represent a data set with a dot plot, a histogram and a box plot, and explain what each display reveals about the distribution.
Instruction
A statistical question is one you expect to answer with data that vary. “How long is my commute?” has one answer. “How long do students at our school take to get to school?” has many answers, and the interesting part is the pattern they make. That pattern is called a distribution.
Suppose 15 students report their one-way commute in minutes (illustrative data):
5, 8, 10, 10, 12, 15, 15, 18, 20, 20, 22, 25, 30, 35, 55
Dot plot. Draw a number line from 0 to 60 and put one dot above each value, stacking repeated values. A dot plot keeps every data point, so it works well for small data sets. You can see a cluster between 10 and 25 minutes and one student far to the right at 55.
Histogram. Group the values into equal-width intervals called bins and draw a bar whose height is the count in each bin. With bins of 10 minutes (0 to under 10, 10 to under 20, and so on) the counts are 2, 6, 4, 2, 0 and 1. A histogram hides individual values but shows the overall shape of large data sets. Here the shape is skewed right: most values are low, with a long tail toward high values. Different bin widths can make the same data look different, so always state the bins you chose.
Box plot. A box plot is built from the five-number summary. Order the data. The median is the middle value: with 15 values it is the 8th, which is 18. The first quartile Q1 is the median of the lower seven values (5 through 15), which is 10. The third quartile Q3 is the median of the upper seven values (20 through 55), which is 25. The minimum is 5 and the maximum is 55.
The interquartile range is IQR = Q3 − Q1 = 25 − 10 = 15 minutes. A common rule flags a value as an outlier if it is more than 1.5 × IQR beyond a quartile. The upper fence is 25 + 1.5(15) = 47.5, so 55 is an outlier. In the box plot below, the whisker stops at 35, the largest value that is not an outlier, and 55 is drawn as a separate point.
Read the box plot as four quarters of the data: about a quarter of students commute 5-10 minutes, a quarter 10-18, a quarter 18-25 and a quarter 25 minutes or more. The long right whisker and the outlier show the same right skew that the histogram showed.
Choosing a display. Use a dot plot when you have few values and want to see each one. Use a histogram for large data sets when shape matters. Use box plots to compare groups side by side on the same scale. Whatever you choose, label the axis with the quantity and its unit (“commute time, minutes”) and start the scale where it lets the reader see the data honestly.
Try it. Choose one data set; all three earn full credit.
- Set A — the commute times above. 5, 8, 10, 10, 12, 15, 15, 18, 20, 20, 22, 25, 30, 35, 55 minutes. Nothing to look up or collect. Its summary is worked out for you above, so use it to check that you can read a display you already understand.
- Set B — a second school (illustrative data). Fifteen students at a rural high school where nearly everyone rides a bus: 12, 15, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 36, 38, 42 minutes. Check your work: the five-number summary is 12, 20, 26, 34, 42, so the IQR is 14 minutes, the fences are −1 and 55, and no value is an outlier. With 10-minute bins the counts are 0, 3, 6, 5, 1, 0.
- Set C — the public record. At Census QuickFacts (census.gov/quickfacts) look up Mean travel time to work (minutes), workers age 16 years+ for fifteen California counties you choose, and use those fifteen numbers. Write down the counties and the date you looked them up. Each value here describes a whole county, not one worker.
Make all three displays for the set you chose, in CODAP or by hand on paper. Then write two sentences: one thing the histogram shows better than the box plot, and one thing the box plot shows better than the histogram.
Formative check
Work through these before moving on. They are not graded — they tell you, and your teacher, whether the standard below has landed yet.
In the commute data, which display would make the 55-minute value easiest to spot as unusual?
For the data 3, 4, 7, 8, 10, 12, 15 (seven values), what are Q1, the median and Q3?
For the commute data, Q1 = 10 and Q3 = 25, so the IQR is minutes and the upper outlier fence is minutes.
Simulation & tools
Three displays of one data set in CODAP
Set up. Open CODAP and choose Tables > New. Name the attribute commute_min and type the 15 values of the set you chose (Set A, B or C from the mini-lecture). Drag the attribute name onto the horizontal axis of a new graph: that is a dot plot.
Observe and record. (1) Use the graph’s ruler menu to show the median and a box plot, and record Q1, the median and Q3 that CODAP reports. For Set A they should be 10, 18 and 25; for Set B, 20, 26 and 34; for Set C, work them out by hand and compare. (2) Use the configuration menu to Group into Bins and then Fuse Dots into Bars to make a histogram. Record the counts with 10-minute bins, then change the bin width to 5 minutes and describe how the shape looks different.
Answer. Which display would you use to show a reader that most commutes are short but a few are long, and why?
No device? Draw the dot plot, the 10-minute histogram and the box plot on the printed number lines and answer the same question.
Open in a new tab ↗ · CODAP, Concord Consortium · MIT open source
How confident are you that you can draw a dot plot, histogram and box plot for a data set and say what each one shows?
Practice
Work these on paper or in your notebook, then open Check your answer. Aim for all of Fluency and Application; try at least one Challenge.
Printable version: this unit’s practice workbook (PDF)
Fluency
Build speed and accuracy with the core skill.
