Business Statistics Quiz: Percentiles Quartiles And Boxplots
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Percentiles Quartiles And BoxplotsQuestion 1 of 20

A retail chain analyzes customer transaction values. Summary statistics from a large sample show that the first quartile (Q1) is $25 and the third quartile (Q3) is $70. Using the standard 1.5 × IQR method, what is the maximum transaction value that would not be classified as a high-value outlier?

$115.00
$137.50
$100.00
$92.50
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Business Statistics Quiz

Business Statistics Quiz: Percentiles Quartiles And Boxplots

Practice Percentiles Quartiles And Boxplots in Business 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 Percentiles Quartiles And Boxplots, giving you a quick way to practice the rules, question types, and explanations that matter most for Business 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.

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Question 1

A retail chain analyzes customer transaction values. Summary statistics from a large sample show that the first quartile (Q1) is $25 and the third quartile (Q3) is $70. Using the standard 1.5 × IQR method, what is the maximum transaction value that would not be classified as a high-value outlier?

  1. $115.00
  2. $137.50 (correct answer)
  3. $100.00
  4. $92.50
Explanation: This is a two-step problem. First, calculate the interquartile range (IQR). Second, use the IQR to find the upper fence for outliers.
  1. IQR = Q3 - Q1 = $70 - $25 = $45.
  2. The upper fence is calculated as Q3 + 1.5 × IQR. Upper Fence = $70 + 1.5 × $45 = $70 + $67.50 = $137.50. Any value above $137.50 would be considered an outlier. Therefore, $137.50 is the maximum value that is not an outlier. Distractor A results from incorrectly calculating the fence as Q3 + IQR ($70 + $45). Distractor C comes from a conceptual error, possibly using Q3 + (Q3 - Median), assuming a median. Distractor D incorrectly calculates the fence from Q1 (Q1 + 1.5 * IQR).

Question 2

A boxplot analysis reveals that a dataset has Q1 = 23, median = 31, Q3 = 42, with no outliers shown. If the whiskers extend to the minimum value of 8 and maximum value of 58, what can be concluded about the percentile distribution and potential data characteristics?

  1. The 10th percentile is approximately 14.5, and the data shows moderate positive skewness based on quartile positions
  2. The data spans exactly 50 units with symmetric distribution, since the whiskers are equal length from their respective quartile boundaries
  3. The absence of outliers confirms that all data points fall within 1.5×IQR of their respective quartiles, with the 95th percentile near 52 (correct answer)
  4. The interquartile range of 19 contains exactly half the data, while the whiskers represent the outer 25% of observations on each side
Explanation: IQR = 42 - 23 = 19. Outlier boundaries: Q1 - 1.5×IQR = 23 - 28.5 = -5.5 (lower), Q3 + 1.5×IQR = 42 + 28.5 = 70.5 (upper). Since min=8 > -5.5 and max=58 < 70.5, no outliers exist, confirming the boundary rule. Choice A cannot determine 10th percentile from boxplot data. Choice B incorrectly assumes symmetry (median closer to Q1 than Q3 suggests skewness). Choice D correctly states IQR contains half the data but whiskers don't necessarily represent exactly 25% each.

Question 3

A quality analyst examines defect rates across 120 production batches. The data shows: minimum = 1.2%, Q1 = 2.8%, median = 4.1%, Q3 = 6.5%, maximum = 12.3%. If the analyst needs to identify batches in the 85th to 95th percentile range for process improvement analysis, approximately how many batches fall in this range, and what does this represent in terms of the overall distribution structure?

