Comparing two distributions
Two groups were measured in the same way. How do you say which runs higher, which is more spread out, and whether the difference is real or just the luck of the sample?
Where you are 13 in Center and spread
When you reach for it
You have measured the same thing in two groups: hours of sleep at two ages, lengths of fish in two lakes, prices of tickets at two theaters. The question is about the groups, not about any one value: which runs higher, which is more spread out, whether the shapes differ, and whether a difference between the samples says anything about the populations behind them. Day 1 describes and compares; Day 2 chooses the measures and judges the difference.
The tools are all met before: dot plots and histograms; the Mean and the median for center; the range and the interquartile range for spread; and shape. Describing one distribution this way is Describing a distribution. Two box plots on one scale compare five-number summaries; here the plots show every value, or every bin.
The procedure
Day 1: describing two distributions side by side
In a made-up town, random samples of 15 people aged 11 to 13 and 15 people aged 16 to 18 gave their hours of sleep on a school night, to the nearest half hour.
Aged 11 to 13: 7.5, 8, 8, 8.5, 8.5, 8.5, 9, 9, 9, 9.5, 9.5, 9.5, 10, 10, 10.5
Aged 16 to 18: 5.5, 6, 6.5, 6.5, 7, 7, 7.5, 7.5, 7.5, 8, 8, 8.5, 8.5, 9, 9.5
Put both on one scale. Draw the two dot plots one above the other, over the same number line.
The eye compares positions, so a position must mean the same number in both plots. Give each plot its own number line, stretched to fit its own data, and both clusters sit in the middle of their plots and look alike. On one scale the shift is the first thing you see.
Describe each distribution by center, spread and shape, with units. Aged 11 to 13: the 8th of the 15 ordered values is 9, so the median is 9 hours, and the values balance around it, so the mean is also 9 hours. They run from 7.5 to 10.5 hours, a range of 3 hours, and the shape is roughly symmetric with one peak, with no gaps or outliers. Aged 16 to 18: median 7.5 hours, mean 7.5 hours, values from 5.5 to 9.5 hours, a range of 4 hours, also roughly symmetric with one peak.
Write the comparison sentence. It names both groups, the measure, both values with their units, and the direction and size of the difference:
- “In these samples, the median sleep of the 11 to 13 year olds, 9 hours, is 1.5 hours more than that of the 16 to 18 year olds, 7.5 hours.”
- “The older group’s sleep times are more spread out, with a range of 4 hours against 3 hours. Both distributions are roughly symmetric with a single peak.”
“The younger group sleeps more” names no measure and no size: ten minutes and three hours both fit it. “In these samples” says the sentence is about the 30 people asked. The whole town is Day 2’s question.
Day 2: choosing the measures, and judging the difference
Choose the measures from the shape. The mean uses every value, so one extreme value can pull it a long way; the median moves only when the middle of the list moves (resistant measures).
- Roughly symmetric, no outliers: the mean, with the range for spread.
- Skewed, or with outliers: the median and the interquartile range (choosing the center).
- Two clusters with a gap: no single center fits, since it may land where no values are. Describe each cluster.
Use the same measure for both groups. If either distribution calls for the median, compare medians. A mean set against a median compares two different things.
Then ask whether the samples speak for the populations. Random samples (simple random samples) stand in for their populations, but not exactly: another random sample would give other numbers (Sampling variability and simulating it). So a gap between two sample centers is partly a real difference and partly the luck of the draw. The informal test sets the shift between the two centers against the spread of each group, with the spread chosen to match the centers: the interquartile range when you compare medians, the range when you compare means.
- A shift that is big against the spread leaves the bulk of one group above the bulk of the other. Luck seldom does that, so it is reasonable to say the populations differ.
- A shift that is small against the spread leaves the plots mostly on top of each other. Another pair of samples could shrink the gap or reverse it, so no conclusion is justified.
The shift should also run through the whole of both samples, not come from one or two far values.
