Describing a distribution
You have a plot and a page of summaries. What do you write so that someone who never saw the data knows what it says, and can check it?
Where you are 13 in Center and spread
When you reach for it
A Statistical questions and variability such as “How long are the lizards at this site?” is answered by data, but a list of fifteen numbers or a dot plot does not answer it by itself. Someone has to say, in sentences, what the data shows, so that a reader who never saw the plot can picture it and check it. That account is a description of the distribution: the context, the shape, a center with a matching spread, and anything unusual, each in the units of the setting. It comes at the end of any question about one variable. Setting two groups side by side is Comparing two distributions.
This lesson takes two days. Part 1 builds a description from a display and its summaries. Part 2 judges and improves short descriptions, and starts from raw data.
The procedure
Part 1: the pieces of a description
| Piece | What it says |
|---|---|
| Context | what was measured, on which individuals, how, in what units, and how many |
| Shape | symmetric, skewed left or right, uniform, one peak or two |
| Center and spread | one matching pair, chosen for the shape |
| Unusual features | outliers tested with the fences, gaps, clusters |
| The answer | a last sentence back to the question |
Context. Name the individuals and the variable (Individuals, variables and units), with the unit: “11” means nothing, “11 cm, snout to tail tip” does. Say how the values were measured, since a lizard measured to the base of its tail and one measured to the tip give different numbers. Say how many there are: the count tells the reader how far to trust the shape.
Shape. Read it from the display, in the words of Symmetric and skewed distributions and Uniform and bimodal shapes. The mean and the median are a cross-check: when they are well apart, the data probably leans toward the mean.
Center and spread. The shape chooses the pair (Mean, median and skew: choosing the center). Skewed data, or data with outliers: the median with the IQR, which are resistant and so stay with the bulk of the values. Roughly symmetric data with no outliers: the mean with the MAD, which use every value. Keep the pair together. The MAD is an average distance from the mean, so beside a median it measures spread about a different number; the IQR comes from the quartiles around the median.
Unusual features. Call a value an outlier only after testing it with the 1.5 IQR rule; a value that looks far away is often inside the fences. A gap is a stretch of the scale with no values, and clusters are groups with gaps between them. Name each with its value and unit, and offer a reason only as a possibility.
The answer. The last sentence answers the question in its own words.
Part 2: checking a description, and starting from raw data
A good description is checkable: every claim in it is something the data can confirm or refute. To judge one, test every claim. Each number should recompute and carry a unit. The shape word should match the display and the order of the mean and median. The spread should belong to the center. Each “outlier”, and each “no outliers”, should survive the fences. And the question should be answered. Every number can be right and the description still mislead, through the wrong pair or a missing outlier.
From raw data, sort first: the count, the median, the quartiles and the shape are all read from the sorted list. Then find the summaries the shape calls for, and the fences.
Worked examples
Examples 1 to 4 go with Part 1, and Examples 5 to 7 with Part 2.
Lizard lengths: skewed, with an outlier
Nerea caught 15 lizards at a reptile survey and measured each with a tape measure, from the tip of the snout to the tip of the tail, to the nearest centimeter (Figure 1). Her summaries: mean 13 cm, MAD about 3.7 cm, median 11 cm, Q_1 = 10 cm and Q_3 = 14 cm. Write a description that answers “How long are the lizards at this site?”
Shape: the dots crowd between 8 and 14, then thin out to the right at 17, 19 and, far off, 30. That is skewed right, and the mean sitting above the median agrees.
Center and spread: for skewed data, the median and the IQR. The tail pulls the mean up to a length greater than 10 of the 15 lizards, so it is not a typical length. The IQR is 14 - 10 = 4 cm.
Unusual features: 1.5 \times 4 = 6, so the fences are 10 - 6 = 4 and 14 + 6 = 20 cm. The 30 is beyond 20, so it is an outlier; the 19 is inside, so it is not. No lizard lies between 19 and 30 cm, a gap.
Description. Nerea measured 15 lizards at the survey site, from snout to tail tip, to the nearest centimeter. The lengths are skewed right. The median length is 11 cm, and the middle half of the lizards are 10 to 14 cm long, an IQR of 4 cm. One lizard, at 30 cm, is an outlier, with a gap from 19 to 30 cm between it and the rest; it may be a different kind of lizard. A typical lizard at this site is about 11 cm long.
