Statistics note: How does sample size affect precision of estimates?

In the previous post we learned that a sample statistic (e.g., a sample mean) is used to estimate a population parameter (e.g., the population mean), and the standard error of the sample statistic indicates the amount of precision around the estimate of the population parameter. A small standard error indicates that a sample statistic estimates a population parameter with high precision. Why is this so, and what determines whether a standard error is large or small?

A standard error indicates how variable a sample statistic is if an experiment is repeated many times. A small standard error indicates the sample statistic only varies by a small amount with many repeats of the experiment, so a small standard error is desirable. A standard error of a mean statistic is calculated from the standard deviation in this way:

statsnotes2_seFrom the formula, the standard error depends on the variability of data in the sample (i.e., standard deviation) and the number of samples in the experiment (i.e., sample size) such that for a given standard deviation, the standard error decreases as sample size increases. This means that a precise estimate of a population parameter is only obtained when sample size is large, or when variability in the sample is small.

To illustrate how sample size affects the calculation of standard errors, Figure 1 shows the distribution of data points sampled from a population (top panel) and associated sampling distribution of the mean statistic (bottom panel) as sample size increases (columns 1 to 3). All samples have a mean of 0 and standard deviation of 1, and all plots share the same x-axis scale.

 


 Figure 1: Distribution of data points (a, b, c) and associated sampling distribution of the mean (d, e, f) for samples of size 10 (a, d), size 30 (b, e) and size 100 (c, f).

 statsnotes2

 

From the figure, it is clear that the standard error decreases with increasing sample size: the standard errors of the mean statistics are 0.32 (column 1), 0.18 (column 2) and 0.10 (column 3). Standard error and sample size also do not change at the same rate because standard error decreases as the square-root of the number of samples increases. For example, when sample size increases from 10 to 100 (a factor of 10), the standard error only decreases by a factor of 3.

Experimental studies (and conclusions based on these studies!) are often conducted on small samples, but it not possible to obtain precise estimates of effects when sample size is small. For this reason, scientists need to conduct experiments on large samples to obtain precise esimates of effects, so they can be confident about their findings.

Summary

The standard error is dependent on sample size: larger sample sizes produce smaller standard errors, which estimate population parameters with higher precision. Scientists need to test more samples in their experiments to increase the certainty of their estimates.

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