due to the probable right-skew, but not guaranteed. Choice B is false; using the 1.5×IQR rule: IQR = 80-20 = 60, upper fence = 80 + 1.5(60) = 170. Since the maximum (180) exceeds 170, it is an outlier.

Question 17

The boxplot represents the battery life, in hours, of 200 smartphones tested. An advertiser claims that at least 75% of these smartphones have a battery life of 12 hours or more. Based on the boxplot, is this claim supported?

  1. Yes, because the third quartile is at 14 hours, which is greater than 12 hours.
  2. Yes, because the median is 12 hours, meaning 50% last 12 hours or more, and the right skew adds at least 25% more.
  3. No, because the first quartile is at 10 hours, meaning approximately 75% of the phones have a battery life of 10 hours or more, not 12. (correct answer)
  4. No, because the median is 12 hours, which means approximately 50% of the phones have a battery life of 12 hours or more.
Explanation: The claim is that at least 75% of phones last 12 hours or more. This is equivalent to saying that the first quartile (Q1, the 25th percentile) must be at least 12 hours. Looking at the boxplot, the first quartile (the left edge of the box) is located at 10 hours. This means that approximately 75% of the smartphones have a battery life of 10 hours or more. Since Q1 is 10, it is not true that 75% have a life of 12 hours or more; the actual percentage lasting 12+ hours is somewhere between 50% (the median) and 75%. Therefore, the claim is not supported. Choice C provides the correct reasoning. Choice D correctly identifies the median but makes the wrong conclusion about the claim.

Question 18

A student reports that for a dataset with 50 values, the five-number summary is {25, 40, 35, 60, 75}. What is the most significant error in this summary?

  1. The median cannot be lower than the first quartile. (correct answer)
  2. The range is too large for a dataset of only 50 values.
  3. The interquartile range is negative.
  4. The maximum value is likely an outlier.
Explanation: When analyzing a five-number summary, you need to verify that the values follow the correct logical order: minimum ≤ Q1 ≤ median ≤ Q3 ≤ maximum. This fundamental property ensures the summary makes statistical sense. Looking at the given summary {25, 40, 35, 60, 75}, let's check the order. If we interpret this as {min, Q1, median, Q3, max}, we have: 25 ≤ 40, but then 40 > 35. This violates the requirement that the first quartile (Q1 = 40) must be less than or equal to the median (35). This is mathematically impossible since Q1 represents the value below which 25% of data falls, while the median represents the value below which 50% falls. Option A correctly identifies this fundamental error - the median cannot be lower than the first quartile because it would contradict the basic definition of quartiles. Option B is incorrect because there's no inherent relationship between dataset size and range. A dataset of 50 values can legitimately have any range. Option C misunderstands the situation. The interquartile range would be 60 - 40 = 20, which is positive, not negative. Option D assumes the data is correctly ordered and focuses on outliers, but the primary issue isn't about outliers - it's about the impossible ordering of the quartiles themselves. Study tip: Always verify that five-number summaries follow ascending order. When you see quartile values that seem out of sequence, immediately check whether the fundamental quartile relationships hold before considering other statistical properties.

Question 19

A histogram of a dataset is strongly skewed to the left. Which of the following boxplots would most likely represent this dataset?

  1. A boxplot where the median is centered in the box and the whiskers are of equal length.
  2. A boxplot where the left whisker is much longer than the right whisker, and the median is closer to Q3. (correct answer)
  3. A boxplot where the right whisker is much longer than the left whisker, and the median is closer to Q1.
  4. A boxplot with a very wide box and very short whiskers.
Explanation: A left-skewed distribution has a long tail of low values. This is represented in a boxplot by a long whisker on the left side. The data values are more concentrated on the high end, so the distance from the median to Q3 and from Q1 to the median will be smaller. Specifically, the median will be closer to Q3 than to Q1. Choice A describes a symmetric distribution. Choice C describes a right-skewed distribution. Choice D describes a distribution with high variability in the middle 50% but low variability in the tails, which does not directly relate to skewness.

Question 20

A dataset consists of 15 distinct integers. The median of the dataset is 50. Two new values, 18 and 62, are added to the dataset. What is the median of the new dataset?

  1. 43.3
  2. 50 (correct answer)
  3. 51
  4. Cannot be determined without the original data.
Explanation: Originally, with n=15n=15, the median is the (15+1)/2=8th(15+1)/2 = 8^{th} value in the ordered dataset. So, the 8th value is 50. When we add two new values, the dataset size becomes n=17n=17. The new median will be the (17+1)/2=9th(17+1)/2 = 9^{th} value. One of the new values, 18, is less than 50, and the other, 62, is greater than 50. When 18 is added, it is inserted somewhere before the original 8th value. This shifts the original 8th value (50) to the 9th position. The value 62 is inserted after it. Therefore, the 9th value in the new ordered list is still 50.