Standard Error Explained: Formula, Meaning and SD vs SE

Standard error is mistakenly reported instead of standard deviation in up to 64% of published articles. Understanding the standard error formula, and exactly how it differs from standard deviation, prevents this common statistical mistake.

If you have ever read a research paper, a psychology textbook, or a set of exam statistics and seen a mean reported with a “±” figure next to it, you may have wondered whether that figure was the standard deviation or the standard error. It is a fair question, because the two terms sound almost identical, use overlapping calculations, and are routinely confused, even in published academic papers. This guide sets out exactly what standard error is, how it differs from standard deviation, how to calculate it from raw data, and why it matters for interpreting research.

Standard error and standard deviation both describe variability, but they describe variability in two different things. Standard deviation tells you how spread out individual scores are within a single sample or population. Standard error tells you how precise your sample mean is as an estimate of the true population mean. Getting this distinction clear is the single most useful thing you can take from this article, so it comes first, before any formulas.

What Is Standard Error?

Standard error, most often written as SE or SEM (standard error of the mean), is a measure of how much a sample mean is likely to differ from the true mean of the population it was drawn from (American Psychological Association, n.d.). It answers a specific question: if you repeated your study many times, each time drawing a new sample from the same population and calculating a new mean, how much would those means vary from one another?

In practice, researchers do not usually repeat a study thousands of times to find this out directly. Instead, standard error is estimated from a single sample using a simple formula, covered in detail below, that relates it to the sample’s standard deviation and its size. The smaller the standard error, the more confident you can be that your sample mean is a good estimate of the population mean.

Standard error belongs to the branch of statistics concerned with making inferences about a wider population from a limited sample — inferential statistics — which is why it turns up constantly in confidence intervals, significance testing, and anywhere a researcher wants to say not just “this is our result” but “here is how much we trust this result.” Standard deviation, by contrast, is a descriptive statistic: it summarises the data you actually collected, without making any claim about a wider population.

That distinction between description and inference is worth holding onto, because it explains why the two statistics exist for different purposes rather than being two ways of measuring the same thing.

Standard Error vs Standard Deviation: The Key Difference

This is the distinction that almost everyone searching for “standard error” actually wants resolved, so here it is in one sentence: standard deviation measures the spread of individual data points around a mean; standard error measures the spread, and therefore the precision, of the sample mean itself.

Put another way:

• Standard deviation answers: “How different are individual scores from each other?”
• Standard error answers: “How different would my sample’s average be if I repeated this study with a new sample?”

That last row matters in practice: because SE is calculated by dividing SD by the square root of the sample size, standard error will always be smaller than standard deviation once a sample has more than one observation. This is precisely why the two are so easy to mix up when misused — a smaller, more flattering-looking error figure is sometimes reported using SE, even in contexts where SD is the statistic that is actually relevant. If a researcher wants to describe how varied their sample was, SD is the right tool. If they want to communicate how confident they are in their estimate of the average, SE is the right tool.

Using one where the other belongs is a genuine and surprisingly common reporting error, not just a beginner’s misunderstanding: a 2025 analysis of manual medicine journals found standard error had been used in place of standard deviation to describe data variability in between 14% and 64% of the articles reviewed, depending on the journal (Manovic et al., 2025). If professional researchers and peer reviewers make this mistake regularly, it is well worth being deliberate about getting it right, whether you are writing up a study, marking a research methods assignment, or simply reading a results section critically.

If you are also working with individual scores and want to know how far a single score sits from the mean in standard deviation units, that is a related but separate concept called a z-score, which is covered in detail in our companion guide.