Both sleep distributions are symmetric, so compare means: 9 and 7.5 hours, a shift of 1.5 hours, half the younger group’s range of 3 hours and nearly half the older group’s range of 4. Each center sits near the edge of the other group: 13 of the 15 older people slept less than 9 hours, and 14 of the 15 younger people slept more than 7.5 hours. With random samples it is reasonable to say that in this town, people aged 11 to 13 tend to sleep more on a school night than people aged 16 to 18. “Tend to” is a claim about groups: the younger person who sleeps 7.5 hours sleeps less than the older one who sleeps 9.5.
There is no fixed cut-off yet for “big against the spread”. Measuring the gap in units of spread is the next step, in comparing populations with the mean absolute deviation.
Worked examples
Day 1: word lengths in two books
Pascal chose 20 words at random from a picture book for young children and 20 from a book on astronomy, and counted the letters in each.
Picture book: 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6
Astronomy book: 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 11, 12
Draw both on one scale, describe each, and compare them.
The values run from 1 to 12 letters, so one number line from 0 to 13 holds both plots.
Picture book. With 20 values the median is halfway between the 10th and 11th. Both are 3, so the median is 3 letters. The range is 6 - 1 = 5 letters. The tallest stack is at 3 and the dots fall away fairly evenly on both sides: roughly symmetric, one peak.
Astronomy book. The 10th and 11th values are both 5, so the median is 5 letters. The range is 12 - 1 = 11 letters. The dots pile up from 2 to 7 letters, then trail away one word at a time out to 12: skewed right, with no gap.
Comparison. “In these samples, the median astronomy word, 5 letters, is 2 letters longer than the median picture book word, 3 letters. The astronomy book’s lengths are far more spread out, with a range of 11 letters against 5, and skewed right by a tail of long words that the picture book does not have.”
“The astronomy book has longer words” would not do: it names no measure and no size, and read literally it is false, since the astronomy sample has a 1 letter word, shorter than 18 of the 20 picture book words. Nor would “the astronomy book is skewed”: the distribution of its word lengths is skewed, and skew has a direction.
Day 1: fish in two lakes, from histograms
Niall measured random samples of 20 fish from each of two lakes, in centimeters.
North lake: 21, 23, 25, 26, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 34, 35, 37, 39
South lake: 14, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 21, 22, 23, 25, 27, 31, 36, 44
The figure shows both as histograms on one scale. Describe and compare the two distributions.
Shape. The north lake’s bars, 2, 7, 8, 3, rise to a peak from 30 to 35 cm and fall again: roughly symmetric. The south lake’s start with one fish, jump to 10 from 15 to 20 cm, then fall away in a long tail of short bars: skewed right. 10 of the 20 south lake fish sit in that one bin.
Center, from the histograms. A histogram hides single values, but counting along the bars finds the bin holding the median, the 10th and 11th lengths. North: 2 + 7 = 9 fish are shorter than 30 cm and 9 + 8 = 17 shorter than 35 cm, so the median is in the 30 to 35 cm bin. South: 1 fish is shorter than 15 cm and 1 + 10 = 11 shorter than 20 cm, so the median is in the 15 to 20 cm bin.
Exact values, from the lists. North: the 10th and 11th lengths are both 30, so the median is 30 cm, and the lengths run from 21 to 39 cm, a range of 18 cm. South: both are 19, so the median is 19 cm, and the lengths run from 14 to 44 cm, a range of 30 cm.
Comparison. “In these samples, the median north lake fish, 30 cm, is 11 cm longer than the median south lake fish, 19 cm.” The south lake has the larger range, but most of it is tail: half its fish crowd into one 5 cm bin.
Day 2: ticket prices with one far-out price
Maelys took random samples of 12 ticket prices, in dollars, from last month’s sales at two theaters.
Riverside: 15, 15, 18, 18, 18, 20, 20, 22, 22, 25, 25, 64
Hilltop: 10, 12, 12, 14, 14, 15, 15, 15, 16, 18, 18, 21
Which measures of center and spread should she report? Write the comparison.
