Title: SPSS and the normal curve
Course: UGQuants (SSPS) > Section: Probability

SPSS and the normal curve

This is a possible result:

The distribution of the sample estimates broadly resembles a normal curve. Indeed the most likely result is for the sample mean to approximate the population mean. The further we get from the population mean, the fewer the number of samples with such an estimate for the population mean.

Now you can probably see why we chose to take samples from the WDI dataset, because unlike most populations, we actually know the parameter for this population: the mean female life expectancy for all 214 cases in the survey is 72.3 (strictly speaking, however, we only have information for 193 cases, but let’s pretend our data is complete).

If we took an infinite number of random samples of a population, and plotted the estimates they gave us for some characteristic of the population (such as mean age), this plot would be a normal curve with its peak at the value of the  population characteristic we were trying to measure.

The standard deviation of this normal curve has a special name: the standard error of the mean. We can also calculate this. It will be equal to the standard deviation of the population parameter) divided by the square root of the sample size. We usually do not know the standard deviation of the population parameter, but - so long as our sample size is about 100 or more - the standard deviation of the sample statistic will be a close approximation to it.

From this formula we can see that the standard error of the mean will increase if the value we were trying to estimate had a lot of spread (that is there is a lot of variation in the population, say a very wide range of ages) and would decrease as we increased the size of the sample.

Think about these properties and they should appear logical. As our sample size increases, we reduce the likelihood of large error in our estimate. If we are trying to estimate the value of something that does not vary much in our population (e.g. the height of adult males) we are likely to get an estimate closer to the mean than would be the case if we were trying to measure something that varied a great deal (e.g. monthly income of adult males).

The bad news is that we need to increase our sample size exponentially in order to improve the accuracy of our results arithmetically. In other words, it takes a massive investment to even modestly improve the accuracy of our estimates once we have gone beyond a certain point.