Year 11 Standard · Unit 1 Describing data · Lesson 5 of 6

How do you say which of two groups is better, and what does better mean?

Two board members look at the same waiting times and disagree. This lesson asks what a fair comparison of two groups has to include, which summaries suit the data, and how to say what the numbers mean in the situation.

Statistics and ProbabilityGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 3: Q1 = 20 and Q3 = 36. What is the IQR?
    Answer16. 36 − 20.
  2. From Year 8: the mean of 6, 8, 10 and 12 is ?
    Answer9. The total is 36, shared among 4.
  3. From Lesson 4: a bar is 4 wide and 2.5 tall on a density axis. What is its frequency?
    Answer10. 4 × 2.5.
Words used in this lesson
centre
a single value that represents the typical value of a data set. For example: the median or the mean.
spread
how far apart the values in a data set are. For example: the IQR or the range.
consistent
not varying much from one value to the next. For example: bus waits that are always close to the same.
skewed
having more values on one side, with a long tail on the other. For example: a few very long waits.
extreme value
a value far from the rest of the data. For example: one house price ten times the others on its street.
compare
say how two things are alike and how they differ, with numbers. For example: “the typical mass is greater for plot P”.

Which bus route is better?

A transport office in Medellin records the waiting time at a stop for two bus routes over a few weeks. A board member looks at the numbers and says that route 2 is the better one. Another says route 1 is.

Predict first: what would you want to know about each route's waiting times before you agreed with either board member? Is “better” one number, or more than one? Use the first figure to move one box plot against another, then the second to see which numbers react to one extreme value.

0
100
7B's median minus A's median
0B's IQR minus A's IQR

B's median is 7 higher than A's (47 and 40), and the IQRs are the same (20 and 20).

Show me the steps
  1. Opening state: B's median is 47 and A's is 40, so B is 7 higher; both IQRs are 20, so the spreads are the same.
  2. Set the spread to 200: B's IQR is 40 against A's 20. B's typical value is still 7 higher, but its middle half is twice as wide, so B is less consistent.
  3. Set the spread to 50 and the shift to −5: B's median is 42, only 2 above A's, and its IQR is 10. B is now more consistent than A and only just higher.

Compare a centre and a spread, and say what they mean.

  • Centre: the median (or the mean). Which group has the larger typical value?
  • Spread: the IQR (or the range). Which group is more consistent?
  • With skewed data or extreme values, use the median and IQR. With roughly symmetric data and no extreme values, the mean and range are fine.
  • A good comparison has two sentences or more: one about the centre and one about the spread, each with numbers and each in the words of the situation.

“Larger” or “smaller” is not enough: say what it means (“the wait is longer and less predictable”).

22
17.1mean
17median

Largest value 22: mean 154 ÷ 9 = 17.1, median 17, range 10, IQR 5.5.

Show me the steps
  1. Largest value 22: the total is 154, so the mean is 154 ÷ 9 = 17.1 to one decimal place. The median is the 5th value, 17, the range is 10 and the IQR is 5.5.
  2. Slide to 42: the total is 174, so the mean is 19.3. The median is still 17 and the IQR is still 5.5, but the range is 30.
  3. Slide to 62: the mean is 21.6, above eight of the nine values, while the median is still 17. One value has pulled the mean and the range, not the median or the IQR.

Do not stop at one number. Two groups can have the same median and very different spreads, and the group with the higher mean may only be higher because of one extreme value. And do not describe the numbers without the situation: “B has a bigger IQR” becomes “B's waiting times are less predictable”.

Worked example: two routes

Waiting times in minutes. Route X: 4, 6, 7, 9, 10, 13, 15. Route Y: 3, 8, 9, 10, 11, 13, 25. Compare them.

  1. X: median 9; the halves are 4, 6, 7 and 10, 13, 15, so Q1 = 6, Q3 = 13 and IQR = 7.
  2. Y: median 10; the halves are 3, 8, 9 and 11, 13, 25, so Q1 = 8, Q3 = 13 and IQR = 5.
  3. Centre: Y's median is 1 minute higher, so a typical wait for Y is slightly longer.
  4. Spread: Y's IQR is smaller (5 and 7), so the middle half of Y's waits is more consistent. But Y has one wait of 25 minutes, the longest of either route: the IQR hides this rare very long wait, while the ranges, 22 for Y and 11 for X, show it.

Your go. The first step is given. Two fields of maize. Field P: median yield 62, IQR 10. Field Q: median yield 62, IQR 24. Write a comparison.

  1. The medians are equal, so the typical yields are the same.
  2. Compare the IQRs, and say what that means for a farmer.
Check the answer

The typical yield is the same, 62 for both fields. Field Q's IQR, 24, is more than twice P's, 10, so Q's yields vary much more from one plot to the next: P is the more reliable field. A common slip is to say the fields are the same because the medians are.

With a GDC (optional)

Enter each data set as its own list and run one-variable statistics on each in turn. Write down the mean, the median and the quartiles for both lists, one under the other, and compare them in a sentence. Some models show a standard deviation too; you do not need it here.

Check one quartile by hand: models differ on how they split a short list. A comparison needs both a centre and a spread, so read two numbers for each group, not one. Everything here can be done by hand.

Check before you go on

1. Group P has an IQR of 9 and group Q has an IQR of 14. What does this tell you?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Group P has median 34 and IQR 12. Group Q has median 29 and IQR 18. How much larger is Q's IQR than P's?

3. A river's daily level has one very large flood reading in the data. Which pair best describes the typical level and how spread out it is?

Level 2 · Building

4. Team A scored 9, 14, 16, 17, 21, 25. Team B scored 11, 13, 17, 19, 23, 25. Using the halves of each ordered list, how much larger is B's IQR than A's?

5. The seven values are 2, 3, 3, 4, 5, 6 and 33. How much larger is the mean than the median?

Level 3 · Stretch

6. Eight values have a mean of 36. A ninth value, 27, is added. What is the new mean?

7. A list has 8 values. Its largest value, 30, is replaced by 70. By how much does the mean increase?

Something to think about

A club says: “Our mean score is higher than theirs, so we are the better team.” Give one reason this might be a weak argument, and one extra piece of information that would make it stronger.

One way to think about it

The mean can be pulled up by one or two very high scores, so a higher mean does not mean a higher typical score. Giving the median would show the typical score, and giving the IQR or the range would show how consistent the team is. Better still, say how many matches were played, because a small number of matches makes any average unreliable.

Exit ticket

8. Two bakeries' daily sales. Bakery P: median 120 loaves, IQR 30. Bakery Q: median 135 loaves, IQR 55. Which conclusion is best supported?