The standard deviation of the mean, also known as the standard error of the mean (SEM), measures the variability of sample means around the population mean. You calculate it by dividing the standard deviation of the sample by the square root of the sample size.
Here's the formula:
SEM = s / √n
where:
- SEM is the standard error of the mean
- s is the sample standard deviation
- n is the sample size
Steps to Calculate SEM:
- Calculate the sample standard deviation (s): This measures the spread of data points around the sample mean. You can use a calculator or statistical software to calculate this.
- Calculate the square root of the sample size (√n): This represents the number of observations in your sample.
- Divide the sample standard deviation (s) by the square root of the sample size (√n): This gives you the standard error of the mean (SEM).
Example:
Let's say you have a sample of 25 students and their average height is 170 cm with a standard deviation of 10 cm.
- s = 10 cm
- √n = √25 = 5
- SEM = 10 cm / 5 = 2 cm
Therefore, the standard error of the mean is 2 cm. This means that the average height of the sample is likely to be within 2 cm of the true population mean.
Practical Insights:
- The smaller the SEM, the more precise the estimate of the population mean.
- SEM is often used in hypothesis testing and confidence interval calculations.
- SEM is a useful statistic for comparing different groups or treatments.
