Unit 1 · Topic 1.8 Beta

Graphical Representations of Summary Statistics for One Quantitative Variable

A boxplot pictures the five-number summary: a box from Q_1 to Q_3 split at the median, whiskers to the most extreme values inside the 1.5×IQR fences, and outliers marked separately.

Practice 3: Analyze DataPractice 4: Interpret Results

Part 1 · Hook

Why this matters

Twenty nurses report their commute times, and one lives 58 minutes away. A dotplot shows every nurse; a boxplot shows just five numbers and the outlier. That sounds like less, and it is, but five numbers are exactly what you need to see the center, the spread of the middle half and the skew at a glance, and to compare groups side by side.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. For 6, 9, 11, 12, 13, 14 what is the median?

  1. 11.5
  2. 11
  3. 12
  4. 10.83
Show the answer

n = 6 is even: average the 3rd and 4th values, (11 + 12)/2.

  • Correct: 11.5:
  • 11:
  • 12:
  • 10.83:

2. With Q1=4Q_1 = 4 and Q3=10Q_3 = 10, what is the upper fence?

  1. 19
  2. 16
  3. 14
  4. 25
Show the answer

IQR=6\text{IQR} = 6, so 10+1.5(6)=1910 + 1.5(6) = 19.

  • Correct: 19:
  • 16:
  • 14:
  • 25:

Part 4 · See it

See it first

Commute times of 20 workers in minutes. Top: a dotplot from 8 to 62. Bottom: the boxplot of the same data on the same scale. The box runs from Q1 = 14.5 to Q3 = 28.5 with the median at 20.5. The upper fence is at 49.5, so 62 is an outlier, drawn as an open circle, and the right whisker stops at 38, the largest value inside the fence. The left whisker reaches the minimum, 8.
The commute times of 20 workers as a dotplot and as a boxplot on the same scale. The outlier at 62 is beyond the fence at 49.5, so the whisker stops at 38. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Find the minimum, Q1Q_1, the median, Q3Q_3 and the maximumthese five numbers split the ordered data into four quarters
  2. Draw a box from Q1Q_1 to Q3Q_3 with a line at the medianthe box holds the middle half of the data, and its length is the IQR
  3. Compute the fences Q1−1.5×IQRQ_1 - 1.5 \times \text{IQR} and Q3+1.5×IQRQ_3 + 1.5 \times \text{IQR}values beyond them are marked as outliers, and each whisker stops at the most extreme value inside its fence
  4. The picture keeps only quarters and outliersit shows center, spread and skew, but it cannot show peaks, gaps or the mean

Part 6 · Key ideas

Key ideas

  • The five-number summary is minimum, Q1Q_1, median, Q3Q_3, maximum. Each quarter of the data lies between two neighbors.
  • A boxplot draws a box from Q1Q_1 to Q3Q_3 with a line at the median, whiskers to the most extreme values inside the fences, and outliers as separate points.
  • Skew shows as a longer whisker and a median closer to one end of the box: skewed right when the right side is stretched.
  • A boxplot cannot show modality, gaps or clusters, so never describe a boxplot as having one or two peaks.

Part 7 · Misconception

A common mistake

The wrong idea: The right whisker always goes to the maximum value.

What actually happens: A whisker stops at the most extreme value inside the fence. If the maximum is beyond Q3+1.5×IQRQ_3 + 1.5 \times \text{IQR}, it is plotted as an outlier and the whisker is shorter.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

Commute times of 20 nurses

The one-way commute times, in minutes, of 20 nurses at a clinic, in order: 6, 9, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 28, 31, 34, 37, 58 (invented data). The boxplot is drawn from these values.
010203040506070minutes

1. What is the first quartile, Q1Q_1, of the commute times, in minutes?

Type a number, in minutes. A table answer and a calculator answer are both accepted.

Show the answer

Lower half: the 10 smallest values, 6 to 19. Its median is 13+142=13.5\frac{13 + 14}{2} = 13.5 minutes.

  • Answer: 13.5 minutes

2. What is the upper fence, Q3+1.5×IQRQ_3 + 1.5 \times \text{IQR}, in minutes?

Type a number, in minutes. A table answer and a calculator answer are both accepted.

Show the answer

Q3=26+282=27Q_3 = \frac{26 + 28}{2} = 27, IQR=27−13.5=13.5\text{IQR} = 27 - 13.5 = 13.5, so the fence is 27+1.5(13.5)=47.2527 + 1.5(13.5) = 47.25 minutes.