Read the shapes first. Eleven Riverside prices lie from 15 to 25 dollars, and one, 64 dollars, sits alone. By the 1.5 IQR rule it is an outlier: the Riverside quartiles are 18 and 23.5 dollars, and 23.5 + 1.5 \times 5.5 = 31.75, far below 64. The Hilltop prices are roughly symmetric with no outlier.
See what the outlier does.
The medians are 20 dollars (the 6th and 7th Riverside prices are both 20) and 15 dollars. Without the 64 dollar ticket the Riverside mean drops to \displaystyle \frac{218}{11} \approx 19.82 dollars, while the median of the other 11 prices is still 20 dollars. With it, the mean is higher than 9 of the 12 prices.
Report medians and interquartile ranges, for both theaters. Riverside calls for the median, so Hilltop gets it too. The Hilltop quartiles are 13 and 17 dollars. The ranges, 49 dollars against 11, would mostly measure one ticket.
Comparison. “In these samples, the median Riverside ticket, 20 dollars, cost 5 dollars more than the median Hilltop ticket, 15 dollars. The middle halves spread over similar widths, 5.5 dollars and 4 dollars, and Riverside had one ticket far above the rest, at 64 dollars.” Comparing means would have made the gap 8.5 dollars, about 3.7 of them from a single ticket.
Day 2: the fish again, with one shape skewed
Use Niall’s fish from Example 2. Which measures should the comparison use, and do the samples give good reason to think fish in the north lake tend to be longer?
SolutionMeasures. The south lake’s lengths are skewed right, and its 36 and 44 cm fish are outliers: its quartiles are 17 and 24 cm, and 24 + 1.5 \times 7 = 34.5. So the south lake needs the median, and one measure for both groups means medians and interquartile ranges for both. Its mean, \displaystyle \frac{436}{20} = 21.8 cm, sits 2.8 cm above its median of 19 cm, pulled up by the tail.
Shift against spread. The medians, 30 and 19 cm, differ by 11 cm. The north lake’s quartiles are 27.5 and 32.5 cm, an interquartile range of 5 cm; the south lake’s is 24 - 17 = 7 cm. The shift is more than twice the north lake’s interquartile range and well over the south lake’s, and the middle halves do not overlap. The shift runs through both samples: 18 of the 20 north lake fish are 25 cm or longer, while 15 of the 20 south lake fish are shorter than 25 cm.
Conclusion. The samples are random and the shift is large against the spread, so it is reasonable to conclude that fish in the north lake tend to be longer, by about 11 cm in typical length. Not every one is: the south lake’s 44 cm fish beats every north lake fish in the sample.
Day 2: goals in two leagues, a difference too small to trust
Leocadio took random samples of 20 matches from this season in each of two soccer leagues.
| Goals in a match | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Coastal League, number of matches | 1 | 3 | 5 | 5 | 3 | 2 | 1 |
| Valley League, number of matches | 2 | 4 | 5 | 4 | 3 | 1 | 1 |
He says the Coastal League scores more. Do the samples support that for the leagues as a whole?
SolutionMeasures. Both distributions have one peak, at 2 or 3 goals, a slightly longer right tail, and no far values: close enough to symmetric for the mean. Each sum multiplies a number of goals by the number of matches that had it:
Shift against spread. The means differ by 0.35 goals, while each league’s matches have a range of 6 goals, from 0 to 6. The two rows of the table are nearly alike: 13 of the Coastal matches and 12 of the Valley matches had from 2 to 4 goals. Two random samples from leagues with the same typical score could easily differ this much.
Conclusion. “In these samples the Coastal League averaged 0.35 goals per match more than the Valley League, but that is small against a range of 6 goals, so it does not show the leagues differ.” Nor does it show they are the same; these samples cannot tell.
Day 2: pulse rates at rest and after a walk
Ragnhild took two separate random samples of 12 adults from a walking club. She measured the pulse rates of the first sample at rest, and of the second just after a 10-minute brisk walk, in beats per minute. No one was measured twice.