  1. Exactly 10 batches fall in this range, located entirely within the fourth quartile between Q3 and the maximum value
  2. Exactly 12 batches fall in this range, representing the upper portion of the fourth quartile where defect rates are highest (correct answer)
  3. Approximately 14 batches fall in this range, spanning from the upper fourth quartile into the maximum boundary region
  4. Exactly 15 batches fall in this range, representing the transition zone between high-normal and exceptional defect rates
Explanation: When you encounter percentile range questions with given sample data, you need to calculate how many observations fall within specific percentile boundaries and understand where those percentiles sit relative to the quartile structure. To find how many batches fall between the 85th and 95th percentiles, calculate the percentage of data in this range: 95% - 85% = 10% of the total sample. With 120 batches, this equals 120×0.10=12120 \times 0.10 = 12 batches exactly. Next, consider where these percentiles fall within the distribution structure. Since Q3 represents the 75th percentile and the maximum represents the 100th percentile, both the 85th and 95th percentiles fall entirely within the fourth quartile (75th to 100th percentile range). The 85th to 95th percentile range captures the upper portion of this fourth quartile, representing batches with the highest defect rates in the dataset. Answer A incorrectly states "exactly 10 batches" - this comes from mistakenly using the 10% range directly rather than applying it to the 120-batch sample size. Answer C suggests "approximately 14 batches," which incorrectly rounds or miscalculates the precise 10% figure. Answer D claims "exactly 15 batches," which appears to confuse this range with a full quartile (25% of 120 = 30 batches) divided by two. Study tip: For percentile range problems, always multiply the percentage range by the total sample size to get the exact count, and use the quartile positions (Q1=25th, Q2=50th, Q3=75th) as reference points to locate where your target percentiles fall within the distribution structure.

Question 4

A retail company analyzes daily sales data for 40 consecutive business days. The sales values (in thousands of dollars) have a median of $85k, Q1 = $72k, and Q3 = $94k. If the company defines "exceptional performance days" as those in the top 15% of all sales days, what is the minimum percentile rank a day must achieve to be considered exceptional, and how does this relate to the quartile structure?

  1. 85th percentile, which falls between Q3 and the maximum value in the upper quartile range (correct answer)
  2. 85th percentile, which is exactly at the median since it represents the top 15% threshold
  3. 94th percentile, which corresponds to Q3 since the top quartile contains the highest 25% of values
  4. 90th percentile, which falls between the median and Q3 in the third quartile range
Explanation: The top 15% corresponds to the 85th percentile (100% - 15% = 85%). Since Q3 is at the 75th percentile ($94k) and the maximum is at the 100th percentile, the 85th percentile falls between Q3 and the maximum value in the fourth quartile. Choice B incorrectly associates the 85th percentile with the median. Choice C confuses Q3 (75th percentile) with the 94th percentile. Choice D incorrectly places the 90th percentile between the median and Q3.

Question 5

A dataset of 150 customer satisfaction scores shows the following percentile values: 25th percentile = 6.2, 50th percentile = 7.1, 75th percentile = 8.3, 90th percentile = 9.1. If a customer's score places them at exactly the 60th percentile, which statement best describes their position relative to the quartile structure?

  1. The score falls between the median and Q3, closer to the median, representing the 2nd decile of the third quartile
  2. The score falls between the median and Q3, representing the 40th percentile within the third quartile range
  3. The score falls in the upper portion of the third quartile, specifically at the 60% mark of the interquartile range
  4. The score falls between Q2 and Q3, positioned at 40% of the distance from the median toward the third quartile boundary (correct answer)
Explanation: The 60th percentile falls between Q2 (median, 50th percentile) and Q3 (75th percentile). The position is (60-50)/(75-50) = 10/25 = 40% of the way from median to Q3. Choice A incorrectly mentions 'decile of third quartile' which is meaningless terminology. Choice B incorrectly calculates position within the quartile. Choice C incorrectly describes it as 'upper portion' and misuses 'within third quartile' - the 60th percentile is still in the second half, not the third quartile itself.

Question 6

Two divisions of a company report the following quarterly earnings data. Division X: Q1=18M,median=18M, median=24M, Q3=35M.DivisionY:Q1=35M. Division Y: Q1=22M, median=26M,Q3=26M, Q3=31M. If both divisions have the same number of quarters in their datasets, which statement about their 75th percentile comparison and distribution characteristics is most precise?

