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.
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.
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.
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.