Cutting a List Into Quarters

Mathematics · Data & Statistics · ages 15-16

Name ______________________ Date ____________

Seven scores in order are 8, 11, 14, 16, 19, 23, 30. What is the interquartile range?
Answer: ______________
You have to put the data in order before finding the quartiles.
Circle one: True False
Where is the upper quartile in an ordered list of 19 values?
- the 14th value
- the 19th value
- the 15th value
- the 5th value
The ordered list 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 has eleven values. What is the lower quartile?
Answer: ______________
Nine values in order are 4, 6, 7, 9, 12, 14, 15, 19, 22. What is the lower quartile?
Answer: ______________
Sam finds the quartiles of 40, 18, 25, 31, 22, 37, 29 by counting straight along the list as written. What has he missed?
- He should have used (n + 1) / 3 for seven values
- The list is not in order, so counting to a position picks whichever value happened to be written there
- Seven values are too few for quartiles to exist
- He should have found the mean before the quartiles
About half the values in a data set lie between the lower and upper quartiles.
Circle one: True False
A new value of 200 joins the data set, far above the rest. Which measure of spread is less affected, and why?
- The interquartile range, because it only looks at the middle half
- The range, because it ignores extremes
- The mean, because it balances everything
- Both are affected equally
Can a set's interquartile range ever be bigger than its range?
- Yes, when one value is extremely large
- Yes, whenever the values are very spread out
- No, because the quartiles both sit inside the smallest and largest values
- Only when the number of values is even
In one set the bottom quarter runs from 40 to 43 and the top quarter runs from 55 to 78. Leo says the three quartile cuts split a set into four groups covering equal stretches of value. What do those two figures show?
- The stretches are 3 and 23, so the groups hold equal numbers of values rather than equal stretches
- The stretches are 3 and 23, which is what equal groups look like
- Nothing, because the two quarters are at opposite ends of the set
- That the set must have been sorted wrongly

Answer key

For grown-ups. Fold this page away before handing over the rest.

Cutting a List Into Quarters W1-mt_vN10d1I9dA-s1

12 · The lower quartile is the 2nd score, 11, and the upper quartile is at position 3 × 8 / 4 = 6, the 6th score, 23. So the interquartile range is 23 - 11 = 12.
True · Both quartiles are found by counting to a position, and a position only means something once the values are lined up smallest to largest.
the 15th value · The position is 3 × (19 + 1) / 4 = 3 × 5 = 15, so the upper quartile is the 15th value. The 5th value is the lower quartile.
3 · The ordered list is 1 to 11, the median is the 6th value (6), and the lower half 1 to 5 has median 3.
6.5 · The position is (9 + 1) / 4 = 2.5, which falls halfway between the 2nd and 3rd values, 6 and 7. The halfway point is (6 + 7) / 2 = 6.5.
The list is not in order, so counting to a position picks whichever value happened to be written there · Counting to position 2 in Sam's list gives 18, but in the ordered list 18, 22, 25, 29, 31, 37, 40 the 2nd value is 22. The position only means something once the values are sorted.
True · A quarter sit below the lower quartile and a quarter above the upper one, so the two middle quarters, which is half the values, lie between them.
The interquartile range, because it only looks at the middle half · One giant value stretches the range but cannot touch the middle half, so the IQR barely moves.
No, because the quartiles both sit inside the smallest and largest values · The lower quartile is at or above the smallest value and the upper quartile is at or below the largest, so the gap between them can never exceed the gap between the ends.
The stretches are 3 and 23, so the groups hold equal numbers of values rather than equal stretches · The bottom quarter covers 3 units and the top quarter covers 23, nearly eight times as wide. The cuts are made by counting values, so the groups match in size and not in the stretch of value they cover.