At rest: 58, 62, 64, 66, 68, 69, 71, 72, 74, 76, 78, 82
After the walk: 74, 78, 80, 82, 84, 86, 88, 90, 92, 96, 100, 106
(a) Compare the two distributions. (b) Is the difference big enough to say that pulse rates after the walk tend to be higher for the whole club? (c) Odalys concludes, “So every adult’s pulse goes up by about 18 beats per minute on a walk.” Is that justified?
Solution(a) At rest the pulses run from 58 to 82, a range of 24, roughly symmetric, with mean \displaystyle \frac{840}{12} = 70 and median 70. After the walk they run from 74 to 106, a range of 32, with a slightly longer upper tail; the mean is \displaystyle \frac{1056}{12} = 88 and the median, halfway between 86 and 88, is 87. The mean after the walk is 18 beats per minute higher, the median 17 higher, and the after-walk pulses are more spread out.
(b) Because of the longer upper tail, use medians and interquartile ranges. The interquartile ranges are 75 - 65 = 10 and 94 - 81 = 13, and the shift of 17 is larger than either. The middle halves, 65 to 75 and 81 to 94, do not overlap, and the two lists meet only between 74 and 82, where 4 resting pulses meet 4 after-walk pulses. Yes: with random samples it is reasonable to conclude that the club’s adults tend to have higher pulse rates just after a walk.
(c) No. These are two independent groups of different people, so the data hold no change for any one person, and even across the groups the resting pulse of 82 is higher than three of the after-walk pulses. “About 18” is a difference between two means. To see how each person’s pulse changes, measure the same people before and after and compare each person with themselves: those are pairs, not two separate groups. A comparison of two groups says nothing about any one person.
Boundaries
The same variable, the same units, one scale. Hours against minutes, or each plot on its own stretched scale, can create or hide a difference.
Different sample sizes. Centers and spreads compare whatever the sizes; histogram bar heights do not, since 20 fish in a bin means more out of 25 than out of 200. Use relative frequency histograms.
Random samples, for a claim about populations. A convenience sample, such as fish netted by one jetty, speaks for itself and not for the lake.
Separate groups. Everything here assumes two independent groups. When the same individuals are measured twice, compare each one with itself instead.
One center needs one peak. With two clusters the mean or median can land in the gap, and comparing such centers compares nothing real.
An informal judgment, not a proof. There is no exact cut-off here, a small sample can show a large shift by luck, and “the samples do not show a difference” is not “the populations are the same”.
A difference does not say why. The sleep samples show the age groups differ, not what causes it. Only a designed experiment can isolate a cause.
Where this goes next
The informal “shift against spread” becomes a number in comparing populations with the mean absolute deviation. How much a sample’s center swings from one random sample to the next is Sampling variability and simulating it. Later that swing is described exactly by the sampling distribution of a difference, and “too big to be luck” becomes a test for a difference of means, or, for experiments, re-randomizing. Back-to-back stem and leaf plots are one more way to put two small data sets side by side.
Exercises
Day 1: describing and comparing
- Random samples of 9 adults gave their hours of sleep on a weeknight. Aged 30 to 40: 6, 6.5, 6.5, 7, 7, 7, 7.5, 7.5, 8. Aged over 65: 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9. Find the median and range of each, and write one comparison sentence about center and one about spread.
- Random samples of 11 words from two books gave these numbers of letters. Book P: 2, 3, 3, 4, 4, 4, 5, 5, 6, 7, 9. Book Q: 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 10. Find the median and range of each, and compare the books.
- Random samples of 20 fish from two lakes were sorted into six 5 cm bins, from 20 to 25 cm up to 45 to 50 cm, a length on a bin edge going into the bin to its right. The counts, bin by bin from the shortest, are East lake: 1, 3, 7, 5, 3, 1 and West lake: 6, 7, 3, 2, 1, 1. Describe the shape of each distribution, find the bin that holds each median, and write a comparison sentence.
