Standard Error Calculator
Enter sample observations to evaluate the standard error of the mean, alongside supporting statistics.
Count
7
Sample mean
21.0000
Sample SD
6.4807
Standard error
2.4495
Interpretation
Precision: Smaller standard errors indicate that repeated samples would produce means closer to the true population mean.
Impact of sample size: The denominator √n shows why larger samples reduce the standard error.
How to Use This Calculator
- Enter sample measurements using spaces or commas.
- Review the sample mean, standard deviation, and standard error displayed in the results.
- Use the standard error when constructing confidence intervals or performing hypothesis tests for the mean.
- Collect more observations to decrease the standard error and improve precision.
Formulas
Sample mean μ̂ = Σx / n
Sample variance = Σ(x − μ̂)² / (n − 1)
Sample standard deviation s = √(sample variance)
Standard error of the mean = s / √n
These formulas align with the unbiased estimator for sample standard deviation, making the standard error suitable for inference.
Frequently Asked Questions
Is standard error the same as standard deviation?
No. Standard deviation measures data spread, while standard error measures the spread of the sample mean across repeated samples.
Can standard error be zero?
Only if every sample value is identical, resulting in zero variance and zero standard error.
How does sample size affect standard error?
Doubling your sample size reduces the standard error by roughly √2, improving estimate precision.
Do I need a normal distribution?
The standard error is defined for any distribution, though inference using it often assumes approximate normality for small samples.