Standard Error of the Mean Calculator
Calculate the Standard Error of the Mean (SEM), 95% confidence interval and margin of error. Enter raw data or provide SD and sample size directly.
What is the Standard Error of the Mean?
The Standard Error of the Mean (SEM) is a statistical measure that quantifies how precisely a sample mean estimates the true population mean. It answers the question: "If I took many samples of the same size, how much would the sample means vary?"
The SEM formula is:
Where:
- SD = the standard deviation of the sample
- n = the sample size (number of observations)
- √n = the square root of n
For example, if a sample of 25 measurements has SD = 10: SEM = 10 ÷ √25 = 10 ÷ 5 = 2.0.
SEM Formula Explained
The SEM formula has an important mathematical property: as sample size (n) increases, SEM decreases in proportion to √n. This means:
- To halve the SEM, you must quadruple the sample size
- To reduce SEM by a factor of 3, you need 9 times as many observations
- Doubling the sample size reduces SEM by only about 29%
This has important implications for research design. Collecting more data always improves precision, but with diminishing returns. A study of n = 100 is four times as precise as n = 25, but a study of n = 400 is only twice as precise as n = 100.
How to Calculate SEM from Raw Data
- Calculate the mean: x̅ = Σx ÷ n
- Calculate variance: Find each deviation from the mean, square it, sum all squared deviations, divide by (n-1) for sample variance
- Calculate SD: SD = √(sample variance)
- Calculate SEM: SEM = SD ÷ √n
Our calculator above performs all these steps automatically when you enter raw data.
Standard Error vs Standard Deviation
These two measures are frequently confused. Here is the key distinction:
| Feature | Standard Deviation (SD) | Standard Error (SEM) |
|---|---|---|
| Measures | Variability of individual data points | Precision of the sample mean |
| Changes with n? | Stays roughly constant | Decreases as n increases |
| Formula | √[Σ(x-x̅)² ÷ (n-1)] | SD ÷ √n |
| Use when | Describing data spread | Estimating population mean |
| Always larger? | SD ≥ SEM | SEM ≤ SD |
A useful rule: SEM is always smaller than SD (unless n = 1, when they are equal). If you see a very small "error bar" in a scientific paper, check whether it represents SD or SEM — SEM bars look deceptively small and can give a misleading impression of low variability in the data.
95% Confidence Interval Using SEM
The most common application of SEM is constructing a 95% confidence interval (CI) for the population mean:
The value 1.96 is the z-score that corresponds to 95% of the normal distribution (1.96 standard deviations from the mean capture 95% of the distribution in each direction).
Worked example: A sample of 36 patients has a mean blood pressure of 125 mmHg with SD = 18 mmHg.
- SEM = 18 ÷ √36 = 18 ÷ 6 = 3.0 mmHg
- Margin of error = 1.96 × 3.0 = 5.88 mmHg
- 95% CI = 125 ± 5.88 = [119.12, 130.88] mmHg
Interpretation: "We are 95% confident that the true population mean blood pressure lies between 119.12 and 130.88 mmHg."
Note: For small samples (n < 30), the t-distribution should be used instead of the z-distribution (1.96). The t-value for 95% CI with 9 degrees of freedom (n=10) is 2.262, for example. Our calculator uses the z-approximation (1.96), which is appropriate for n ≥ 30.
When to Use Standard Error of the Mean
Research and Science
SEM is widely used in scientific papers, clinical trials, psychology experiments and surveys. When a study reports "mean ± SEM," the SEM communicates how reliable the mean estimate is. Journals often require 95% confidence intervals rather than SEM bars in figures, as CIs are more interpretable.
Quality Control
In manufacturing and quality assurance, SEM helps determine whether a production process is producing items within specification. A narrow confidence interval around the mean measurement indicates high precision and consistency.
Survey Analysis
In opinion polls and surveys, SEM is used to calculate the margin of error. A poll reporting "Labour: 42% ± 3%" uses a form of standard error (in this case the standard error of a proportion) to express the precision of the estimate.
A-Level and University Statistics
SEM appears in A-level Statistics and is a core concept in university-level statistics courses. It underpins hypothesis testing, power analysis and sample size calculations across medicine, biology, psychology, economics and social sciences.
Frequently Asked Questions
What is the standard error of the mean formula?
SEM = SD ÷ √n, where SD is the sample standard deviation and n is the sample size. SEM measures how precisely the sample mean estimates the true population mean.
What is the difference between standard deviation and standard error?
SD measures the spread of individual data points. SEM measures how precisely the sample mean estimates the population mean. SD stays roughly constant as n grows; SEM decreases as n increases. SD describes your data; SEM describes the uncertainty of your mean estimate.
How do I calculate a 95% confidence interval using SEM?
95% CI = x̅ ± 1.96 × SEM. For example, mean = 50, SEM = 2.5: CI = 50 ± (1.96 × 2.5) = 50 ± 4.9 = [45.1, 54.9]. For small samples (n < 30), use the t-distribution instead of 1.96.
When should I report SEM instead of SD?
Use SEM when communicating the precision of your mean estimate (inferential statistics). Use SD when describing the variability within your dataset (descriptive statistics). In scientific publications, 95% confidence intervals are increasingly preferred over SEM as they are more directly interpretable.
How does sample size affect the standard error?
SEM decreases as n increases: SEM = SD / √n. To halve the SEM, quadruple n. With SD = 10: n=25 gives SEM=2.0, n=100 gives SEM=1.0, n=400 gives SEM=0.5.
What is a good standard error value?
There is no universal threshold. Evaluate SEM relative to the mean — SEM/mean × 100% should ideally be under 10% for acceptable precision. More importantly, consider whether the resulting confidence interval is narrow enough to be practically meaningful for your research question.