A Mean When You Only Have the Bands

Mathematics · Data & Statistics · ages 14-15

Name ______________________ Date ____________

A class of 18 members has a midpoint of 15. How much does that class contribute to the total?
Answer: ______________
Why does a grouped table need a midpoint at all?
- Because midpoints always come out as whole numbers
- Because every class in a table has to be the same width
- Because no individual value was recorded, so each class needs one value to stand for it
- Because the mean always lands inside the middle class of the table
Frequencies down a grouped table read 5, 7, 9 and 4 for the classes 0-5, 5-10, 10-15 and 15-20. The median lies in 5 ≤ x < 10.
Circle one: True False
A parcel weighs exactly 15 kg. Which class does it belong to?
- 10 ≤ w < 15
- 15 ≤ w < 20
- either one, it does not matter which
- neither, 15 falls in the gap between the two classes
Heights are grouped 150-160, 160-170, 170-180 and 180-190 with frequencies 4, 9, 5 and 2. Estimate the mean height in cm.
Answer: ______________
From a grouped table you can say exactly how long the longest visit was.
Circle one: True False
Frequencies down a grouped table read 12, 18, 6 and 4 for the classes 0-10, 10-20, 20-30 and 30-40. Which class contains the median?
- 20 ≤ s < 30
- 0 ≤ s < 10
- 10 ≤ s < 20
- 30 ≤ s < 40
In one class of 18 shoppers, nearly everybody actually spent about 32 pounds. The class is 30 ≤ s < 50. What does that do to the estimate?
- Nothing, because the midpoint is correct by definition
- It makes the estimate come out higher than the truth, because all 18 are counted as 40-pound shoppers
- It makes the estimate come out lower than the truth, because 32 is above the midpoint
- It makes the modal class wrong, because the frequencies shift
An estimated mean comes out at 62 kg. What is the safest way to report it?
- The mean mass is 62 kg, since that is what the working gave
- The mean mass is about 62 kg, estimated from the classes on the form
- Every item weighs 62 kg, since that is the average
- The mean mass is 62.0000 kg, to show the working was careful
A grouped table's modal class is 0 ≤ x < 10 and its median lies in 10 ≤ x < 20. Is that possible?
- No, because the modal class and the median class are always the same band
- Yes, because one asks which band is fullest and the other asks which band holds the middle value
- No, because the median must lie in the fullest band
- Yes, but only when the classes have different widths

Answer key

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

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270 · Each of the 18 is treated as 15, so the class contributes 18 × 15 = 270 to the running total.
Because no individual value was recorded, so each class needs one value to stand for it · A grouped table says how many fell in each band and nothing about where inside it. Adding up values you do not have is impossible, so each band is given a stand-in.
False · There are 25 values, so the median is at position (25 + 1) / 2 = 13. The running totals are 5, 12, 21 and 25, and the 13th is past 12, so it sits in 10 ≤ x < 15 rather than 5 ≤ x < 10.
15 ≤ w < 20 · The < sign means 10 ≤ w < 15 stops just short of 15, and the ≤ sign means 15 ≤ w < 20 takes 15 itself. So the parcel belongs to the second class.
167.5 · Midpoints are 155, 165, 175 and 185. The total is 4 × 155 + 9 × 165 + 5 × 175 + 2 × 185 = 620 + 1485 + 875 + 370 = 3350, across 20 people, so 3350 / 20 = 167.5 cm.
False · The top class gives a band the longest visit fell inside, not the visit itself. A table with a top class of 80 ≤ t < 100 tells you the longest was under 100 minutes and nothing sharper.
10 ≤ s < 20 · There are 40 values, so the median is at position (40 + 1) / 2 = 20.5. Running totals are 12, 30, 36, 40, and the 20th and 21st are past 12 but inside the block ending at 30, which is the class 10 ≤ s < 20.
It makes the estimate come out higher than the truth, because all 18 are counted as 40-pound shoppers · The midpoint of 30 ≤ s < 50 is 40, so the method credits all 18 with 40 pounds each when most spent nearer 32. That class contributes too much, and the estimated mean sits above the real one.
The mean mass is about 62 kg, estimated from the classes on the form · The number is worth reporting and so is where it came from. Dropping the word estimated claims a precision the classes never held, and extra decimal places claim more still.
Yes, because one asks which band is fullest and the other asks which band holds the middle value · They answer different questions. A fat band low down can be the fullest while more than half the values still sit above it, which puts the middle value in a band further along.