Title: Repeated samples are normally distributed
Course: UGQuants (SSPS) > Section: Probability

Sampling and the normal distribution: repeated samples are normally distributed

Using the example of 'sampling' a repeated series of coin tosses, we saw that the estimates of a population parameter produced by a large number of repeated random samples takes the form of a normal distribution. In that distribution the value of the population parameter will be the mean. Given this, we can calculate the probability of any of the estimates that we get from a sample being near or far from the true population value (i.e. the mean/highest point of our curve).

For any one random sample we will have about a 68% chance of our estimate being within one standard deviation unit above or below the population parameter (the mean of our distribution), and about a 95% chance of being within two standard deviations. The standard deviation of a normal curve is thus called the standard error of the mean. The standard error of the mean will be equal to the standard deviation of the population parameter divided by the square root of the size of the sample.

Try it for yourself next...

The WDI 2011 dataset has 214 cases in it. Since it is a dataset comprising every country in the world, we could say it has information about the complete population (with the population units being countries).

Let us imagine that we are researchers interested in the average female life expectancy, but we only have the resources to collect data from 50 cases of this population. We can simulate this by getting SPSS to pick a random sample of 50 of the 214 cases and calculate the mean life expectancy for this sample. We could then repeat this exercise forty-nine times as if we had undertaken fifty successive sample surveys of 50 members of this population drawn at random.

You could do this by using the select cases command to ask SPSS to take a random sample of 50 cases and then using the descriptives command to calculate the mean female life expectancy of the selected cases. And then repeat that 49 times …

This would be unnecessarily laborious. Instead we can use command syntax to tell SPSS to do this sequence of operations fifty times: that is to say, it will take fifty different random samples and calculate the mean for each one. The command syntax for this is located in a text file.

To do this exercise, open the WDI2011 dataset file, open the text file as  syntax file, select all of the syntax, and then run the instructions in the syntax file either typing command-R or selecting the command 'run' from the SPSS menu. The result will be fifty results for the mean of the variable 'LE_female' (from each of our fifty, separate, randomly drawn samples).

SPSS will produce the result for the mean female life expectancy of the sample obtained from each of 50 successive random samples. Take a sheet of squared paper and draw an X and Y axis with the X-axis taking the values 69 to 76 in divisions of 0.5 years.

On the Y axis plot the number of times the sample mean produced by SPSS for each 50 country random sample falls within each of the half year divisions on the X axis.

Try it now and then have a look at the results you get...