Sled run times: roughly symmetric
Nello timed 20 runs down the sled hill with a stopwatch, from the push at the top to the flag at the bottom, to the nearest second (Figure 2). His summaries: mean 24 seconds, MAD 1.4 seconds, median 24 seconds, Q_1 = 23 and Q_3 = 25 seconds. Describe the run times.
Shape: one peak, at 24 seconds, with the counts falling away on both sides, 4 seconds to the left and 3 to the right. Roughly symmetric, and the mean and the median agree. So the mean and the MAD, which is measured from the mean.
Unusual features: the IQR is 25 - 23 = 2 and 1.5 \times 2 = 3, so the fences are 23 - 3 = 20 and 25 + 3 = 28. The quickest run, 20 seconds, sits exactly on the lower fence, and an outlier has to lie beyond a fence, so there are none. Every whole second from 20 to 27 has at least one run, so there are no gaps.
Description. Nello timed 20 runs down the sled hill with a stopwatch, to the nearest second. The times are roughly symmetric, with one peak at 24 seconds. The mean time is 24 seconds, and the runs are on average 1.4 seconds away from it (the MAD). The times go from 20 to 27 seconds, with no gaps and no outliers. A typical run takes about 24 seconds, give or take a second or two.
Ages at the climbing wall: two clusters
Norberta copied the ages, in whole years, from the sign-in sheet of the 16 climbers at a climbing wall’s family session (Figure 3). The mean age is about 23.6 years and the median 22.5 years. Describe the ages.
The dots form two clusters, eight climbers aged 7 to 11 and eight aged 34 to 43, with nobody aged 12 to 33: the data is bimodal, most likely children and their parents.
Both centers fall in the gap. The mean, \displaystyle \frac{378}{16} = 23.625, and the median, halfway between the eighth age, 11, and the ninth, 34, are ages that no climber had. So describe each cluster. The children’s ages, sorted, are 7, 8, 8, 9, 9, 9, 10, 11, with median 9; the adults’ are 34, 36, 37, 38, 38, 40, 41, 43, with median 38.
Description. The sign-in sheet gives the ages of 16 climbers at the family session, in whole years. They form two clusters: 8 children aged 7 to 11, with a median of 9 years, and 8 adults aged 34 to 43, with a median of 38 years. Nobody was aged 12 to 33. The overall mean, about 23.6 years, and median, 22.5 years, both fall in that gap, so neither describes a typical climber.
Lizard masses from a five-number summary
Nadja weighed 40 lizards at the reptile survey on a digital scale, to the nearest gram, and kept only the five-number summary: minimum 10, Q_1 = 16, median 19, Q_3 = 24 and maximum 41 grams. Describe the masses as fully as the summary allows, and say what it cannot tell you.
SolutionShape, from the parts of the box plot: the halves of the box are 19 - 16 = 3 and 24 - 19 = 5 grams wide, and the whiskers are 16 - 10 = 6 and 41 - 24 = 17 grams long. Everything stretches farther right: skewed right. That calls for the median and the IQR, 24 - 16 = 8 grams, which the summary gives.
Outliers: 1.5 \times 8 = 12, so the fences are 16 - 12 = 4 and 24 + 12 = 36. The maximum, 41, is beyond 36, so at least the heaviest lizard is an outlier. The minimum, 10, is above 4.
Description. Nadja weighed 40 lizards to the nearest gram. The masses are skewed right, with a median of 19 grams; the middle half weighed 16 to 24 grams, an IQR of 8 grams. The heaviest lizard, at 41 grams, is an outlier.
The summary cannot give the mean, the MAD, how many outliers there are, or any gaps, clusters or peaks inside the box or along a whisker.
Judging three descriptions
Judge three descriptions of Nerea’s lizards (Example 1) against the data.
- (a) “The lizards averaged 13, with a MAD of 3.7.”
- (b) “Nerea’s 15 lizards were 8 to 30 cm long, with a median of 11 cm and a MAD of 3.7 cm.”
- (c) “The 15 lizards, measured from snout to tail tip, are skewed right, with a median of 11 cm and an IQR of 4 cm. The 30 cm lizard is an outlier.”
