Sampling

However, often we do not known both the mean and standard deviation values for the population we are investigating. Instead, we have investigated a sample from this population. Since our sample is unlikely to be representative of the population, we cannot use the formulas for the mean and standard deviation above. These formulas measure population parameters (that is, they measure the population mean and population standard deviation) and not sample statistics (that is, they do not measure the sample mean and sample standard deviation).

Population parameters should be replaced by sample statistics when we are using a sample to investigate a population because they do not take into account the sampling error (that is, the extent to which the sample does not reflect the population). This can be seen in our example where a Government statistician looked at the marks of 100 students from a single school (our sample) to assess the maths performance of 16 year olds nationwide (our population). In this case: What if the school that the 100 students’ marks came from was one of the best in the country. Here, we would probably expect the mean score of our sample to be higher when compared against the mean score achieved by 16 year olds countrywide (our population). If the Government statistician took these 100 students marks and treated them as population parameters (that is, the population mean and population standard deviation) rather than sample statistics (that is, the sample mean and sample standard deviation), the statistician would over-estimate the maths ability of 16 year olds nationwide. Rather, sample statistics should be used because they take into account the potential error in a sample. These statistics are called the sample mean and sample standard deviation.

In principle, the sample mean and sample standard deviation are very similar to the population mean and population standard deviation. Indeed, if we add up all the scores of a population, X, and divide it by the number of scores, N, then we arrive at the population mean. From the population mean, we can then work out the population standard deviation. Where the scores are normally distributed (see the statistical guide on Frequency Distributions) we can then calculate a z-score (see the statistical guide on Standard Score (z-score)) to work out probability values using our standard normal distribution tables.



Just as the sample mean is an estimation of the population mean, the sample standard deviation, s, is also an estimation of the population standard deviation, σ . If we look at the formulas below for calculating the sample standard deviation there a clear similarities to the formula used to calculate the population standard deviation.



In the case of the population standard deviation, σ , this is the square root, , of the population sum of squares, (note that we are interested in the score minus the population mean), divided by the total number of scores, N. However, in the case of the sample standard deviation, this is the square root, , of the sample sum of squares, (note that we are interested in the score minus the sample mean), divided by the degrees of freedom (df), n-1 (here, n is the sample size). [Note: we use the second formula because it allows us to calculate the sample standard deviation without first having to calculate the sample mean].