To find a missing frequency from a mean, put the unknown in both the total of the scores and the number of scores. Then use total of the scores = mean × number of scores.
For example, suppose three students score 1, two score 2 and the rest score 4. If the mean is exactly 3, eight students scored 4. Here is how to show the calculation and check that the answer is a valid count.
- Multiply each score by its frequency to build the total of the scores.
- Add the frequencies to find the number of observations.
- Substitute the answer into the original table to check the mean.
CHAPTER 1 OF 4 · Back to contents
Read each frequency as a count
A score of 1 with frequency 3 contributes 1 + 1 + 1 = 3 to the total. A score of 2 with frequency 2 contributes 2 + 2 = 4. Together these are five observations worth 7 points.
Let x be the frequency of the score 4. This row contributes x observations and 4x points. The table now represents 5 + x observations and 7 + 4x points.
| Score | Frequency | Contribution to score total |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 2 | 4 |
| 4 | x | 4x |
CHAPTER 2 OF 4 · Back to contents
Worked example | Set the mean equal to 3
Write 7 + 4x = 3(5 + x). Expanding gives 7 + 4x = 15 + 3x. Subtract 3x and then 7 to find x = 8.
Check the original mean. There are 3 + 2 + 8 = 13 observations. Their total is 3 + 4 + 32 = 39, so the mean is 39 ÷ 13 = 3.
A second way to see it is to measure each score relative to 3. The three 1s contribute a shortfall of 6 points and the two 2s contribute a shortfall of 2 points. Each score of 4 contributes a surplus of 1 point, so eight 4s balance the shortfall.
CHAPTER 3 OF 4 · Back to contents
Try another table
A table lists values 1, 3 and 5. Their frequencies are 2, 4 and x respectively. The mean is exactly 3. Find x.
The total is 2 + 12 + 5x = 14 + 5x. The number of observations is 6 + x. Set 14 + 5x = 3(6 + x). This simplifies to 2x = 4, so x = 2. A check gives 24 ÷ 8 = 3.
Ask your child to explain why the denominator is 6 + x. If they count the three rows, have them write the first few scores individually and count how many observations each row stands for.
CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
What if the answer is a decimal?
An exact frequency must be a non-negative whole number. Check your arithmetic and whether the question gives an exact or rounded mean.
What if the missing row’s score equals the mean?
Adding scores equal to the mean leaves the mean unchanged if the other scores already have that mean. In that case, you need more information, such as the total frequency, to find a unique count.
Can I do the same for grouped intervals?
For a grouped-data estimate, use each interval midpoint as the representative value. An estimated or rounded mean can affect which counts are possible.
Contents · Official syllabus and next practice
Official syllabus and next practice
Read the official SEAB 2027 syllabus
Continue with combining two cumulative frequency tables.