- Random samples of 20 matches were taken from two leagues. The numbers of matches with 0, 1, 2, 3, 4, 5 and 6 goals were, in that order, Northern League: 2, 5, 6, 4, 2, 1, 0 and Southern League: 1, 2, 4, 6, 4, 2, 1. Find the mean and the median of each, and write a comparison sentence using the means.
Day 1: comparison sentences
- The medians of random samples of ticket prices at two theaters are 12 dollars and 16 dollars. Ragnhild wrote: “Theater 2 is cheaper, 12 compared with 16.” Say what the sentence is missing, and rewrite it.
- Niall drew the two sleep samples of the procedure on two separate number lines, each running from its own smallest value to its own largest. Explain why his plots look much more alike than the plots in the figure, and why that misleads.
Day 2: choosing the measures
- A random sample of 11 ticket prices at one theater, in dollars, is 8, 8, 9, 9, 10, 10, 10, 24, 25, 25, 26. Find the mean and the median. Explain why neither describes a typical ticket well, and describe the prices better.
- Pulse rates, in beats per minute, of two separate random samples of 9 adults. At rest: 60, 64, 66, 68, 70, 72, 74, 76, 80. After a walk: 80, 82, 84, 84, 86, 88, 90, 92, 124. Find the mean and the median of each. Which pair should be compared, and what does the comparison say?
- A random sample of 11 words from a recipe book has these numbers of letters: 1, 2, 2, 3, 3, 3, 4, 4, 5, 9, 13. Find the mean and the median. Which better describes a typical word, and why?
Day 2: different enough?
- Each part gives the medians and interquartile ranges of random samples of fish lengths from two lakes. Decide whether the samples give good reason to think the lakes’ fish differ in typical length. (a) Medians 26 cm and 27 cm; interquartile ranges 8 cm and 9 cm. (b) Medians 40 cm and 52 cm; interquartile ranges 4 cm and 5 cm.
- Two separate random samples of 10 adults, in beats per minute. At rest: 62, 64, 65, 67, 68, 70, 71, 73, 75, 78. After a walk: 84, 86, 88, 90, 91, 93, 95, 96, 99, 104. Find the median and interquartile range of each, and decide whether pulse rates after the walk tend to be higher.
- Random samples of 9 ticket prices, in dollars. First theater: 12, 14, 15, 17, 18, 18, 20, 22, 25. Second theater: 13, 15, 16, 19, 20, 21, 22, 24, 27. Find the median and interquartile range of each, and decide whether tickets at the second theater tend to cost more.
Reasoning
- Leocadio netted fish from the jetty of each of two lakes. The medians of his two catches differ by 10 cm, with little overlap. Explain why he still cannot conclude that the lakes’ fish differ.
- Two random samples have medians 22 cm and 25 cm. Odalys says, “So the medians of the two populations differ by exactly 3 cm.” Explain what is wrong with that.
- In Exercise 4, Maelys says, “Every Southern League match has more goals than every Northern League match, because its mean is higher.” Explain what is wrong, using the data.
- Pascal took random samples of 5 words from each of two books and found medians of 3 and 5 letters. Maelys took random samples of 40 words from the same books and found the same medians, with similar spreads. Whose samples give stronger evidence that the books differ, and why?
Answers
- Aged 30 to 40: median 7 hours, range 2 hours. Aged over 65: median 7 hours, range 4 hours. The median is the 5th of 9 values. “In these samples the two groups have the same median sleep, 7 hours, but the over 65 group’s times are more spread out, with a range of 4 hours against 2 hours.”
- Book P: median 4 letters, range 7 letters. Book Q: median 6 letters, range 7 letters. The median is the 6th of 11 values. In these samples the median word in Book Q is 2 letters longer than in Book P, and both books’ lengths have a range of 7 letters.
