How to use the method of moments to find the mean and variance on R?

Question

I have a set of 50 independent observations and need to model these observations as a normal distribution. I'm assuming I need to use the method of moments but not entirely sure how to find the mean and variance?

Answer 1

Here is a quick demonstration using simulated data. Maybe it will help you get oriented for working your exercise.

First, I use R statistical software to simulate 50 observations from the distribution $\mathsf{Norm}(\mu = 100,\, \sigma = 15).$ I find the sample mean $\bar X = 98.33$ and sample standard deviation $S = 15.54$; note that $\bar X$ and $S^2$ are the method-of-moments estimates of $\mu$ and $\sigma^2.$ (Look in your text or notes for details of finding $\bar X$ and $S.$)

set.seed(417)
x = rnorm(50, 100, 15)
a = mean(x);  s = sd(x)
a; s
## 98.3272
## 15.53877

Here is a list of the sample (rounded for compactness), the numbers in brackets show the index of the first observation in each row. There are ten observations per row.

round(x)
 [1] 126  87  80 103  73 121  83  68 124  74
[11]  85 101 105  89 101  71  93  96 102  97
[21] 111  67  93 110  84  98 110  86 113  86
[31] 123  99 105  93 106  93 111  96  86 110
[41]  78 113 114 109 111 104 123 121  99  84

Here is a histogram of the fifty observations. The tick marks along the horizontal axis show the positions of the individual observations.

The red curve is the density function of the 'best-fitting' normal distribution. It is obtained by using mean $\hat \mu = \bar X$ and $\hat \sigma = S$ (where 'hats' on parameters indicate they have been estimated).

This is not a bad fit for only $n = 50$ observations. $\hat \mu = \bar X$ is not far from $\mu = 100$ and $\hat \sigma = S$ is not far from $\sigma=15.$ But if you used a larger sample, you would get better results. Here is the same kind of figure as above, but for a sample of size $n = 500.$