  1. Division X outperforms Division Y at the 75th percentile by $4M, though Division Y demonstrates more consistent median-quartile relationships
  2. Division X's 75th percentile exceeds Division Y's by $4M, with Division X showing greater variability in earnings performance (correct answer)
  3. The $4M advantage in Division X's third quartile position indicates superior performance, while both divisions show identical skewness patterns
  4. Division X's Q3 advantage of $4M is offset by Division Y's smaller interquartile range, suggesting comparable overall performance distributions
Explanation: When analyzing quartile data and distribution characteristics, you need to understand what each quartile position tells you and how to interpret variability measures like the interquartile range (IQR). Let's examine the data systematically. The 75th percentile is equivalent to Q3, so Division X's 75th percentile is $35M while Division Y's is $31M, giving Division X a $4M advantage. Now compare the interquartile ranges: Division X has an IQR of $35M - $18M = $17M, while Division Y has $31M - $22M = $9M. Since Division X's IQR is nearly twice as large, it shows greater variability in earnings performance. This makes option B correct. Option A incorrectly claims Division Y has "more consistent median-quartile relationships." Actually, Division X shows greater variability with its larger IQR. Option C states both divisions show "identical skewness patterns," but you cannot determine skewness from quartile data alone - you'd need the full distribution or additional statistics. Option D suggests the performances are "comparable" because Division Y has a smaller IQR, but this misinterprets what IQR measures. A smaller IQR indicates less variability, not necessarily better performance. Study tip: Remember that Q3 equals the 75th percentile, and IQR (Q3 - Q1) measures spread or variability, not performance quality. Larger IQR means more variability, while quartiles alone cannot determine skewness. Always calculate the specific values rather than making assumptions about distribution characteristics.

Question 7

A manager reviews side-by-side boxplots of daily sales for two store locations. Location A has a median of $200 and an IQR of $100. Location B has a median of $260 and an IQR of $70. Which statement is the most accurate comparison for a manager focused on both sales performance and consistency?

  1. Location B has higher median sales and demonstrates more consistency in the middle 50% of its sales. (correct answer)
  2. Location A has a greater interquartile range, indicating that its average sales are generally higher than Location B.
  3. Both locations have approximately the same sales consistency, but Location B has a higher overall range of sales.
  4. At least 50% of sales days at Location A were lower than the lowest 25% of sales days at Location B.
Explanation: The median represents the central tendency (typical performance), and the IQR represents the variability or consistency of the middle 50% of the data. Location B has a higher median ($260 vs. 200),indicatingbettertypicalsalesperformance.LocationBalsohasasmallerIQR(200), indicating better typical sales performance. Location B also has a smaller IQR (70 vs. $100), indicating less variability and thus more consistency in its daily sales. Therefore, statement A is the most accurate summary. Statement B incorrectly links IQR to average sales level. Statement C is incorrect because the IQRs are different, indicating different consistency. Statement D makes a specific data comparison that cannot be confirmed without knowing Q1 for Location B.

Question 8

A standard box-and-whisker plot provides a graphical summary of a dataset. While useful, it has limitations. Which of the following characteristics of a dataset's distribution cannot be reliably determined from a standard boxplot?

  1. The presence of potential outliers based on the 1.5 × IQR criterion.
  2. The variability of the central 50% of the data.
  3. The general skewness of the distribution.
  4. The modality (i.e., the number of peaks) of the distribution. (correct answer)
Explanation: A standard boxplot displays the five-number summary (Min, Q1, Median, Q3, Max) and can show outliers. This information allows for the determination of variability (IQR, range) and general skewness (by comparing whisker lengths and the median's position within the box). However, a boxplot cannot show the shape of the distribution in detail, specifically its modality. For example, a bimodal (two-peaked) distribution could produce a boxplot that appears perfectly symmetric, hiding the true underlying structure.

Question 9

The table below shows the distribution of daily commute times for 200 employees. In which interval is the third quartile (Q3) of commute times located?

Commute Time (minutes)Number of Employees
0 - 1520
16 - 3050
31 - 4580
46 - 6040
61+10
  1. 16 - 30 minutes
  2. 31 - 45 minutes (correct answer)
  3. 46 - 60 minutes
  4. 61+ minutes
Explanation: This is a multi-step problem. First, find the position of the third quartile (Q3). Q3 is the value at the 75th percentile. The position is 0.75 × n = 0.75 × 200 = 150. We need to find the interval containing the 150th employee when they are ordered by commute time. Second, calculate the cumulative frequency for each interval:
  • 0 - 15: 20
  • 16 - 30: 20 + 50 = 70
  • 31 - 45: 70 + 80 = 150 The cumulative frequency reaches exactly 150 at the end of the '31 - 45 minutes' interval. This means the 150th employee's commute time falls within this interval. The 16-30 minute interval contains the median (100th employee), and the 46-60 minute interval begins with the 151st employee.

