Standard error is the standard deviation of the sampling distribution of means. It shows how much a sample mean is expected to vary from the true population mean. A smaller standard error indicates that your sample mean is a more precise estimate of the real average.
How to Calculate Standard Error
Standard error is calculated by dividing the standard deviation of the sample by the square root of the sample size.
- Calculate the mean of the sample.
- Calculate each measurement’s deviation from the mean.
- Square each deviation from the mean.
- Add the squared deviations from Step 3.
- Divide the sum of the squared deviations by one less than the sample size (n-1).
- Calculate the square root of the value obtained from Step 5. This result gives you the standard deviation.
- Divide the standard deviation by the square root of the sample size (n). This result gives you the standard error.
- Subtracting the standard error from the mean / adding the standard error to the mean will give the mean ± 1 standard error.
Standard Error Formula
SE = standard error of the mean
σ = population standard deviation (usually estimated by the sample standard deviation, s)
n = number of independent observations.
The formula above is for a sample mean, but every statistic has its own standard error. For a difference between two independent means, the two standard errors combine as the square root of the sum of their squares. Standard errors of 2.0 and 2.4 give 3.12, not 4.4.
| Statistic | Standard error |
|---|---|
| Mean | s / √n |
| Proportion | √(p(1 − p) / n) |
| Difference between two independent means | √(SE₁² + SE₂²) |
| Correlation (on the Fisher z scale) | 1 / √(n − 3) |
Standard Error Example
The values in your sample are 52, 60, 55, and 65.
- Calculate the mean of these values by adding them together and dividing by 4. (52 + 60 + 55 + 65)/4 = 58 (Step 1).
- Next, calculate the sum of the squared deviations of each sample value from the mean (Steps 2-4).
- Using the values in this example, the squared deviations are (58 – 52)^2= 36, (58 – 60)^2= 4, (58 – 55)^2=9, and (58 – 65)^2=49. Therefore, the sum of the squared deviations is 98 (36 + 4 + 9 + 49).
- Next, divide the sum of the squared deviations by the sample size minus one and take the square root (Steps 5-6). The standard deviation in this example is the square root of [98 / (4 – 1)], which is about 5.72.
- Lastly, divide the standard deviation, 5.72, by the square root of the sample size, 4 (Step 7). The resulting value is 2.86, which gives the standard error of the values in this example.
Adding and subtracting one standard error gives 55.14 to 60.86. That range is a handy summary, but it is not a 95% interval. Small samples need wider ones.
With only four scores, the t-distribution sets the 95% multiplier at 3.18, not 1.96. It allows for the extra uncertainty when the standard error is estimated from so few values. The 95% confidence interval is therefore 58 ± 3.18 × 2.86, or about 48.9 to 67.1.
Key Takeaways
- Definition: The standard error (SE) is the standard deviation of a statistic across repeated samples. It shows how precisely a sample estimate pins down the population value.
- Formula: For a sample mean, SE = SD / √n. It falls as samples get larger and rises as scores vary more.
- Square-Root Law: Halving the standard error needs four times as many observations, so extra data helps less and less.
- SE vs. SD: The standard deviation describes the scores and stays steady as n grows. The standard error describes the estimate and shrinks.
- Interpretation: A small standard error means a replication would give a similar mean. It cannot detect bias in the sample.
- Independence: The formula assumes independent observations. Counting repeated measurements as separate cases makes the standard error too small.
- Error Bars: Standard-error bars that just touch do not mean p < .05. Always label what error bars show.
Large vs. Small Standard Error
Standard error estimates how accurately the mean of any given sample represents the true mean of the population.
A larger standard error signals more spread. The sample means scatter widely, so your own mean is more likely to misrepresent the true population mean.
On the other hand, a smaller standard error means the sample means sit closer together. A replication would then probably give a similar result, and your sample mean is more likely to be accurate.
Small does not mean unbiased. The standard error captures random sampling variation only, so a biased sample can still produce a tiny standard error.
The standard error rises when the scores vary more and falls as the sample grows. The fall has diminishing returns.
The formula divides by the square root of n, so halving the standard error needs four times as many observations. With a standard deviation of 15, a sample of 25 gives a standard error of 3.0, and a sample of 100 gives 1.5.
Standard Error vs. Standard Deviation
The two are easy to confuse, and the difference is the whole point. The standard deviation summarises the data in hand. The standard error summarises the estimate.
How the Two Measures Differ
Altman and Bland (2005) put the practical rule briefly. To say how widely scattered some measurements are, use the standard deviation. To show the uncertainty around the estimate of the mean, quote the standard error.
| Standard deviation (SD) | Standard error (SE) | |
|---|---|---|
| What it describes | How spread out the individual scores are | How much the estimate varies across repeated samples |
| Question it answers | How widely scattered are the measurements? | How precisely does this sample estimate the population value? |
| Effect of a larger sample | Settles near the population value and does not shrink | Shrinks in proportion to 1/√n |
| Used for | Describing the sample | Confidence intervals and tests |
The standard error is always smaller than the standard deviation once a sample has more than one observation.
A Worked Contrast
Take a sample of 25 adults with a mean of 100 and a standard deviation of 15. The standard error is 15 / √25, or 3.0.
