Standard Error Calculator

Standard Error (SE) measures the variability or uncertainty in the estimate of a population mean based on a sample. It tells you how much the sample mean is expected to differ from the true population mean. A smaller SE indicates a more precise and reliable estimate.

The Standard Error is calculated as SE = s / √n, where s is the sample standard deviation and n is the sample size. For raw data, you first compute the mean and standard deviation, then divide by the square root of the number of observations.

Select 'Raw Data' to enter a comma-separated list of numbers, or choose 'Summary Data' to input your standard deviation and sample size directly. The calculator will automatically compute the Standard Error, Mean, Sum of Squares, and Standard Deviation.

Standard Deviation measures the spread or variability of individual data points within a dataset. Standard Error measures the precision of the sample mean as an estimate of the population mean. SE is always smaller than SD and decreases as sample size increases.

Standard Error decreases as sample size increases because SE = s / √n. Larger samples produce a more reliable estimate of the population mean, resulting in a smaller SE. Doubling the sample size reduces the SE by a factor of √2 (approximately 1.41).

Use Summary Data mode when you already know the standard deviation and sample size but don't have access to the individual data points. This is common when working with published research results, pre-aggregated datasets, or large datasets where only descriptive statistics are available.

Sum of Squares is the sum of the squared differences between each data point and the sample mean: SS = Σ(x − x̄)². It is an intermediate step in calculating the variance and standard deviation. A higher SS indicates more spread in the data.

Standard Error cannot be negative since it is calculated from a square root. It equals zero only when all data values are identical (standard deviation is zero) or theoretically when the sample size approaches infinity. In practice, any real dataset with variation will produce a positive SE.