Interpreting Standard Deviation in Experimental Data

An average does not tell the whole story

Imagine measuring the volume of gas produced after a fixed time in five trials. Two sets of results, in $\mathrm{c}{\mathrm{m}}^3$, are:

Both sets have a mean of $20\mathrm{c}{\mathrm{m}}^3$. Yet the measurements in A stay closer to 20, while those in B vary more. The mean describes the centre; standard deviation describes the spread around that centre.

Watch this short comparison before we put numbers to that idea. It uses scores rather than laboratory measurements, but the interpretation of spread is the same.

A standard deviation is never negative: spread cannot be less than zero. It is zero if every measurement is identical. Its units are the same as those of the measurements. For gas volume in $\mathrm{c}{\mathrm{m}}^3$, standard deviation is also reported in $\mathrm{c}{\mathrm{m}}^3$, not in $\mathrm{c}{\mathrm{m}}^6$ or as a percentage.

Where the number comes from

To see why standard deviation describes spread, begin with each measurement’s deviation from the mean. In set A, whose mean is $20\mathrm{c}{\mathrm{m}}^3$, those deviations are $-2,-1,0,1,2\mathrm{c}{\mathrm{m}}^3$. Adding them would give zero: measurements below and above the mean cancel. Instead, the calculation squares each deviation, combines the squared values, and finally takes a square root to return to the original units.

Read the worked example below with that purpose in mind. You do not need to memorise every arithmetic step on a first reading.

For repeated trials viewed as a sample of all the results you might obtain under the same conditions, the sample standard deviation is

where $x_i$ is a measurement, $\bar{x}$ is the sample mean, and $n$ is the number of measurements. The symbol $\sum$ means “add up.” As the reading explains, the divisor $n-1$ is used for the sample calculation.

For set A, the squared deviations add to $4+1+0+1+4=10$. There are five trials, so

For set B, the deviations from 20 are $-4,-2,0,2,4$, whose squares add to 40. Thus

The calculation confirms what the table suggested: set B has twice the sample standard deviation of set A, despite having the same mean. Its measurements are more spread out around $20\mathrm{c}{\mathrm{m}}^3$.

A calculator may display both a sample standard deviation, often labelled $s_x$, and a population standard deviation, often labelled ${\sigma }_x$. The population calculation divides by the number of values rather than one fewer. For example, treating A’s five listed values as the complete population would give $\sqrt{10/5}\approx 1.41\mathrm{c}{\mathrm{m}}^3$, rather than $1.58\mathrm{c}{\mathrm{m}}^3$. Check which convention a question requires, and use the same convention when comparing data sets. The central interpretation—smaller means more clustered—does not change.

Interpret the spread in the experiment

A useful report gives the mean alongside the standard deviation. For these sample results, you could write:

At the fixed measurement time, both sets had a mean gas volume of $20\mathrm{c}{\mathrm{m}}^3$. Set A had a sample standard deviation of about $1.58\mathrm{c}{\mathrm{m}}^3$, compared with $3.16\mathrm{c}{\mathrm{m}}^3$ for set B, so A’s repeated measurements were more closely clustered around their mean.

That is a statement about variation between trials. If the trials really used the same method and conditions, A shows greater repeatability in these results. It does not establish that A is more accurate: a measurement method could give tightly clustered results that are all offset from the true value. Nor does a standard deviation identify why trials vary. Differences in timing, apparatus readings, or experimental conditions would need separate investigation.

Standard deviation is a measure of overall spread, not a boundary containing every value. Set A’s mean plus one sample standard deviation is approximately $21.58\mathrm{c}{\mathrm{m}}^3$, yet A includes a measurement of $22\mathrm{c}{\mathrm{m}}^3$. There is no contradiction: “one standard deviation from the mean” describes a distance, not a rule that all observations must lie inside it.

Small data sets and unusual results

With only a few trials, one unusual measurement can noticeably change both the mean and standard deviation. The final part of the reading shows this effect and, importantly, how to respond to it.

In an experiment, an unusually high gas-volume reading might be a recording error—or a genuine result. Check your records and method before deciding how to handle it; do not discard a value solely because it increases the standard deviation. When reporting a small data set, the individual measurements remain valuable context for the summary number.

You may also encounter standard deviation marked on a bell-shaped normal distribution, as in the diagram below.

The percentages in this picture describe a normal distribution. They are not a promise that a particular percentage of five experimental trials will fall within one or two standard deviations of their mean. For a small set of repeated measurements, first use standard deviation for its direct purpose: describing how widely the observed values are spread.

Conclusion

Standard deviation puts a number, in the same units as the measurements, on variation around the mean. A smaller value indicates more closely clustered results; a larger value indicates greater spread. Interpret it with the mean and the individual measurements, especially when the data set is small or contains an unusual value. It can support a statement about repeatability, but not by itself a claim about accuracy or the cause of variation.

Next, you will bring data and models together by comparing predictions from a linear, quadratic, or exponential model with observed results.