- For the data 11, 14, 16, 18, 19, 20, 24, 29: find the mean, median, range and interquartile range. (Find the quartiles as the medians of the lower and upper halves, leaving out the overall median when the count is odd.)
Check your answer
Answer: mean = 18.88; median = 18.5; range = 18; IQR = 7.0 (Q1 = 15.0, Q3 = 22.0) - For the data 7, 9, 13, 31, 32, 33, 34, 38, 39: find the mean, median, range and interquartile range. (Find the quartiles as the medians of the lower and upper halves, leaving out the overall median when the count is odd.)
Check your answer
Answer: mean = 26.22; median = 32.0; range = 32; IQR = 25.0 (Q1 = 11.0, Q3 = 36.0) - For the data 4, 11, 24, 25, 28, 30, 31, 32, 39, 40: find the mean, median, range and interquartile range. (Find the quartiles as the medians of the lower and upper halves, leaving out the overall median when the count is odd.)
Check your answer
Answer: mean = 26.40; median = 29.0; range = 36; IQR = 8.0 (Q1 = 24.0, Q3 = 32.0) - For the data 7, 9, 19, 31, 31, 39, 40: find the mean, median, range and interquartile range. (Find the quartiles as the medians of the lower and upper halves, leaving out the overall median when the count is odd.)
Check your answer
Answer: mean = 25.14; median = 31.0; range = 33; IQR = 30.0 (Q1 = 9.0, Q3 = 39.0) - For the data 8, 8, 10, 15, 26, 26, 35, 40: find the mean, median, range and interquartile range. (Find the quartiles as the medians of the lower and upper halves, leaving out the overall median when the count is odd.)
Check your answer
Answer: mean = 21.00; median = 20.5; range = 32; IQR = 21.5 (Q1 = 9.0, Q3 = 30.5) - For the data 8, 9, 9, 16, 23, 26, 30, 37, 39, 40: find the mean, median, range and interquartile range. (Find the quartiles as the medians of the lower and upper halves, leaving out the overall median when the count is odd.)
Check your answer
Answer: mean = 23.70; median = 24.5; range = 32; IQR = 28.0 (Q1 = 9.0, Q3 = 37.0) - For the data 12, 15, 15, 18, 20, 22, 24, 25, 48, find the five-number summary, the IQR and the outlier fences, and list any outliers.
Check your answer
Answer: Min 12, Q1 15, median 20, Q3 24.5, max 48; IQR 9.5; fences 0.75 and 38.75; outliers: 48Fences are Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. - For the data 3, 8, 9, 10, 11, 12, 13, 14, 30, find the five-number summary, the IQR and the outlier fences, and list any outliers.
Check your answer
Answer: Min 3, Q1 8.5, median 11, Q3 13.5, max 30; IQR 5; fences 1 and 21; outliers: 30Fences are Q1 − 1.5 × IQR and Q3 + 1.5 × IQR.
Application
Use the skill in context. Show your reasoning.
- A sixteenth student joins the Lesson 1.1 commute survey with a 42-minute commute. Find the new five-number summary and decide whether 42 or 55 is an outlier.
Check your answer
Answer: Min 5, Q1 11, median 19, Q3 27.5, max 55; IQR 16.5; upper fence 52.25. 55 is an outlier; 42 is not an outlier.With 16 values, the median is the mean of the 8th and 9th values; each half has 8 values. - Daily highs (°F) for 14 July days in an inland California city (illustrative): 92, 95, 97, 99, 101, 103, 104, 98, 96, 94, 100, 105, 108, 97. Make a frequency table with bins 90-94, 95-99, 100-104 and 105-109, and describe the shape.
Check your answer
Answer: Counts 2, 6, 4, 2; the distribution is mound-shaped, peaking in the 95-99 bin, with a slightly longer tail to the right.Count each value into exactly one bin, then read the heights of the bars left to right. - Five-number summaries of daily air-quality index (AQI) values for one summer month (illustrative): City A (22, 35, 42, 51, 88); City B (30, 38, 44, 49, 63). Which city has the larger IQR, and is either maximum an outlier?
Check your answer
Answer: City A: IQR 16, upper fence 75, so 88 is an outlier. City B: IQR 11, upper fence 65.5, so 63 is not.IQR = Q3 − Q1; upper fence = Q3 + 1.5 × IQR.
Challenge
Stretch problems. Expect to think before you write.
- Write a data set of seven whole numbers with median 12, IQR 6 and range 20. Check your set.
Check your answer
Answer: One answer: 4, 9, 10, 12, 14, 15, 24 (Q1 = 9, Q3 = 15, range 20).Fix the median first, then choose the 2nd and 6th values to set Q1 and Q3, and the extremes to set the range.
Review
Keep earlier skills sharp.
- Solve: −7x + 5 = 47
Check your answer
Answer: x = −6Subtract 5 from both sides: −7x = 42. Divide by −7. - Compute −3/4 + 5/6.
Check your answer
Answer: 1/12Common denominator 12: −9/12 + 10/12. - 18 minutes is what percent of 24 minutes?
Check your answer
Answer: 75%18 ÷ 24 = 0.75.