  • Answer: 47.25 minutes

3. Why does the right whisker end at 37 minutes and not at the maximum, 58?

  1. Whiskers stop at Q3Q_3 plus the IQR, which is about 40.5
  2. 58 is beyond the upper fence, so it is drawn as an outlier
  3. The whisker is drawn to the same length as the left whisker
  4. 37 is the 95th percentile of the 20 commute times
Show the answer

Whiskers go to the most extreme values that are not outliers. Outliers, beyond the fences, are marked separately.

  • Whiskers stop at Q3Q_3 plus the IQR, which is about 40.5: The whisker stops at an actual data value, the largest one inside the fence.
  • Correct: 58 is beyond the upper fence, so it is drawn as an outlier: Right: a whisker reaches the most extreme value inside the fences, here 37; 58 is above 47.25.
  • The whisker is drawn to the same length as the left whisker: Whiskers can have different lengths; that is how a boxplot shows skew.
  • 37 is the 95th percentile of the 20 commute times: Whiskers are not set at a percentile; they end at the most extreme value inside the fences.

4. About what percent of the nurses have commutes longer than Q3Q_3, 27 minutes?

  1. About 75%
  2. About 50%
  3. About 5%
  4. About 25%
Show the answer

The five numbers split the data into quarters: about 25% of the values lie above Q3Q_3. Here 5 of 20 nurses do.

  • About 75%: About 75% are at or below Q3Q_3; the part above is a quarter.
  • About 50%: The box holds the middle half; the upper whisker and outliers hold about a quarter.
  • About 5%: The outlier is one of five values above 27 minutes.
  • Correct: About 25%: Right: Q3Q_3 has about three quarters of the data at or below it, so about a quarter above.

5. A student says, "The boxplot shows that the commute times have one peak." What is wrong with this claim?

  1. A boxplot shows five numbers and outliers, so it cannot show peaks
  2. The boxplot shows two peaks, one on each side of the median
  3. Commute times are categorical, so they have no peaks
  4. The peak of a boxplot is the outlier at 58 minutes
Show the answer

Never read modality (one peak, two peaks) from a boxplot. Use a dotplot or histogram for shape details.

  • Correct: A boxplot shows five numbers and outliers, so it cannot show peaks: Right: a boxplot hides how values are spread inside the box and whiskers, so it cannot show modality, gaps or clusters.
  • The boxplot shows two peaks, one on each side of the median: A boxplot shows no peaks at all; the halves of the box are quarters of the data, not peaks.
  • Commute times are categorical, so they have no peaks: Commute times are quantitative. The issue is what a boxplot can show.
  • The peak of a boxplot is the outlier at 58 minutes: An outlier is a single far value, not a peak.

6. What is the five-number summary of 3, 5, 7, 8, 12, 13, 14?

  1. 3, 6, 8, 12.5, 14
  2. 3, 5, 8, 13, 14
  3. 3, 5, 8.86, 13, 14
  4. 3, 7, 8, 12, 14
Show the answer

Min 3; Q1Q_1 = median of 3, 5, 7 = 5; median 8; Q3Q_3 = median of 12, 13, 14 = 13; max 14.

  • 3, 6, 8, 12.5, 14: That keeps the median, 8, in both halves. With n odd, leave it out: lower half 3, 5, 7.
  • Correct: 3, 5, 8, 13, 14: Right: median 8; the lower half 3, 5, 7 has median 5; the upper half 12, 13, 14 has median 13.
  • 3, 5, 8.86, 13, 14: 8.86 is the mean. The five-number summary uses the median, 8.
  • 3, 7, 8, 12, 14: The quartiles are the medians of each half, not the neighbors of the median.

7. In a boxplot, the median is close to Q1Q_1 and the right whisker is much longer than the left. What is the most likely shape?

  1. Skewed left
  2. Skewed right
  3. Roughly symmetric
  4. Bimodal
Show the answer

Long right whisker and the median near Q1Q_1: the upper values are spread out, so the distribution is skewed right.

  • Skewed left: A median near Q1Q_1 means the lower values are bunched, and the upper values stretch out to the right.
  • Correct: Skewed right: Right: the upper half of the data stretches farther, a long tail toward the larger values.
  • Roughly symmetric: Symmetry would show a median near the middle of the box and whiskers of similar length.
  • Bimodal: A boxplot cannot show peaks.

Part 9 · Summary

Summary

A boxplot pictures the five-number summary: a box from Q1Q_1 to Q3Q_3 split at the median, whiskers to the most extreme values inside the 1.5×IQR1.5 \times \text{IQR} fences, and outliers marked separately. It shows center, the spread of the middle half and skew, and about a quarter of the data lies in each section, but it hides peaks and gaps.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

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