(a) The numbers are right, but there is no unit, count, method or shape, and the pair is wrong for skewed data with an outlier: the mean, 13, is greater than 10 of the 15 lengths.
(b) The range 8 to 30 is set by an outlier that is never named, and the median is paired with the MAD, which measures distances from the mean, not from the median. There is no shape.
(c) Every claim checks, and the pair suits the shape. It lacks the gap from 19 to 30 cm and a sentence for the question; with those it is close to the description in Example 1.
From raw data: routes finished
A climbing wall’s beginners’ evening has 10 routes, and the instructor ticks a card for each route a climber finishes. Miroslav counted the ticks for 13 climbers: 10, 9, 10, 8, 10, 9, 4, 7, 9, 10, 6, 9, 8. Describe how many routes the climbers finished.
SolutionSorted: 4, 6, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10. The count, 13, is odd, so the median is the seventh value, 9 routes, and the quartiles come from the six values on each side: \displaystyle Q_1 = \frac{7 + 8}{2} = 7.5 and \displaystyle Q_3 = \frac{10 + 10}{2} = 10. The IQR is 2.5 routes.
The values pile up at 9 and 10, against the ceiling of 10 routes, and trail off left as far as 4: skewed left. The mean, \displaystyle \frac{109}{13}, about 8.4, is below the median, as the shape predicts. So use the median with the IQR.
Is the 4 an outlier? 1.5 \times 2.5 = 3.75, so the lower fence is 7.5 - 3.75 = 3.75. The 4 is just inside it: the end of the tail, not an outlier. No count can be above 10.
Description. Miroslav counted the routes finished, out of 10, by each of 13 climbers at the beginners’ evening, from the instructor’s tick card. The counts are skewed left. The median is 9 routes, and the middle half of the climbers finished 7.5 to 10 routes, an IQR of 2.5 routes. Four climbers finished all 10. The lowest count, 4, is not an outlier. A typical beginner finished about 9 of the 10 routes.
Two descriptions, one data set
On an icy morning Nadja and Nello timed 10 sled runs with a stopwatch, in seconds: 21, 19, 31, 20, 18, 22, 21, 19, 23, 20. Nadja wrote: “Skewed right with one outlier, 31 seconds. Median 20.5 seconds, IQR 3 seconds.” Nello wrote: “Roughly symmetric. Mean 21.4 seconds, MAD 2.36 seconds. No outliers.” Check every number, then decide which description is right.
SolutionSorted: 18, 19, 19, 20, 20, 21, 21, 22, 23, 31. The median is \displaystyle \frac{20 + 21}{2} = 20.5, the halves give Q_1 = 19 and Q_3 = 22, and the IQR is 3. The total is 214, so the mean is 21.4, and the distances from it total 23.6, a MAD of 2.36. Every number is right.
The claims are not. The upper fence is 22 + 4.5 = 26.5 and 31 is beyond it, so “no outliers” is false. Nine runs sit between 18 and 23 and the tenth stands far off to the right, so “skewed right” fits better than “roughly symmetric”. The outlier pulls the mean above 7 of the 10 runs, and its own distance, 9.6, is a large share of the MAD’s total.
Nadja’s description is the right one. It still needs the count, how the runs were timed, and a last sentence: a typical run that morning took about 20.5 seconds.
Boundaries
- This data, not everyone. A description reports the values collected. Whether 15 lizards stand for all the lizards in a region depends on how they were chosen.
- Few values. With about six values or fewer, say the data is too small to call a shape.
- A summary hides things. A box plot cannot show gaps, clusters, peaks, the mean, or how many outliers there are. A histogram hides the exact values, so the median can only be placed in a bin (Reading a histogram).
- Two clusters. One center for the whole set can land where there are no values; describe each cluster.
- Symmetric with no outliers. Either pair is honest; the mean with the MAD is usual. If the question needs a total, the mean is needed whatever the shape.
- Numbers only. Shape, center and spread need a numerical variable. A categorical one, such as the kind of lizard, is described by the count in each category.
Where this goes next
Comparing two distributions writes two descriptions on the same scale and sets them side by side: which group is typically larger, which is more spread out, and by how much. The matching pair matters even more there, since one group’s median against another’s mean compares two kinds of center.