- East lake: roughly symmetric, one peak from 30 to 35 cm; 1 + 3 = 4 fish are shorter than 30 cm and 4 + 7 = 11 shorter than 35 cm, so the median (the 10th and 11th lengths) is in the 30 to 35 cm bin. West lake: skewed right, peak from 25 to 30 cm; 6 fish are shorter than 25 cm and 13 shorter than 30 cm, so the median is in the 25 to 30 cm bin. “In these samples East lake fish tend to be longer: the median lies between 30 and 35 cm, against 25 to 30 cm in the West lake.”
- Northern League: mean 2.1 goals, median 2 goals. Southern League: mean 3 goals, median 3 goals. The means are \displaystyle \frac{42}{20} and \displaystyle \frac{60}{20}. “In these samples Southern League matches had a mean of 3 goals, 0.9 goals more than the Northern League mean of 2.1 goals.”
- It lacks the measure (medians), the unit (dollars), the size of the difference and the word “samples”. “In these samples, the median ticket at Theater 2, 12 dollars, is 4 dollars cheaper than the median at Theater 1, 16 dollars.”
- Each of Niall’s lines starts at its own group’s smallest value, so each cluster fills its own plot and sits in the middle of it, and both look like the same hump. The 1.5 hour shift, the thing being compared, disappears, because one position on the page now means different hours in the two plots.
- Mean about 14.91 dollars (\displaystyle \frac{164}{11}), median 10 dollars. The prices form two clusters with a gap: 7 tickets from 8 to 10 dollars and 4 from 24 to 26 dollars. No ticket costs anything near 14.91 dollars, and the median describes only the lower cluster. Better: “Most tickets cost 8 to 10 dollars, and about a third cost 24 to 26 dollars.”
- At rest: mean 70, median 70 beats per minute. After the walk: mean 90, median 86. The 124 is an outlier (quartiles 83 and 91, and 91 + 1.5 \times 8 = 103) and pulls the after-walk mean 4 above its median. Compare medians: in these samples the median pulse after the walk is 16 beats per minute higher than at rest, more than either interquartile range, 10 and 8 beats per minute.
- Mean about 4.45 letters (\displaystyle \frac{49}{11}), median 3 letters. The median: 8 of the 11 words have 4 letters or fewer, and the two long words, 9 and 13 letters, pull the mean above all but three of the values. The distribution is skewed right.
- (a) No: the shift, 1 cm, is tiny against interquartile ranges of 8 and 9 cm, so the middle halves overlap almost completely and another pair of samples could reverse the order. (b) Yes: the shift, 12 cm, is more than twice either interquartile range, so the middle halves are far apart.
- At rest: median 69, interquartile range 73 - 65 = 8 beats per minute. After the walk: median 92, interquartile range 96 - 88 = 8. The shift, 23 beats per minute, is nearly three times either interquartile range, and the lowest after-walk pulse, 84, is above the highest resting pulse, 78. Yes: pulse rates after the walk tend to be higher.
- First theater: median 18 dollars, interquartile range 21 - 14.5 = 6.5 dollars. Second theater: median 20 dollars, interquartile range 23 - 15.5 = 7.5 dollars. A shift of 2 dollars is small against those, the middle halves overlap from 15.5 to 21 dollars, and the prices interleave from 12 to 27 dollars, so the samples do not show that the second theater’s tickets tend to cost more.
- His samples are not random. A net at a jetty catches only fish that come near it, of sizes the net holds, so the difference could come from the two jetties rather than the two lakes.
- The 3 cm is a difference between two sample medians. Other random samples would give another difference, so the population medians need not be exactly 3 cm apart, and if the spreads are large even their order is not settled.
- A mean describes a typical value, not every value. In the samples, 3 Northern League matches had 4 or 5 goals, more than each of the 3 Southern League matches with 0 or 1 goals. The data say only that Southern League matches tend to have more goals.
- Maelys’s. With 5 words per sample the medians swing a lot from one sample to the next, so a 2 letter shift could easily be luck. With 40 words per sample the medians are much steadier, so the same shift against the same spread is much harder to explain by luck.