Why These Two Get Confused

The confusion is understandable, and it is not really about carelessness. Both statistics:

• Are called “standard [something]”
• Are built from the same underlying ingredient, the standard deviation
• Are usually reported as a single number attached to a mean, often with a “±” symbol
• Appear in the same statistical software output, frequently right next to each other, sometimes in adjacent columns labelled only “SD” and “SE” with no further explanation

The difference only becomes obvious once you ask “the spread of what, exactly?” Standard deviation is about the spread of your actual data points. Standard error is about the spread of an imagined collection of sample means, drawn from repeatedly sampling the same population — a concept statisticians call the sampling distribution of the mean. Because that second idea is more abstract and cannot be seen directly in a single dataset the way individual data points can, it is easy for the two to blur together, especially for anyone encountering both terms for the first time in the same statistics module.

Understanding the Sampling Distribution

To really grasp why standard error works the way it does, it helps to picture the idea it is built on: the sampling distribution.

Imagine you wanted to know the average height of every ten-year-old in the country. In practice, you cannot measure every ten-year-old, so you take a sample — say, 50 children — and calculate the mean height of that sample. Now imagine taking a second, completely separate sample of 50 different ten-year-olds and calculating their mean height too. It would almost certainly be slightly different from the first sample’s mean, purely by chance, because you happened to select a different group of children. If you kept doing this — taking sample after sample of 50 children and recording each sample’s mean — you would eventually build up a distribution of sample means. That distribution is the sampling distribution.

Three things are consistently true about a sampling distribution of means, and together they explain most of what makes standard error useful:

1. Its own mean equals the true population mean. Average enough sample means together and they converge on the real answer, even though any single sample might be slightly off.
2. Its spread is smaller than the spread of the original data. Sample means are more stable than individual scores, because extreme individual values get balanced out by everyone else in the sample.
3. Its shape tends toward a normal (bell-shaped) distribution as sample size increases, even when the underlying population data is not normally distributed. This result is known as the central limit theorem, and it is one of the reasons standard error calculations work reliably across so many different types of data.

Standard error is simply the standard deviation of that sampling distribution. Researchers do not usually generate a sampling distribution by hand — the formula below estimates it directly from a single sample, using the central limit theorem as its mathematical justification.

The Standard Error Formula and How to Calculate It

The Formula

The standard error of the mean is calculated as:

SE = SD / √n

Where:

• SD is the standard deviation of your sample
• n is your sample size (the number of observations)
• √n is the square root of the sample size

This formula shows precisely why SE and SD are related but not the same: SE is simply SD scaled down by the square root of how many observations went into the mean.

Step-by-Step Worked Example, Starting From Raw Data

Most explanations of standard error start with the standard deviation already calculated. It is worth seeing the whole process from raw scores, since that is what a real dataset actually looks like.

Suppose a researcher records the reaction times, in milliseconds, of five participants completing a simple task: 240, 250, 255, 245, 260.

Step 1: Calculate the mean. (240 + 250 + 255 + 245 + 260) ÷ 5 = 1250 ÷ 5 = 250

Step 2: Find each score’s deviation from the mean, and square it.

• 240 − 250 = −10, squared = 100
• 250 − 250 = 0, squared = 0
• 255 − 250 = 5, squared = 25
• 245 − 250 = −5, squared = 25
• 260 − 250 = 10, squared = 100

Step 3: Sum the squared deviations, then divide by n − 1 (this is the sample variance). 100 + 0 + 25 + 25 + 100 = 250 250 ÷ (5 − 1) = 250 ÷ 4 = 62.5

Step 4: Take the square root of the variance to get the standard deviation. √62.5 ≈ 7.91

Step 5: Divide the standard deviation by the square root of the sample size to get the standard error. SE = 7.91 ÷ √5 = 7.91 ÷ 2.236 ≈ 3.54

So this sample has a mean reaction time of 250ms, a standard deviation of 7.91ms, and a standard error of approximately 3.54ms. The standard deviation tells us individual reaction times in this sample varied by around 7.91ms from the mean. The standard error tells us that if this study were repeated many times with new samples of five participants, the resulting sample means would typically vary by around 3.54ms from one sample to the next — a noticeably tighter range than the spread among the individual scores.