If the scores are roughly normal, about 95% of individuals fall within two standard deviations of the mean, between 70 and 130. The 95% confidence interval for the population mean runs from 93.8 to 106.2.
The first range describes people. The second describes the estimate.
Quoting 100 ± 3 describes the precision of the mean. A reader who takes it to describe the people will wrongly believe that almost everyone scored between 97 and 103.
Journals therefore ask authors to write mean (SD) or mean (SE) instead of an unlabelled ± value (Altman & Bland, 2005).
The standard error also masks variability. With a standard error of 3, samples of 25, 100 and 400 have standard deviations of 15, 30 and 60.
Misuse in Published Research
Mixing the two up is common. Nagele (2003) checked how often four leading anaesthesia journals used the standard error to describe a sample.
- Aim: To measure how often leading anaesthesia journals used the standard error of the mean where the standard deviation was needed to describe the sample.
- Method: Nagele searched four anaesthesia journals by hand for 2001. He counted inappropriate uses of the standard error of the mean among 860 articles.
- Results: It was used inappropriately in 198 of the 860 articles (23%). Rates ranged from 11.5% to 27.7% across the four journals.
- Conclusion: About one article in four described its sample this way. Because the standard error is always smaller, readers underestimate how much individuals differed.
The count is plain and the finding is striking, but it covers one year and one specialty. Whether a use was inappropriate also rests on the author’s judgement of what each paper intended.
Look-Alike Terms
Several terms share the abbreviation or the name. Each answers a different question.
- Standard error of the mean (SEM): the standard error of a sample mean. “Standard error” on its own can mean the standard error of any statistic.
- Standard error of measurement: also abbreviated SEM. A psychometric quantity equal to SD × √(1 − reliability) that shows how much one person’s observed test score varies around their true score.
- Residual standard error: the typical size of the residuals around a regression line. It measures spread in the scores, not sampling variability.
- Probable error: an older measure equal to about 0.67 standard errors.
With a standard deviation of 15 and a reliability of .91, the standard error of measurement is 4.5 points. It describes uncertainty in one person’s score, not uncertainty from sampling people.
Limitations of the Standard Error
A standard error is only as trustworthy as the conditions behind it. Three problems matter most in practice.
Observations Must Be Independent
The formula SD / √n treats every observation as a separate piece of information. Pupils in one classroom, patients of one clinician or repeated trials from one participant share a common influence. They carry less independent information than their number suggests.
Statisticians measure the loss with the design effect. It equals 1 + (k − 1) × ρ, where k is the cluster size. Here ρ is the intraclass correlation, the share of the variation that lies between clusters.
Imagine 20 classrooms of 10 pupils each, with an intraclass correlation of .30. The design effect is then 3.7.
The true standard error is therefore about 1.9 times the naive one. The 200 pupils carry the information of only about 54. A naive 95% interval captured the true mean only about 69% of the time in 40,000 simulated studies.
Counting the wrong unit as n is common. In a random sample of 200 animal studies, only 22% used the correct unit of replication.
Another 46% showed pseudoreplication, which means treating non-independent measurements as independent replicates (Lazic et al., 2018).
Bias Is Invisible to the Standard Error
The standard error measures random sampling variation only. It shrinks as n grows whether or not the sample is representative. A huge sample with sampling bias can therefore produce a tiny standard error around a wrong estimate.
- Aim: To test whether very large surveys can still give badly wrong estimates, using first-dose COVID-19 vaccine uptake in US adults as the test case.
- Method: Two large surveys (Delphi-Facebook and the Census Household Pulse) were checked against US Centers for Disease Control and Prevention data. Delphi-Facebook collected about 250,000 responses a week, and the comparison ran from January to May 2021.
- Results: Delphi-Facebook overestimated uptake by 17 percentage points and the Household Pulse by 14, despite minuscule margins of error. A survey of 250,000 respondents can be no more accurate than a random sample of 10 people.
- Conclusion: Data quality matters more than data quantity. Compensating for poor quality with large size is, in the authors’ words, “a mathematically provable losing proposition.”
Bradley and colleagues (2021) could check the surveys against an external benchmark, which is unusual for this question.
The benchmark was itself uncertain. The result also covers one outcome in one country over a few months.
Misreading Error Bars
Error bars are easy to draw and often misread. In a web task, 473 authors from psychology, behavioural neuroscience and medicine adjusted two means until they judged them significantly different (p < .05).
Belia and colleagues (2005) concluded that many researchers had severe misconceptions about how error bars relate to significance. They did not reliably distinguish standard-error bars from confidence-interval bars, or independent means from repeated measures.
Cumming et al. (2007) offer a rule of thumb. It applies to independent means with 10 or more per group.
A gap of about one standard error between standard-error bars indicates p of about .05. By the same arithmetic, bars that just touch correspond to p of about .16, nowhere near significance.
The habit is widespread. Weissgerber and colleagues (2015) reviewed 703 articles in top physiology journals. Most of those with a bar graph showed mean ± standard error rather than mean ± standard deviation.
Always say in the figure legend what the bars show and how many independent samples each represents.