Exercises
Part 1: from a display and its summaries
- Miroslav timed 12 runs on the short slope of the sled hill with a stopwatch, to the nearest second: 1 run of 11 seconds, 2 of 12, 3 of 13, 3 of 14, 2 of 15 and 1 of 16. His summaries: mean 13.5 seconds, MAD about 1.2 seconds, median 13.5 seconds, Q_1 = 12.5 and Q_3 = 14.5 seconds. Write a description of the run times.
- Nello measured 11 lizards from snout to tail tip, to the nearest centimeter. Sorted, the lengths are 12, 13, 13, 13, 14, 14, 15, 16, 18, 20, 22, and the mean is about 15.5 cm. Find the median and the IQR, test for outliers, and write the description.
- Over 16 days a climbing wall counted its climbers: 2 days each with 21, 22, 23, 24, 25, 26, 27 and 28 climbers. Name the shape, find the mean and the MAD, and write the description.
- Thirty runs on the long sled hill, timed with a stopwatch, have the five-number summary 41, 48, 51, 54, 58 seconds. Describe the times, test for outliers, and say what the summary cannot show.
- Norberta timed 22 climbers on the long route and kept only a histogram: 4 climbers took 20 to 30 seconds, 9 took 30 to 40, 6 took 40 to 50, 2 took 50 to 60, none took 60 to 70 and 1 took 70 to 80. Each bin includes its left edge and not its right. Which bin holds the median? Describe the shape and any unusual features. Why can you not give the median exactly, or say whether the slowest climber is an outlier?
Part 2: judging a description
- “The sled runs took about 30.” List what this description is missing.
- Norberta timed 12 climbers on the speed route with a stopwatch, in seconds: 30, 31, 33, 34, 35, 36, 36, 38, 40, 42, 45, 52. She wrote: “Twelve climbers took 30 to 52 seconds, with a median of 36 seconds. The 52 second climb is an outlier.” Check each claim and improve the description.
- Nadja weighed 9 young lizards, to the nearest gram: 10, 12, 13, 13, 14, 15, 16, 18, 33. She wrote: “Nine lizards, median mass 14 grams, MAD about 4.2 grams.” Both numbers are right. What is wrong, and what should replace the MAD?
- A description says: “The runs on the long track are roughly symmetric, with a mean of 70 seconds and a median of 52 seconds.” Can all of it be right? Explain.
Part 2: from raw data
- Miroslav counted how many runs each of 10 children made down the sled hill in one hour: 7, 9, 6, 7, 8, 5, 7, 8, 6, 7. Describe the counts.
- Nerea measured 11 newly hatched lizards from snout to tail tip, in millimeters: 53, 45, 90, 40, 62, 48, 43, 57, 46, 50, 42. Describe the lengths.
- Nello measured how far 8 sleds slid past the finish flag, in meters: 6.5, 7, 8, 7.5, 8.5, 7, 9, 6.5. Describe the distances.
Explaining
- Give 7 lizard lengths, in whole centimeters, whose description would say: median 12 cm, IQR 4 cm, and exactly one outlier. Check your answer.
- Give 8 sled run times, in whole seconds, that are symmetric with a mean of 30 seconds and a MAD of 2 seconds.
- Explain why a description that gives the median should give the IQR, not the MAD, as its spread.
- One survey measured lizards from the snout to the base of the tail, another from the snout to the tip of the tail. Explain why a description must say how the values were measured.
- Two descriptions both say “skewed right, median 11 cm”. One comes from 6 lizards and the other from 60. Explain why a description should say how many values there are.
- A description made from a box plot says “there are no gaps”. Explain why the box plot cannot support that claim.
Answers
- Check: the total is 162 and \displaystyle \frac{162}{12} = 13.5; the distances total 14, and \displaystyle \frac{14}{12} is about 1.2. The IQR is 2, so the fences are 9.5 and 17.5: no outliers. Description: 12 stopwatch runs on the short slope are symmetric about 13.5 seconds, with a mean of 13.5 seconds and a MAD of about 1.2 seconds, from 11 to 16 seconds, no gaps or outliers.
- Median 14 cm; halves 12, 13, 13, 13, 14 and 15, 16, 18, 20, 22 give Q_1 = 13 and Q_3 = 18, so the IQR is 5 cm. Fences 13 - 7.5 = 5.5 and 18 + 7.5 = 25.5: no outliers. Crowded from 12 to 16, thinning to 22, mean above median: skewed right. A typical lizard of the 11 is about 14 cm long, the middle half 13 to 18 cm.
