Optional extension: Unknown frequencies, grouped estimates and weighted spread
Example 1
The table has 11 observations with mean 1169. Find the frequency x.
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Answer: 3
- The counts give x+y=8.
- The weighted total gives 4x+6y+27=69.
- Solve simultaneously to obtain x=3.
Example 2
Find the population variance of the frequency distribution. Use variance =ΣfΣf(x−mean)2. Give an exact fraction if needed.
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Answer: 29
- Symmetry gives mean 14.
- The two outer groups each contribute 4×32.
- Divide their combined contribution by total frequency 16, obtaining 29.
Example 3
The table has 8 observations with mean 433. Find the frequency x.
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Answer: 2
- The counts give x+y=6.
- The weighted total gives 6x+8y+22=66.
- Solve simultaneously to obtain x=2.