Averages without writing the list out

Mathematics · Data & Statistics · ages 14-15

Name ______________________ Date ____________

A table's largest frequency is 9. That information alone proves that the mode is 9.
Circle one: True False
A goals table reads 0 goals in 5 matches, 1 goal in 8, 2 goals in 6, 3 goals in 4 and 4 goals in 2. What is the median number of goals?
Answer: ______________
Values 0, 3, 6 and 9 occur 8, 3, 5 and 2 times. What is the mode?
- 3
- 9
- 0
- 8
Values 2, 5 and 8 have frequencies 6, 4 and 1. The mode is 8 because it is the largest listed value.
Circle one: True False
Values 4, 8 and 12 occur 3, 2 and 5 times. What is the median?
- 8
- 10
- 12
- 20
A survey of 25 students recorded siblings: 0 for 7 students, 1 for 11, 2 for 4, 3 for 3. What is the mean number of siblings?
Answer: ______________
Frequencies down a table read 6, 4, 9, 3, 3 for the values 0 to 4. Which value holds the 20th answer?
- 1
- 3
- 4
- 2
A table has frequencies 8, 5, then one missing, then 7, and the total frequency is 31. What is the missing frequency?
Answer: ______________
Why does multiplying value by frequency give the same mean as writing all the answers out and adding them?
- Because multiplying is faster than adding
- Because a value repeated f times adds up to exactly value × f
- Because the frequencies always add to the number of rows
- Because the table is written in order
Values 2, 5 and 8 have frequencies 4, 7 and 1. Find the mean.
Answer: ______________

Answer key

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

False · The largest frequency tells us how many times the most common value occurs. We still need the value beside it to identify the mode. That value could be 9, but the frequency alone does not prove it.
1 · There are 25 matches, so the median is the 13th. Running totals are 5, 13, 19, 23, 25, so matches 6 to 13 all scored 1 goal, and the 13th is one of them.
0 · The largest frequency is 8, belonging to value 0. Zero is an observed value and is the mode.
False · The largest frequency is 6, beside value 2. The mode is 2; being the largest value does not make 8 the mode.
10 · There are 10 observations. Position 5 contains 8 and position 6 contains 12. Average those values: (8 + 12) / 2 = 10.
1.12 · The siblings total 0 × 7 + 1 × 11 + 2 × 4 + 3 × 3 = 28, across 25 students, so the mean is 28 / 25 = 1.12.
3 · The running totals are 6, 10, 19, 22, 25. The 20th passes 19 and is caught by 22, which is the row for the value 3.
11 · The frequencies always add to the number of people, so the three known ones total 8 + 5 + 7 = 20 and the missing one is 31 - 20 = 11.
Because a value repeated f times adds up to exactly value × f · Adding 5 eight times is what 5 × 8 means, so the table's shortcut is the long addition already done. Nothing about the data changes, only the writing.
4.25 · The weighted total is 8 + 35 + 8 = 51. The count is 4 + 7 + 1 = 12, so the mean is 51 / 12 = 4.25.