A larger, rounder example makes the relationship even easier to see. Imagine instead a sample of 64 participants with a standard deviation of 16ms. Using the same formula:

SE = 16 ÷ √64 = 16 ÷ 8 = 2.0

Note that when the true population standard deviation is unknown, which is almost always the case in real research, the sample standard deviation is used as an estimate in this formula. This is standard practice, and the reason this figure is sometimes labelled an estimated standard error in statistical software output.

How Sample Size Affects Standard Error: A Worked Comparison

One of the most useful things to understand about standard error is its relationship with sample size: larger samples produce smaller standard errors, which means more precise estimates of the population mean. The table below holds the standard deviation constant at 16 and varies only the sample size, to show exactly how much difference sample size alone makes.

Two patterns are worth noticing here. First, the standard error keeps shrinking as the sample grows, which is exactly what “more data gives a more precise estimate” looks like in numbers. Second, the relationship is not a straight line, because of the square root in the formula. Going from a sample of 16 to a sample of 64 — four times as many participants — halved the standard error from 4.0 to 2.0. Going from 64 to 256, another fourfold increase, halved it again to 1.0. This is a general rule: to halve your standard error, you need four times as many observations; to reduce it by a factor of ten, you would need roughly one hundred times as many.

Why Standard Error Decreases as Sample Size Increases

It is worth understanding why larger samples produce smaller standard errors, not just that they do. As a sample grows larger, it becomes a more complete and representative slice of the population. Unusually high or low individual scores get balanced out by the sheer number of other scores in the sample, so the sample mean settles closer to the true population mean. Repeat that larger sample many times, and the resulting means cluster more tightly together — hence a smaller standard error.

This is also a well-known practical constraint in research design. Because of the square-root relationship shown in the table above, there are strong diminishing returns to simply adding more participants once a study already has a reasonably large sample. Doubling a sample from 500 to 1,000 participants produces a much smaller improvement in precision than doubling it from 10 to 20. This is one reason researchers weigh sample size decisions carefully against time, cost, and recruitment difficulty rather than always defaulting to “bigger is better” — beyond a certain point, the precision gained per additional participant becomes small.

How Standard Error Is Used: Confidence Intervals and Hypothesis Testing

Standard error is not usually the end goal of an analysis in itself — it is a building block for two of the most common tools in inferential statistics.

Confidence intervals. A confidence interval gives a range within which the true population mean is likely to fall, based on the sample data. For a 95% confidence interval with a reasonably large sample, the standard formula is:

Sample mean ± (1.96 × SE)

Using the n = 64 example above, if the sample mean reaction time was 250ms with an SE of 2.0, the 95% confidence interval would run from roughly 246.08ms to 253.92ms (250 − 1.96×2 to 250 + 1.96×2). This is interpreted as: if the study were repeated many times, 95% of the confidence intervals calculated this way would contain the true population mean. A smaller standard error produces a narrower, more precise confidence interval; a larger standard error produces a wider, less precise one.

The 1.96 multiplier comes from the standard normal distribution and is most appropriate for larger samples. For smaller samples, researchers typically use the t-distribution instead of the normal distribution, with the exact multiplier depending on the sample’s degrees of freedom (n − 1). The t-distribution has slightly heavier tails than the normal distribution, which produces a slightly wider, more cautious confidence interval to account for the extra uncertainty that comes with estimating variability from a small sample. As sample size grows, the t-distribution converges toward the normal distribution, which is why the 1.96 shortcut becomes an increasingly good approximation for larger samples.

