However, notice how the blue distribution (N=100) clusters more tightly around the actual population mean, indicating that sample means tend to be closer to the true value. Both distributions center on 100 because that is the population mean. The probability distribution plot displays the sampling distributions for sample sizes of 25 and 100. We know that the larger sample size produces a smaller standard error of the mean (1.5 vs. To calculate the SEM, I’ll use the standard deviation in the calculations for sample sizes of 25 and 100.Īs expected, quadrupling the sample size cuts the SEM in half. These scores have a mean of 100 and a standard deviation of 15. The SEM equation quantifies how larger samples produce more precise estimates! Mathematical and Graphical Illustration of Precisionįor this example, I’ll use the distribution properties for IQ scores. The total effect is that the standard error of the mean declines as the sample size increases.īecause the denominator is the square root of the sample size, quadrupling the sample size cuts the standard error in half. However, the denominator increases because it contains the sample size. During this process, the numerator won’t change much because the variability in the underlying population is a constant. Imagine that you start a study but then increase the sample size. The denominator is the square root of the sample size (N), which is an adjustment for the amount of data. The numerator (s) is the sample standard deviation, which represents the variability present in the data. Here’s the equation for the standard error of the mean. The reason becomes apparent when you understand how to calculate the standard error of the mean. I’m sure you’ve always heard that larger sample sizes are better. Related posts: Sample Statistics are Always Wrong (to Some Extent)! and How Hypothesis Tests Work Standard Error of the Mean and Sample Size As I mentioned, the SEM is the doorway that opens up to these standard tools of inferential statistics. Often, these statistics are more helpful than the standard error of the mean. However, statistical software uses SEMs to calculate p-values and confidence intervals. Again, larger values correspond to wider distributions.įor a SEM of 3, we know that the typical difference between a sample mean and the population mean is 3. Larger values correspond with broader distributions and signify that data points are likely to fall farther from the sample mean.įor the standard error of the mean, the value indicates how far sample means are likely to fall from the population mean using the original measurement units. The value for the standard deviation indicates the standard or typical distance that an observation falls from the sample mean using the original data units. Let’s return to the standard deviation briefly because interpreting it helps us understand the standard error of the mean. Related posts: Populations, Parameters, and Samples in Inferential Statistics and Interpreting P-values Interpreting the Standard Error of the Mean Consequently, you can assess the precision of your sample estimates without performing the repeated sampling. Statisticians know how to estimate the properties of sampling distributions mathematically, as you’ll see later in this post. Hello SEM!įortunately, you don’t need to repeat your study an insane number of times to obtain the standard error of the mean. You want your sample mean to be close to the population parameter. Specifically, you’re hoping that the sampling error is small. When using a sample to estimate the population, you want to know how wrong the sample estimate is likely to be. Sampling error is the difference between the sample and population mean. When you have a sample and calculate its mean, you know that it won’t equal the population mean exactly. That’s crucial information for inferential statistics!
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