- Uniform: every count from 21 to 28 appears twice, no peak. Total 392, so the mean is \displaystyle \frac{392}{16} = 24.5 climbers; the distances 3.5, 2.5, 1.5, 0.5 each appear four times, total 32, so the MAD is 2 climbers. Daily counts ran evenly from 21 to 28, no gaps or outliers.
- Box halves 3 and 3, whiskers 7 and 4: roughly symmetric, the left whisker a little longer. Median 51 seconds, IQR 6 seconds. Fences 48 - 9 = 39 and 54 + 9 = 63, and 41 and 58 are inside: no outliers. It cannot show the mean, the MAD, gaps, clusters or peaks.
- The median averages the 11th and 12th values; the running totals are 4 and then 13, so both lie in the bin 30 to 40 seconds. Skewed right; the empty bin 60 to 70 is a gap, with one climber apart at 70 to 80. The bins hide the exact values, so the median, the quartiles and the fences are unknown.
- The count, how the runs were timed, the unit, which center “about 30” is, a spread, the shape, and any unusual runs.
- Count, range and median, \displaystyle \frac{36 + 36}{2} = 36, are right. \displaystyle Q_1 = \frac{33 + 34}{2} = 33.5 and \displaystyle Q_3 = \frac{40 + 42}{2} = 41, so the IQR is 7.5 and the upper fence 41 + 11.25 = 52.25: the 52 is just inside, not an outlier. Improved: twelve stopwatch times, 30 to 52 seconds, leaning right; median 36 seconds, IQR 7.5 seconds, no outliers.
- The MAD measures distances from the mean, 16 grams, not from the median, and the 33 swells it. Use the IQR: Q_1 = 12.5, Q_3 = 17, IQR 4.5 grams. The fences are 12.5 - 6.75 = 5.75 and 17 + 6.75 = 23.75, so the 33 is an outlier and should be named, with the shape, skewed right.
- Unlikely. In roughly symmetric data the mean and median are close; a mean 18 seconds above the median points to a long right tail or large outliers.
- Sorted 5, 6, 6, 7, 7, 7, 7, 8, 8, 9: total 70, mean 7, median 7; distances total 8, MAD 0.8 runs. Q_1 = 6, Q_3 = 8, fences 3 and 11: no outliers. The 10 counts are symmetric about 7 runs in the hour, from 5 to 9.
- Sorted 40, 42, 43, 45, 46, 48, 50, 53, 57, 62, 90. Median 48 mm, Q_1 = 43, Q_3 = 57, IQR 14 mm; fences 43 - 21 = 22 and 57 + 21 = 78, so 90 is an outlier. Mean \displaystyle \frac{576}{11}, about 52.4, above the median: skewed right. Report the median and IQR and name the 90 mm hatchling.
- Sorted 6.5, 6.5, 7, 7, 7.5, 8, 8.5, 9: total 60, mean 7.5 m; distances total 6, MAD 0.75 m. Median 7.25, IQR 8.25 - 6.75 = 1.5, fences 4.5 and 10.5: no outliers. Roughly symmetric (only 8 values), so report the mean and MAD, range 6.5 to 9 m.
- One answer: 9, 10, 11, 12, 13, 14, 25. The median is the fourth value, 12; Q_1 = 10 and Q_3 = 14, so the IQR is 4; the fences are 4 and 20, and only 25 is beyond them.
- One answer: 26, 28, 28, 30, 30, 32, 32, 34. Symmetric about 30, so the mean is 30; the distances total 16, and \displaystyle \frac{16}{8} = 2.
- The MAD is an average distance from the mean, so beside the median it describes spread about a different center, swollen by the values that pulled the mean away. The IQR comes from quartiles split at the median, so the two describe the same middle.
- The methods give different numbers for the same lizard, so without the method a reader cannot tell what “11 cm” means or compare it with other surveys.
- Six values can look skewed by chance; sixty show a pattern more reliably. The count says how far to trust the shape and the summaries.
- A box plot shows only five numbers. A gap inside the box or along a whisker would leave it looking the same.