Question 10

A sales manager wants to identify the performance benchmark for the 80th percentile of sales. Given the 14 weekly sales figures (in thousands of dollars) below, what is this value?

Data: 22, 25, 25, 28, 30, 32, 34, 35, 38, 40, 41, 45, 48, 50

  1. $41,800
  2. $41,000
  3. $45,000 (correct answer)
  4. $48,000
Explanation: First, ensure the data is sorted (it is). There are n=14 data points. To find the 80th percentile (P80), we first find its location (L) in the dataset using the formula L = (n+1) * (p/100). L = (14+1) * (80/100) = 15 * 0.80 = 12. Since L is an integer, the 80th percentile is the 12th value in the sorted dataset. Counting from the beginning, the 12th value is 45. Therefore, the 80th percentile of sales is $45,000. Distractor A, $41,800, results from using an alternative formula L = n * (p/100) = 14 * 0.8 = 11.2, then interpolating between the 11th and 12th values. Distractors B and D are off-by-one errors, representing the 11th and 13th values, respectively.

Question 11

While compiling a report on executive salaries for a large corporation, a data analyst mistakenly enters a CEO's $2,500,000 salary as $25,000,000. Which of the following descriptive statistics would be least affected by this data entry error?

  1. Range
  2. Mean
  3. Standard Deviation
  4. Interquartile Range (correct answer)
Explanation: The interquartile range (IQR) is a 'robust' or 'resistant' measure of spread. This means it is not significantly influenced by extreme values or outliers. The error creates a large outlier at the upper end of the data. Since the IQR is calculated based on the 25th (Q1) and 75th (Q3) percentiles, it only considers the middle 50% of the data. A single outlier at the extreme high end will not change the values of Q1 and Q3. In contrast, the range (Max - Min), the mean (which uses every value), and the standard deviation (which is based on distance from the mean) are all highly sensitive to outliers and would be greatly affected by this error.

Question 12

An operations manager records the time (in seconds) for a process to be completed. The eight recorded times are: 110, 115, 120, 125, 130, 140, 150, 160. Using the Tukey hinge method (where the median is included in both halves if n is odd), what is the interquartile range (IQR) for this dataset?

  1. 27.5 (correct answer)
  2. 31.25
  3. 50.0
  4. 25.0
Explanation: With an even number of data points (n=8), we first find the median, which is the average of the 4th and 5th values: (125 + 130) / 2 = 127.5. Using the Tukey hinge method (also called the exclusive method), we find the median of the lower and upper halves of the data. The lower half of the data is [110, 115, 120, 125]. The first quartile (Q1) is the median of this set, which is (115 + 120) / 2 = 117.5. The upper half of the data is [130, 140, 150, 160]. The third quartile (Q3) is the median of this set, which is (140 + 150) / 2 = 145. The IQR is Q3 - Q1 = 145 - 117.5 = 27.5. Distractor B uses a different method involving interpolation. Distractor C is the range (160 - 110). Distractor D is the difference between the 6th and 3rd values (140 - 115).

Question 13

The five-number summaries for delivery times (in days) for two shipping companies, ShipFast and QuickHaul, are given below:

  • ShipFast: Min=2, Q1=4, Median=5, Q3=7, Max=12
  • QuickHaul: Min=3, Q1=5, Median=6, Q3=7, Max=9

Based on these summaries, which company offers more consistent delivery times for the central 50% of its shipments?

  1. ShipFast is more consistent because its overall range of delivery times is smaller.
  2. ShipFast is more consistent because its interquartile range is smaller.
  3. QuickHaul is more consistent because its interquartile range is smaller. (correct answer)
  4. QuickHaul is more consistent because its maximum delivery time is lower.
Explanation: Consistency, especially for the 'typical' or central part of a distribution, is best measured by the interquartile range (IQR). A smaller IQR indicates less variability and therefore greater consistency.
  • ShipFast's IQR = Q3 - Q1 = 7 - 4 = 3 days.
  • QuickHaul's IQR = Q3 - Q1 = 7 - 5 = 2 days. Since QuickHaul has a smaller IQR (2 < 3), its delivery times for the middle 50% of shipments are more consistent. Distractors A and D use other measures (range and maximum) which are less robust for judging the consistency of typical performance as they are affected by extreme values.

