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The Elm Tree golf course in Cortland, NY is a par 70 layout with 3 par fives, 5 par threes, and 10 par fours. Find the mean and variance of par on this course.

Mean was calculated as follows: Mean = 70/18 = 3.8888
Variance was found to be: second moment = (75 + 160 + 45)/18 = 280/18 = 15.555
variance = (5040 − 4900)/324 = 140/324 = 0.432

I am not sure what happened from the second moment part up to the calculation of variance. I never calculated variance like this. Can someone elaborate how second moment was found and how it led to the calculation of variance?

Kurapika
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1 Answers1

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The second moment is $$\mathbb E(x^2)=\frac1{n}\sum_{i=1}^m n_i\cdot x_i^2$$

with $m=3, n=18, n_1=3, n_2=5, n_3=10, x_1=5, x_2=3, x_3=4$.

And we have the relation $\sum\limits_{i=1}^3 n_i=n$

And then the variance is $Var(x)=\mathbb E(x^2)-[\mathbb E(x)]^2$

callculus42
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