Reading Five Figures Off a Curve

Mathematics · Data & Statistics · ages 15-16

Name ______________________ Date ____________

Sam plots each cumulative total against the midpoint of its class instead of the upper boundary. Why is that wrong?
- Midpoints are only used for estimating a mean, and the rule is a matter of habit
- Midpoints give a curve that goes downwards
- Nothing is wrong, either point works
- The running total is not reached until the end of the class, so plotting it at the middle claims a count too early
What does the height of a cumulative frequency curve at a value of 35 tell you?
- how many values are exactly 35
- how many values are 35 or less
- how many values are more than 35
- how many classes end at 35
The last point on a cumulative frequency curve is at a height of 80. What does that say?
- The largest value in the data is 80
- There are 80 classes
- There are 80 values altogether
- The largest class holds 80 values
A cumulative frequency curve has a total frequency of 60. At what running total do you read off the median?
Answer: ______________
A curve of 160 values gives a lower quartile of 22 and an upper quartile of 39. What is the interquartile range?
Answer: ______________
Leo is asked how many of 60 runners took longer than 35 minutes. He reads the curve's height at 35, which is 48, and writes 48. What has gone wrong?
- 48 is how many finished in 35 minutes or less, so his answer is 60 - 48 = 12
- He should have read the height at 60 instead
- He should have halved 48
- Nothing, 48 is correct
A median read off a cumulative frequency curve is exact.
Circle one: True False
The curve shows 46 of 60 values at or below 20. About how many values sit above 20?
Answer: ______________
A class runs 15 ≤ w < 20 and its cumulative total is 34. Where exactly is the point plotted, and why?
- At 15, because that is where the class starts counting
- At 17.5, because that is the class's midpoint
- At 20, because 34 is the number known to be counted by the end of the class
- Anywhere in the class, since the curve is an estimate anyway
Two curves are drawn on one grid. Curve A ends at a height of 40 and curve B ends at 200. Both have a median of 55. What can you say?
- B's values are more spread out than A's
- The two groups are different sizes but their typical value is the same
- B's typical value is five times A's
- A must have been drawn to a different scale

Answer key

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

Reading Five Figures Off a Curve W1-mt_5U70qgbnST-s1

The running total is not reached until the end of the class, so plotting it at the middle claims a count too early · The total of 20 means 20 runners were home by 50 minutes, not by 45. Plotting at 45 shifts the whole curve left and makes every reading too small.
how many values are 35 or less · The curve carries a running total, so its height at 35 is everything counted up to 35. Values above 35 have not been added in yet.
There are 80 values altogether · By the end of the last class everything has been counted, so the final height is the total frequency. The largest value is read off the other axis, and no single class is named by it.
30 · Half of 60 is 30, so you read across at a running total of 30.
17 · The interquartile range is upper quartile minus lower quartile: 39 - 22 = 17. The total of 160 was only needed to find the heights to read across at.
48 is how many finished in 35 minutes or less, so his answer is 60 - 48 = 12 · The curve counts upwards from the left, so its height is always how many are below. Leo has reported the fast runners as if they were the slow ones.
False · The data arrived in classes, so no individual value was recorded. The smooth curve assumes an even spread inside each class, and the reading is only as good as that assumption.
14 · 60 - 46 = 14 values lie above 20.
At 20, because 34 is the number known to be counted by the end of the class · The 34 is a fact about 20: by then, 34 had been counted. Earlier in the class the total was smaller, and at 17.5 nothing was recorded at all, so 20 is the one place the point can honestly sit.
The two groups are different sizes but their typical value is the same · The final height is the total frequency, so the groups hold 40 and 200 values. The median is read at half of each group's own total, so both typical values landing on 55 is a fair comparison.