Question 14

An analyst uses a modified boxplot where the central box shows the standard Q1 to Q3 range, but the whiskers extend to the 5th and 95th percentiles. For a particular stock, this plot shows the box from $120 to $150 and whiskers extending from $100 to $180. Which of the following statements must be true based on this plot?

  1. The interquartile range for the stock price is $80.
  2. Approximately 90% of the observed stock prices fall between $100 and $180. (correct answer)
  3. Any stock price above $195 would be considered a statistical outlier.
  4. The median stock price must be exactly $135.
Explanation: The question requires careful interpretation of the modified plot's definition. The whiskers extend to the 5th and 95th percentiles. This means 5% of prices are at or below $100, and 95% of prices are at or below $180. The percentage of prices between these two values is 95% - 5% = 90%. Therefore, statement B is correct. Statement A is incorrect; the IQR is Q3 - Q1 = $150 - $120 = $30. The value 80istherangeofthewhiskers.StatementCisincorrectbecauseitappliesthe1.5×IQRrule(150+1.5∗30=195),whichtheproblemstatesisnotbeingused.StatementDisincorrectbecausethemediancanbeanyvaluebetweenQ1(80 is the range of the whiskers. Statement C is incorrect because it applies the 1.5 × IQR rule (150 + 1.5 * 30 = 195), which the problem states is not being used. Statement D is incorrect because the median can be any value between Q1 (120) and Q3 ($150); it is not necessarily the midpoint.

Question 15

A company's logistics department tracks the weight of all outgoing packages. As part of a new billing system, a standard 0.5 kg is added to the recorded weight of every package to account for packaging materials. How will this adjustment affect the third quartile (Q3) and the interquartile range (IQR) of the package weights?

  1. Q3 will increase by 0.5 kg, and the IQR will also increase by 0.5 kg.
  2. Q3 will increase by 0.5 kg, and the IQR will remain unchanged. (correct answer)
  3. Both Q3 and the IQR will remain unchanged because the adjustment is uniform.
  4. The IQR will remain unchanged, but the effect on Q3 cannot be determined without the original data.
Explanation: Adding a constant value to every data point in a set is a linear transformation that shifts the entire distribution. Measures of position or location (like the minimum, maximum, percentiles, quartiles, and median) will shift by that constant amount. Therefore, Q3 will increase by 0.5 kg. However, measures of spread or variability (like the range, IQR, and standard deviation) are unaffected by adding a constant, because the distance between any two data points remains the same. The new IQR will be (Q3_old + 0.5) - (Q1_old + 0.5) = Q3_old - Q1_old = IQR_old. Thus, the IQR will remain unchanged.

Question 16

The performance scores for a team of 19 employees are analyzed, and the interquartile range (IQR) is found to be 12 points. A 20th employee, a top performer, is added to the team. Their score is well above the original maximum score. How will the addition of this single high score most likely affect the IQR of the scores?

  1. The IQR will increase substantially because the overall range of the data has increased.
  2. The IQR will decrease because the total number of data points has increased.
  3. The IQR will be unchanged or change only minimally, as it is a resistant measure of spread. (correct answer)
  4. The IQR will become equal to the new median, as the distribution is now more skewed.
Explanation: The IQR is calculated from the first (Q1) and third (Q3) quartiles. For n=19, Q1 is the 5th value and Q3 is the 15th value. When a 20th data point is added at the high end, the new Q1 position is (20+1)/4 = 5.25 (between the 5th and 6th original values) and the new Q3 position is 3(20+1)/4 = 15.75 (between the 15th and 16th original values). Since the new extreme value does not change the values or positions of the original data points that determine the new quartiles, the IQR will remain the same or change very slightly due to interpolation. It is a robust statistic, meaning it is resistant to the influence of outliers. The range and standard deviation would be greatly affected, but not the IQR.