Hypothesis testing. Standard error is also the denominator in many common test statistics, including the one-sample t-statistic, which is calculated as the difference between the sample mean and a hypothesised population mean, divided by the standard error. This is the mathematical link between standard error and the world of p-values and statistical significance: a smaller standard error, all else being equal, produces a larger test statistic and therefore a smaller p-value, making a result more likely to reach statistical significance. This is also why studies with very large samples can sometimes find a result “statistically significant” even when the actual difference is tiny and of little practical importance — a very small standard error can make even a small effect stand out clearly from zero. Readers who want the fuller picture of how this feeds into significance testing, Type I and Type II errors, and effect sizes can find that covered in our guide to statistical significance, p-values and effect sizes.

Standard Error and Error Bars in Graphs

Standard error is one of the most common things represented by the small vertical lines, called error bars, that appear above and below bars or data points on research graphs. But error bars can represent several different statistics — standard deviation, standard error, or a confidence interval — and the graph itself rarely makes clear which one you are looking at.

This matters because the same dataset will produce visually different error bars depending on which statistic was chosen, since standard error is always smaller than standard deviation. A chart using standard error bars will look more precise and convincing, with tighter, less overlapping bars, than the exact same data plotted with standard deviation bars. Neither choice is automatically wrong, but they answer different questions: SD bars show how spread out the raw data was; SE bars show how precisely the mean has been estimated. When reading a graph with error bars, it is always worth checking the figure caption or methods section to see which one is being shown, rather than assuming — and when creating your own graphs, it is worth stating clearly which measure you have used, exactly for the reasons covered in the next section.

Common Mistakes When Reporting Standard Error

Given how often standard error and standard deviation are mixed up in practice, it is worth being specific about the mistakes to watch for, whether you are writing up your own research or evaluating someone else’s.

• Reporting SE when describing sample variability. If the goal is to describe how spread out a set of scores was, SD is the correct statistic. Reporting SE here understates the true variability in the data, because SE will always be the smaller number.
• Reporting SD when the goal is to express precision of an estimate. If the goal is to communicate how confident a researcher is in an estimated mean, SE (or a confidence interval built from it) is the appropriate statistic, not SD.
• Leaving the reader to guess which one is being shown. A number reported simply as “mean ± X” without stating whether X is SD or SE forces the reader to assume, and assumptions are exactly where errors creep in.
• Assuming a smaller SE always means a more meaningful result. A very small standard error, often achieved simply through a very large sample, tells you the estimate is precise, not that the underlying effect is large or practically important.

Being explicit about which statistic is being reported, and why, is a small habit that prevents a genuinely common and easily avoidable source of confusion in both academic writing and everyday reporting of results.

Standard Error in Psychology Research

Standard error appears throughout psychological research wherever a sample mean is being used to estimate something about a wider population — which is to say, in most published studies. It is commonly reported in the results sections of journal articles, often visualised as error bars on bar charts and line graphs. When you see error bars on a psychology research figure, it is worth checking the figure caption or methods section to confirm whether they represent standard error or standard deviation, since, as covered above, this is one of the most frequently mislabelled or misreported details in published research, and the two convey meaningfully different information to the reader.

Understanding standard error also matters for evaluating research findings critically, a core skill in psychology research methods teaching. A study reporting a very small standard error alongside its mean is signalling a precise estimate, typically because it used a large and well-designed sample; a study reporting a large standard error is signalling more uncertainty about how well its sample mean reflects the true population, regardless of how interesting the headline finding might sound. This is particularly relevant when comparing findings across studies of different sizes: a smaller study reporting the same mean as a larger one will typically have a wider standard error, and therefore a less precise estimate, even if both studies reached the same headline conclusion.

Standard Error of Other Statistics

Everything above has focused on the standard error of the mean, since that is the version most commonly encountered in introductory statistics and psychology research. However, the same underlying logic, the spread of a sampling distribution, applies to other statistics too. A standard error can be calculated for a sample proportion (used, for example, when reporting what percentage of a sample responded a certain way), for the difference between two means (used in comparing two groups), and for individual regression coefficients in regression analysis. Each has its own formula, but the interpretation is the same in every case: it describes how much that particular statistic would be expected to vary across repeated samples from the same population, and therefore how precise the current estimate is.

