Mean when values repeat
A value that occurs 11 times must be counted 11 times.
How big is a family?
A class of 36 students counted the people in their families. Here are the results.
| Family size | Students |
|---|---|
| 3 | 3 |
| 4 | 11 |
| 5 | 9 |
| 6 | 7 |
| 7 | 3 |
| 8 | 1 |
| 9 | 1 |
| 10 | 1 |
Roughly, what is the average family size in this class?
What this lesson covers
The idea
When values occur with frequencies, the mean = (sum of each value × its frequency) ÷ (sum of the frequencies), since every occurrence of a value must be added.
How big is a family?
A class of 36 students counted the people in their families. Here are the results.
Roughly, what is the average family size in this class?
- Family size | Students
- 3 | 3
- 4 | 11
- 5 | 9
- 6 | 7
- 7 | 3
- 8 | 1
- 9 | 1
- 10 | 1
- About 4
- About 5
- About 6.5
Count every student
Every student is a weight on the beam: a family size that occurs 11 times is a stack of 11 weights. The pivot starts at the quick way, the average of the different sizes only. Does the beam balance? Tap every row to count its students. Do both lists.
Count every occurrence
When values occur with frequencies, the mean = (sum of each value × its frequency) ÷ (sum of the frequencies), since every occurrence of a value must be added.
The size 4 occurs 11 times, so 4 is added 11 times: 4 × 11 = 44. In the family table: 9 + 44 + 45 + 42 + 21 + 8 + 9 + 10 = 188 3 + 11 + 9 + 7 + 3 + 1 + 1 + 1 = 36 The first line is the total of all the family sizes. The second line is the number of students.
The mean is 188 ÷ 36 = 5.22 The quick way gave 6.5 because it counted each size only once, as if all sizes occurred equally often.
Notes
When values occur with frequencies, the mean = (sum of each value × its frequency) ÷ (sum of the frequencies), since every occurrence of a value must be added.
Check yourself
10 students counted their pencils: 1 pencil: 4 students, 2 pencils: 3 students, 3 pencils: 2 students, 4 pencils: 1 student. What is the mean number of pencils?
Answer: 2
Total pencils: 1 × 4 + 2 × 3 + 3 × 2 + 4 × 1 = 20. Students: 4 + 3 + 2 + 1 = 10. Mean: 20 ÷ 10 = 2.
The value 2 occurs 3 times and the value 5 occurs 2 times. Which expression gives the mean?
20 students counted the books they borrowed this week: 0 books: 7 students, 1 book: 6, 2 books: 4, 3 books: 2, 4 books: 1. What is the mean number of books?
Answer: 1.2
Total books: 0 × 7 + 1 × 6 + 2 × 4 + 3 × 2 + 4 × 1 = 24. Students: 7 + 6 + 4 + 2 + 1 = 20. Mean: 24 ÷ 20 = 1.2.
In the family table the true mean is 5.22, less than the quick answer 6.5. Why?
- (2 + 5) ÷ 2. That counts each value once. 2 occurs 3 times and 5 occurs 2 times.
- (2 × 3 + 5 × 2) ÷ 2. The top is right, but there are 3 + 2 = 5 values, not 2.
- (2 × 3 + 5 × 2) ÷ (3 + 2) — correct. Yes! The total of the 5 values is 2 × 3 + 5 × 2 = 16 and there are 3 + 2 = 5 values: 16 ÷ 5 = 3.2.
- (2 + 5) ÷ (3 + 2). That divides by 5 values but adds each value only once. Every occurrence must be added.
- Larger families occur most often.. The sizes 4 and 5 occur most often (11 and 9 students), and they are small.
- Because 36 is bigger than 8.. The number of students does not make the mean smaller. It is the total ÷ the number of students.
- The small sizes 4 and 5 occur many times, so they pull the mean down. — correct. Yes! 20 of the 36 students have a family of 4 or 5. The quick way gave each size one vote; counting every student gives the small sizes more weight.
- Because the frequencies are added to the values.. The frequencies multiply the values: 4 × 11, not 4 + 11.