Standard error is a measure of variability in a set of data that describes the range of values that the sample mean could take, given that the sample is randomly selected from the population. It is a quantifiable measure of the uncertainty associated with a sample mean.

Formula:

Standard Error (SE) = Standard Deviation (SD) / Square Root(Sample Size)

where:

• SE is the standard error of the mean
• SD is the standard deviation of the sample
• n is the sample size

Interpretation:

• The standard error is a measure of how much the sample mean deviates from the true population mean.
• A low standard error indicates that the sample mean is very close to the population mean, with a high degree of certainty.
• A high standard error indicates that the sample mean is further from the population mean, with a lower degree of certainty.

Applications:

• Confidence intervals: Standard error is used to calculate confidence intervals, which provide a range of values within which the true population mean is likely to fall.
• Hypothesis testing: Standard error is used in hypothesis testing to determine whether there is a significant difference between sample means.
• Standard error of the mean (SEM): The SEM is a standardized measure of standard error, calculated by dividing the standard error by the square root of the sample size. It is used to compare standard errors across different samples.

Example:

A sample of 25 students’ test scores has a standard deviation of 10 points. Calculate the standard error of the mean:

SE = 10 / sqrt(25) = 2 points

This means that the sample mean is likely to be within 2 points of the true population mean.

Key Points:

• Standard error is a measure of variability in a sample.
• It is calculated using the standard deviation and sample size.
• Standard error is used in confidence intervals, hypothesis testing, and SEM calculations.