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Practise the method.
Combined means, variance and missing observations
Optional extension: Combined means, variance and missing observations
Example 1
Two equally sized groups have the summaries shown. Find the population variance after combining them. You may use variance = mean(x2)− mean(x)2.
Group summaries
Group
Count
Mean
Population variance
A
30
6
4
B
30
18
4
Show worked answer
Answer: 40
The combined mean is 12.
Each group mean is 6 from it.
The combined variance is within-group variance 4 plus 62 from the separation of the means: 40.
Example 2
Three observations have population mean 17 and population variance 3104. One observation is 25. Find the other two values. Use variance =3Σ(x−mean)2.
Distribution summary
Count
Mean
Population variance
Known value
3
17
104/3
25
Show worked answer
Answer: x=11x=15
The missing values sum to 26.
Their squared deviations from 17 sum to 40.
Solving these two conditions gives 11 and 15.
Example 3
Two equally sized groups have the summaries shown. Find the population variance after combining them. You may use variance = mean(x2)− mean(x)2.
Group summaries
Group
Count
Mean
Population variance
A
40
9
16
B
40
19
16
Show worked answer
Answer: 41
The combined mean is 14.
Each group mean is 5 from it.
The combined variance is within-group variance 16 plus 52 from the separation of the means: 41.
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