Statistics
Mean and standard error calculator
Paste your replicate measurements to get the two numbers most lab reports and figures quote: the mean and its standard error. The standard deviation and sample size behind the SEM are shown too, since a reader needs them to interpret it.
Separate numbers with commas, spaces or new lines, or paste a spreadsheet column.
Decimals and negatives are fine; write 10:3 for a value that occurs 3 times.
Your results will appear here: the standard deviation first, then the rest of the summary and a chart.
Standard error of the mean
0.0500278
If you drew another sample of 6 from the same population, its mean would typically land within about 0.05 of this one's (4.258). The individual values vary more than that: their standard deviation is 0.1225.
- Count (n)
- 6
- Mean (x̄)
- 4.25833
- Standard deviation (s)
- 0.122543
- Variance (s²)
- 0.0150167
- Minimum
- 4.08
- Q1 (25%)
- 4.195
- Median
- 4.255
- Q3 (75%)
- 4.3375
- Maximum
- 4.42
- Range
- 0.34
More statistics (5)
- Relative SD (%RSD)
- 2.87771%
- Coefficient of variation
- 0.0287771
- Sum (Σx)
- 25.55
- Sum of squares, Σ(x − x̄)²
- 0.0750833
- IQR (Q3 − Q1)
- 0.1425
The wide, faint curve is the spread of the individual values (s = 0.1225). The narrow curve is the spread of the sample mean (SE = 0.05003) — √6 times tighter, because averaging cancels noise.
Show the working, step by step
The two formulas
x̄ = Σx / n SEM = s / √n
s is the sample standard deviation, computed with n − 1. The SEM estimates how
far the sample mean would typically move if you repeated the whole experiment with a fresh
set of n measurements.
A worked example: six enzyme assays
Six replicate assays of an enzyme preparation give activities of
4.21, 4.35, 4.08, 4.42, 4.19, 4.30 U/mL.
- Σx = 25.55 over n = 6, so the mean is x̄ = 4.25833 U/mL.
- Σ(x − x̄)² = 0.0750833, and dividing by n − 1 = 5 gives s² = 0.0150167.
- s = √0.0150167 = 0.122543 U/mL.
- SEM = 0.122543 ÷ √6 = 0.0500278 U/mL.
Rounded for a report: 4.26 ± 0.05 U/mL (mean ± SEM, n = 6). The SEM is rounded to one significant figure and the mean to the same decimal place.
From mean ± SEM to a 95% confidence interval
The familiar shortcut is that mean ± 2 SEM covers the true mean 95% of the time. That is the large-sample version, with 2 standing in for 1.96. With six replicates the correct multiplier is the t value on 5 degrees of freedom, 2.571:
95% CI = x̄ ± t* × SEM = 4.25833 ± 2.57058 × 0.0500278 = 4.12973 to 4.38693 U/mL
The shortcut would have given ± 0.100 instead of ± 0.129 — an interval about 22% too narrow. Below roughly 30 observations, use t; the confidence interval calculator does it directly, and the margin of error calculator gives the ± half-width alone.
SEM error bars in figures
Bar charts in biology and chemistry papers often show mean ± SEM. That is appropriate when the point of the figure is to compare means, but it makes the data look far less variable than it is. A few conventions keep figures honest:
- Say which bar it is in the legend: SD, SEM or 95% CI.
- State n for each group, so readers can convert SEM back to SD (s = SEM × √n).
- Use SD when the message is about variability between individual samples or subjects.
- Consider 95% CI bars when the message is about the mean; they already include the t multiplier.
How the SEM shrinks with n
The SEM falls with the square root of the sample size, not the sample size itself. If the assay's SD stays at about 0.1225, then:
| Replicates (n) | SEM (U/mL) |
|---|---|
| 6 | 0.0500 |
| 24 | 0.0250 |
| 96 | 0.0125 |
Each halving of the SEM costs four times the replicates. The standard deviation, by contrast, does not shrink with more data — it settles towards the true spread of the method. The general standard error calculator also works backwards, from a published SEM to the SD behind it.
Related calculators
-
Standard error calculator
The general SEM tool, and SD recovered from a published SE.
-
SD and standard error
Both measures of spread from one data set, side by side.
-
Confidence interval
The exact t-based interval for the mean.
-
Margin of error
The ± half-width from the SD, n and confidence level.
Common questions
How do I write mean ± SEM?
Give the mean, the ± sign, the standard error, and say which it is, for example "4.26 ± 0.05 U/mL (mean ± SEM, n = 6)". Without the label and the n, a reader cannot tell a standard error from a standard deviation, and they differ by a factor of √n.
How many decimal places should the mean and SEM have?
Round the SEM to one or two significant figures, then round the mean to the same decimal place. 4.25833 ± 0.0500278 becomes 4.26 ± 0.05. Reporting the mean to more places than its standard error supports implies precision you do not have.
Is mean ± 2 SEM a 95% confidence interval?
Approximately, for large samples. The exact multiplier comes from the t distribution and is larger for small n: 2.571 with 6 values, 2.093 with 20, 2.009 with 50. With only a handful of replicates, ± 2 SEM understates the uncertainty noticeably.
Why do SEM error bars look smaller than SD error bars?
Because the SEM is the SD divided by √n. With six replicates it is 41% of the SD; with 25 it is a fifth. SEM bars describe how precisely the mean is known, not how variable the individual measurements are.
If two SEM error bars overlap, is the difference not significant?
Overlap of SEM bars does suggest the difference is not significant at 5%, but the converse fails: bars that just fail to overlap do not mean p < 0.05. Roughly, for two similar groups the gap between SEM bars needs to be about one bar-length before p reaches 0.05. Run the t-test rather than judging by eye.