Statistics
Standard deviation and standard error calculator
Standard deviation and standard error answer different questions about the same data. Enter your values to get both: the SD for how much the values vary, the SE for how precisely their mean is known. The chart shows the two side by side.
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
1.76383
If you drew another sample of 12 from the same population, its mean would typically land within about 1.76 of this one's (123.7). The individual values vary more than that: their standard deviation is 6.11.
- Count (n)
- 12
- Mean (x̄)
- 123.667
- Standard deviation (s)
- 6.1101
- Variance (s²)
- 37.3333
- Minimum
- 115
- Q1 (25%)
- 118.75
- Median
- 123
- Q3 (75%)
- 127.5
- Maximum
- 135
- Range
- 20
More statistics (5)
- Relative SD (%RSD)
- 4.94078%
- Coefficient of variation
- 0.0494078
- Sum (Σx)
- 1484
- Sum of squares, Σ(x − x̄)²
- 410.667
- IQR (Q3 − Q1)
- 8.75
The wide, faint curve is the spread of the individual values (s = 6.11). The narrow curve is the spread of the sample mean (SE = 1.764) — √12 times tighter, because averaging cancels noise.
Show the working, step by step
The formulas
SD: s = √( Σ(x − x̄)² ÷ (n − 1) ) SE: SE = s ÷ √n
The standard error is not measured separately. It is the standard deviation divided by the square root of the sample size, so every SE carries an SD inside it.
A worked example: systolic blood pressure
Twelve readings in mmHg: 118, 124, 131, 115, 127, 122, 135, 119, 126, 121, 129, 117.
- Mean: 1484 ÷ 12 = 123.667 mmHg.
- Squared deviations sum to 410.667, so the sample variance is 410.667 ÷ 11 = 37.3333.
- Standard deviation: s = √37.3333 = 6.11010 mmHg.
- Standard error: 6.11010 ÷ √12 = 6.11010 ÷ 3.46410 = 1.76383 mmHg.
Read together: individual readings typically sit about 6.1 mmHg from the mean, while the mean of 123.7 is itself uncertain by about 1.8 mmHg. A 95% confidence interval, mean ± t × SE with t = 2.201 on 11 degrees of freedom, runs from 119.78 to 127.55 mmHg.
What happens as n grows
Suppose the same spread held with four times as many readings, n = 48. The SD would stay near 6.11, but the SE would halve to 6.11 ÷ √48 = 0.882. More data sharpens the estimate of the mean without making blood pressure any less variable.
| n | SD (stays put) | SE = SD ÷ √n |
|---|---|---|
| 12 | 6.11 | 1.76 |
| 48 | 6.11 | 0.88 |
| 192 | 6.11 | 0.44 |
Which one to report
- Describing a sample (baseline characteristics, a table of participants): report the SD.
- Comparing means or estimating an effect: report the SE, or better a confidence interval built from it — see the margin of error calculator.
- Error bars on a chart: either can be right, but label them. SE bars look about √n times tighter than SD bars for the same data.
Swapping one for the other is one of the most common reporting errors in published research, and it usually flatters the data. The standard error calculator covers the SE on its own, including recovering an SD from a paper that only gives SE.
Related calculators
-
Standard deviation calculator
The general calculator, with both modes and a dot plot.
-
Standard error calculator
SEM from data, and the SD recovered from a published SE.
-
Mean and standard error
The mean with its precision, for reporting.
-
Confidence interval
Turn the SE into an interval for the mean.
Common questions
What is the difference between standard deviation and standard error?
The standard deviation describes the individual values: how far a typical one sits from the mean. The standard error describes the mean: how far it would typically move if you repeated the whole study with a new sample of the same size. SE = SD ÷ √n, so the SE is always the smaller of the two once n is above 1.
Why does the standard error shrink as the sample grows but the SD does not?
More data does not make people, parts or plants less variable, so the SD settles towards the population value rather than falling. But averages of more values wobble less, so the SE keeps shrinking — in proportion to 1/√n. Four times the sample halves the SE.
Which should my error bars show?
It depends on the claim. SD bars show the spread of the data. SE bars show the uncertainty in each mean but are narrow and easy to over-read; non-overlapping SE bars do not mean a significant difference. 95% confidence-interval bars are the most directly interpretable. Whatever you choose, say so in the caption.
How do I get the SD from a reported standard error?
Multiply by the square root of the sample size: SD = SE × √n. A paper reporting SE = 1.76 with n = 12 implies SD = 1.76 × 3.464 = 6.10. The standard error calculator has a tool for exactly this conversion.
Is SEM the same as SE?
Yes, when the statistic is a mean. SEM is "standard error of the mean". Other statistics have standard errors too — of a proportion, of a regression slope — each with its own formula.