Combined Means and Changing Data
A mean does not show the total or the group size. To combine or change a group, recover the total first.
Recover totals before combining means
group A
20 values × mean 8.75 = total 175
group B
30 values × mean 8.4 = total 252
pooled
(175 + 252) ÷ (20 + 30) = 8.54
The answer lies between 8.4 and 8.75, nearer 8.4 because group B is larger.
A mean hides a total
Symbols
- = total of the data values (same unit as x)
- = number of values (none)
- = mean of the group (same unit as x)
- Recover each total. Multiply each group mean by its number of values.
- Change the totals and counts. Add or subtract the data values named in the question. Change the count only when a value is added or removed.
- Divide once at the end. Use the final total divided by the final count.
| Change | Total | Count |
|---|---|---|
| add value v | + v | + 1 |
| remove value v | − v | − 1 |
| replace wrong w by correct v | − w + v | no change |
A replacement changes one observation but does not add or remove an observation. That is why its count stays the same.
Worked example
Combine two groups from a real paper
In 9709/51 May/June 2025 Q3(c), 20 Eagles have total time 175.0 s. Thirty Swifts have mean time 8.4 s, so their total is s.
Key idea
Common mistake
Now you try
Group A contains 18 values with mean 42. Group B contains 12 values with mean 51. Find the combined mean. One new value is then added and the mean of all 31 values becomes 46. Find the new value.
Show worked answer
The first total is , so the combined mean is . The new total is .