Question 17

A software company's data shows that the 25th percentile for project completion time is 45 days and the 75th percentile is 80 days. The company wants to give a bonus to project managers in the top 25% of performance (shortest completion times). What is the interquartile range (IQR), and what does it signify in this context?

  1. The IQR is 125 days; this is the total time span from the fastest to slowest projects.
  2. The IQR is 35 days; it represents the range of completion times for the middle 50% of projects. (correct answer)
  3. The IQR is 35 days; any project completed in less than 35 days qualifies for a bonus.
  4. The IQR is 45 days; this represents the cutoff time for the best-performing 25% of projects.
Explanation: The 25th percentile is Q1, and the 75th percentile is Q3. The interquartile range (IQR) is the difference between them: IQR = Q3 - Q1 = 80 - 45 = 35 days. The IQR represents the spread of the central or 'typical' half of the data. Therefore, it is the range of completion times for the middle 50% of projects. The bonus cutoff for the top 25% of performance (shortest times) would be the 25th percentile (Q1), which is 45 days, not the IQR. The IQR is a measure of spread, not a cutoff point.

Question 18

An analyst finds that for a dataset of customer ages, the 25th percentile is 35 years, the 50th percentile is 40 years, and the 75th percentile is 60 years. Which of the following statements is the most likely description of the age distribution?

  1. The distribution is symmetric, with the mean age approximately equal to 40 years.
  2. The distribution is negatively skewed, with a higher concentration of older customers.
  3. The distribution is positively skewed, with a tail of older customers extending to the right. (correct answer)
  4. The interquartile range is too large to make any valid conclusion about the distribution's skewness.
Explanation: Skewness can be assessed by comparing the distances between the median and the other quartiles. The median (50th percentile) is 40. The distance from the median to Q1 (25th percentile) is 40 - 35 = 5 years. The distance from the median to Q3 (75th percentile) is 60 - 40 = 20 years. Since the median is much closer to Q1 than to Q3, the upper half of the data is more spread out. This indicates a positive (or right) skew, meaning the distribution has a long tail towards the higher ages.

Question 19

A manufacturing quality control dataset contains 200 measurements. An analyst reports that the 90th percentile is 47.3 and the 95th percentile is 52.1. If the company wants to flag the top 12 measurements for detailed inspection, what percentile threshold should they use, and how many measurements fall between the 90th and 95th percentiles?

  1. 94th percentile threshold; exactly 10 measurements fall between the 90th and 95th percentiles (correct answer)
  2. 94th percentile threshold; exactly 12 measurements fall between the 90th and 95th percentiles
  3. 93rd percentile threshold; exactly 8 measurements fall between the 90th and 95th percentiles
  4. 88th percentile threshold; exactly 10 measurements fall between the 90th and 95th percentiles
Explanation: To flag the top 12 measurements out of 200, use 200-12=188, so 188/200=0.94 or 94th percentile. Between 90th and 95th percentiles: 95th percentile = 0.95×200 = 190th value, 90th percentile = 0.90×200 = 180th value. So 190-180 = 10 measurements fall between these percentiles. Choice B uses correct threshold but wrong count. Choice C uses wrong threshold (93rd would flag 14 measurements). Choice D uses completely wrong threshold.

Question 20

A boxplot visualizing customer service call durations shows that the rectangular box portion of the plot spans from 10 minutes to 35 minutes. This box visually represents:

  1. the interval containing the central 50 percent of the call durations. (correct answer)
  2. the range of all call durations, excluding any identified outliers.
  3. the call durations that occurred with the highest frequency.
  4. the value of the interquartile range, which is 25 minutes.
Explanation: The rectangular box in a boxplot is defined by the first quartile (Q1) at its lower end and the third quartile (Q3) at its upper end. By definition, Q1 is the 25th percentile and Q3 is the 75th percentile. The region between these two quartiles contains the middle 50% of the data. Therefore, the box represents the interval where the central half of the call durations lie. Distractor B describes the range covered by the whiskers (in the absence of outliers). Distractor C is incorrect as a boxplot does not show frequency (mode). Distractor D describes the width of the box (the IQR value), not what the box represents geographically on the axis.