Conclusion

Standard error and standard deviation are easy to confuse because they share a formula, a symbol, and a habit of appearing side by side in research output — but they answer different questions. Standard deviation describes how spread out your individual data points are; standard error describes how precise your sample mean is as an estimate of the true population mean. Once that distinction is clear, the rest follows naturally: SE is calculated by dividing SD by the square root of the sample size, it shrinks as samples grow larger, and it underpins both confidence intervals and hypothesis testing throughout psychological research.

Frequently Asked Questions

What is an example of standard error?

If a sample of 64 participants has a standard deviation of 16, the standard error is calculated as SD ÷ √n, which is 16 ÷ √64 = 16 ÷ 8 = 2.0. This means that if the study were repeated many times with new samples of the same size, the sample means would typically vary by around 2.0 from one sample to the next — a much tighter range than the variation seen among individual scores within any single sample.

What is standard error vs SD?

Standard deviation (SD) measures how spread out individual scores are within a sample or population. Standard error (SE) measures how precise the sample mean is as an estimate of the true population mean — in other words, how much sample means would vary if a study were repeated many times. SE is calculated from SD, and because it is scaled down by the square root of the sample size, it is always smaller than SD once a sample has more than one observation.

How to calculate SD and SE?

Standard deviation is calculated by finding how far each individual score is from the sample mean, squaring those differences, averaging them, and taking the square root. Standard error is calculated afterwards, using the formula SE = SD ÷ √n, where n is the sample size. In practice, statistical software calculates both automatically, but understanding the relationship helps with interpreting the output correctly.

What does a standard error of 1.5 mean?

A standard error of 1.5 means that if the study were repeated many times with new samples of the same size from the same population, the resulting sample means would typically vary by about 1.5 units from the true population mean. A standard error of 1.5 indicates a relatively precise estimate when compared against a much larger standard error, though what counts as “small” always depends on the scale of the data being measured and the size of the sample.

Further Reading and Research

Recommended Articles

• What Is Standard Error? How to Calculate (Guide with Examples) — Scribbr’s overview of standard error, including how it differs from standard deviation and how it is used for other statistics such as proportions.
• Standard Error of the Mean (SEM) — Statistics By Jim’s detailed walkthrough connecting standard error to sampling distributions and statistical significance.
• How To Find The Standard Error: Formula & Calculation — Simply Psychology’s calculation-focused guide, including guidance on interpreting what counts as a “good” standard error.

Suggested Books

• Statistics for the Behavioural Sciences by Frederick J. Gravetter and Larry B. Wallnau — a widely used psychology statistics textbook covering standard error, sampling distributions, and hypothesis testing in depth, aimed at undergraduate psychology students.
• Discovering Statistics Using IBM SPSS Statistics by Andy Field — a popular, approachable guide to applied statistics for psychology and social science students, covering standard error alongside practical software instruction.
• Statistics for People Who (Think They) Hate Statistics by Neil J. Salkind — an accessible introduction to core statistical concepts, including standard error and standard deviation, aimed at readers who find statistics intimidating.

Recommended Websites

• APA Dictionary of Psychology — the American Psychological Association’s authoritative reference entry defining standard error of the mean.
• Simply Psychology — a widely used psychology education resource with statistics guides written for students and general readers.
• Scribbr — a student-focused statistics and research methods resource covering standard error and related inferential statistics concepts.

References

American Psychological Association. (n.d.). Standard error of the mean. In APA dictionary of psychology.

Manovic, A., Immelsjö, E., Axén, I., & Palmgren, P. J. (2025). Reporting the standard error of the mean: A critical analysis of three journals in manual medicine. Chiropractic & Manual Therapies. Advance online publication.

Kathy Brodie

Kathy Brodie is an Early Years Professional, Trainer and Author of multiple books on Early Years Education and Child Development. She is the founder of Early Years TV and the Early Years Summit.

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