Error Bars: Standard Deviation, Standard Error and Confidence Intervals

A bar chart carries error bars, they are small, and the result looks solid. What those bars represent is stated in the caption, or it is not stated at all, and the difference between the three possibilities changes the figure’s meaning by a factor of two or more.

Three different quantities

• Standard deviation describes the spread of the individual observations. It is a property of the data and does not shrink as more observations are collected — with a larger sample it is estimated more precisely, but the spread itself is whatever it is.
• Standard error of the mean describes the precision of the estimated mean. It is the standard deviation divided by the square root of the sample size, so it shrinks as the sample grows, regardless of whether the underlying variability changed.
• Confidence interval is a range constructed so that, over repeated experiments, a stated proportion of such intervals would contain the true value. It is roughly twice the standard error either side of the mean for a large sample, and wider for a small one.

They answer different questions. The first asks how variable the system is. The second and third ask how well the average is known.

Why standard error is the one usually chosen

Because it is the smallest. Standard error is always narrower than standard deviation, by a factor of the square root of the sample size, so a figure drawn with standard error looks more convincing than the same data drawn with standard deviation.

That is not necessarily improper — if the claim concerns the mean, the precision of the mean is the relevant quantity. It becomes misleading when the bars are unlabelled, because a reader accustomed to one convention will misread the other. An unlabelled error bar on a figure from four replicates could be either of two quantities differing twofold.

What n refers to, which is the load-bearing question

The arithmetic for both quantities depends on the sample size, and the sample size depends on what is being counted as an independent observation.

Three wells from the same dilution of the same preparation are three measurements of one thing. Treating them as n = 3 makes the standard error a third smaller than it should be, and the error bars shrink accordingly. This is the distinction covered in technical and biological replicates, and it is where most misleading error bars originate — not in the choice of statistic but in the count feeding it.

Reading overlap, which does not work the way it looks

Two common intuitions are both wrong. Non-overlapping standard error bars do not establish a significant difference, and overlapping standard error bars do not rule one out. The relationship between bar overlap and any test depends on the sample sizes and the test used.

Confidence intervals behave slightly better — two ninety-five percent intervals that do not overlap usually correspond to a difference that would test as significant — but the reliable version is to look at the interval on the difference between the groups, which is the quantity the claim is actually about.

What small samples do to all three

With three or four observations, the standard deviation is itself estimated very imprecisely, so the standard error computed from it is unreliable, and the confidence interval built on it is wide and unstable.

This is worth stating plainly because n = 3 is close to universal in this literature. Error bars drawn from three observations look like precision and are not; they are a rough indication that the measurement was repeated. The same caution applies to the sample as represents-the-batch question in sampling plans — a small sample constrains what can be claimed regardless of how it is displayed.

What a figure should state, and rarely does

What the bars represent. What n is, and what an individual observation was. Whether the observations were independent. And, for a summary statistic, whether the centre is a mean or a median — a distinction that matters as soon as the data are skewed, which concentration-response data frequently are.

Where the replicate count is small, plotting the individual points rather than a bar and a bar alone is more honest and takes no more space. A reader can see the spread directly instead of inferring it from a summary that may have been chosen for its appearance.

The connection to analytical reporting

The same principle governs a certificate. A purity figure without an uncertainty is a point estimate presented as if it were exact, which is the subject of measurement uncertainty, and a figure reported to more decimal places than the method supports claims a precision it does not have, per significant figures and rounding.

In both settings the failure is the same: a display choice that makes a number look better determined than it is. The remedy is also the same, which is to state what the spread represents and how many observations produced it.