Standard deviation and standard error

Question

I have a homework in which I am using data from NFL teams. This information was taken from the ESPN site and I was asked to calculate the following: $mean$, $standard$ $deviation$ and $standard$ $error$ of certain data. I know that there are 32 teams so I am taking data $(32$ $numbers)$ which I suppose represent the entire population and not a sample. This means that I have 32 "numbers" to take into account in order to make the calculations.

The mean is easy to get because I only add the 32 $numbers$ and divide the total by 32.

Standard deviation can be calculated for the population and for a sample as follows:

$\sigma = \sqrt{\frac{\sum_1^N(x_i - \mu)^2}{N}}$

$s = \sqrt{\frac{\sum_1^n(x_i - \bar{x})^2}{n - 1}}$

I am also asked to calculate $standard$ $error$ which I understand is the standard deviation of the sampling distribution.

I am not sure how to proceed.

Which formula should I use to calculate standard deviation, population or sample? I am actually using the population mean formula because I think I have all the data and not a sample.

How will I calculate standard error?

Answer 1

Technically, the standard error is the standard deviation of an estimator. Most commonly, this refers to sample mean $\bar X$ as an estimator of the population mean $\mu.$

So the 'standard error of the mean' is $SD(\bar X) = \sigma/\sqrt{n}.$ If $\sigma$ is unknown, it is estimated as the sample standard deviation $S.$ This means that the '(estimated) standard error' is $S/\sqrt{n}.$

Minitab's describe procedure for 32 observations, including SE Mean, can be illustrated as follows:

 MTB > rand 32 c1;    # put 32 observations from NORM(100, 10)
 SUBC> norm 100 10.   #  into C1 of the Worksheet
 MTB > round c1 c1    # round results to integers
 MTB > print c1

 Data Display 

 C1
    104    98    99   105    98    82   111    97   121   110    91   104    97
    110   103    93   102   110   103    90    85   104   102    90    79    98
     93   104    95    93    94   106

 MTB > describe c1

 Descriptive Statistics: C1 

 Variable   N  N*   Mean  SE Mean  StDev  Minimum     Q1  Median      Q3
 C1        32   0  99.09     1.59   8.97    79.00  93.00   98.50  104.00

 Variable  Maximum
 C1         121.00

In this instance the 32 observations are treated as a sample, using your formula for $S$ to get 8.97. Then SE Mean is $8.97/\sqrt{32} = 1.585687 \approx 1.59.$

In your case, where you are using the entire sample, I would use $\sigma = 8.83$ from your formula for $\sigma$ and divide by $\sqrt{32}$ to get $1.65.$ But I would also explain that I used the formula for $\sigma$ instead of the formula for $s$ because I am working with the whole population instead of a sample.