In the previous post we learned that inferential statistics uses information from individuals observed in a sample to provide information about other individuals in the population that were not observed. A description of the population provides information about all individuals. But how is a sample or population described, and how are sample data used to make inferences about a population?

Since data from individuals fall over a range of values, a sample is usually described by an average value and a value of how much the data vary around that average (e.g., continuous data such as height or muscle tension can be described by means and standard deviations.) When statistics are used to describe a population, they are known as parameters. A sample statistic usually provides the best guess of a population parameter, but it is only an estimate and could be larger or smaller than the actual population parameter. That is, there is uncertainty about how well a sample statistic estimates a population parameter, and the amount of this uncertainty needs to be known.

One way to conceptualise this uncertainty is to think about how a sample statistic changes if an experiment is repeated many times. Knowing the range of values that a statistic takes when an experiment is repeated many times will indicate how precisely a sample statistic estimates a population parameter. In the example from the previous post, the physiologist DK Hill measured muscle tension in 8 frogs and toads. The mean muscle tension calculated from these animals is an estimate of mean muscle tension in all frogs and toads. If Hill repeated his experiment many times, the mean values from each repeated experiment would be distributed over a range centered around the population mean. The key idea is that this is the distribution of the sample mean statistic, not of data from individual animals. By knowing how much the mean statistic varies with repeated experiments, Hill could determine how precisely mean muscle tension in his sample estimates mean muscle tension in all frogs and toads.

To illustrate these ideas, Figure 1 shows the distribution of 1000 data points (blue bars, line of best fit in red) sampled from a population. These data have a mean of 0, a standard deviation of 1, and a bell-shaped Normal distribution where the area in blue adds up to 1.

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Figure 1:

If another 1000 data points were sampled and the process repeated many times, the means of these samples would be distributed as shown in Figure 2, plotted on the same x-axis scale. The average value of all the means is also 0. However, the means are distributed over a much smaller range compared to the individual data points. In fact, the standard deviation of these means is only 0.03. The standard deviation of a statistic (e.g., a mean) is given a special name: a standard error. Standard errors are needed to calculate the range of values a statistic is expected to take in most of the repeated experiments. This range of values is known as a confidence interval. Narrow confidence intervals indicate a statistic estimates a population parameter with high precision.

Standard deviation vs standard error. Standard deviations and standard errors are frequently mentioned in scientific reports but they refer to totally different things: standard deviations indicate variability of individual data, while standard errors indicate variability of statistics.

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Figure 2:

Summary

Sample statistics estimate a population parameter while standard errors tell us how precise this estimate is. If an estimate falls within a narrow range of values (i.e., narrow confidence intervals), we can be confident the sample provides a precise estimate of the population parameter.

For further reading on the difference between standard deviations and standard errors, see this short statistics note by Douglas Altman and Martin Bland.

In the next post, we will learn how sample size affects the precision of estimates.