# How can the standard deviation for a set of experimental value be interpreted in terms of precision?

I have read from many websites that the sample standard deviation can be used to measure the precision of experimental values. I understand that the lower the standard deviation the better, as it would mean that the values are close together (precise). But what standard deviation is small enough to be considered precise?

As an example, in a redox titration experiment I obtained titrant values of $16.3$, $16.4$, $16.5$ and $\pu{17.1 mL}$.

From these four measurements, I thus calculate an average of $\pu{16.58 mL}$ and a standard deviation of $\pu{0.35 mL}$. The uncertainty of the burette is $\pm\,\pu{0.05 mL}$.

From this standard deviation, would this experiment's results be considered precise?

If not, why not, and how would I interpret it?

Is there a better method for quantifying precision?

• No, it's not precise, 'cause 4th value is outlier. – Mithoron Apr 29 '17 at 14:08
• One std dev corresponds to obtaining data that is within $\approx 68$ % of the mean, ($\pm34$ %) this means that $32$ %of the time results will be outside this range by random chance and you seem to have observed one of these. If you calc 2 std dev then your values should be outside this range only $\approx 5$ % of the time. Of course your results may not be accurate but that is a different matter. – porphyrin Apr 29 '17 at 18:04

## 1 Answer

But what standard deviation is small enough to be considered precise?

"Sufficiently precise," ironically enough, is something that can't really be defined precisely in general terms.

In other words, a level of precision that is perfectly fine in one context might be unacceptably low in others.

For example, if I'm trying to estimate how much table salt I need to buy to supply my catering business for the next six months, I might be perfectly happy with a not-very-precise calculation at $\pm\,30\%$, since salt is cheap and will keep just fine if I buy too much.

On the other hand, if I'm a pharmacist compounding a prescription containing a drug where the level in the body leading to nasty side effects is not all that much higher than the level that provides the desired pharmacological activity, I probably want to use (and am probably required by law to use!) methods and apparatus that'll give me, say, $\pm\,3\%$ precision on my measurements.

Is there a better method for quantifying precision?

Not really. Pretty much every metric I've seen for quantifying precision involves the standard deviation in some way. Depending on the situation, one might use:

• $s$, the standard deviation by itself

• $s/\bar x$, the standard deviation divided by the mean, also called the relative standard deviation or coefficient of variation (CV)

• $s/\!\left(\bar x \sqrt n\right)$, the relative standard deviation divided by the square root of the number of measurements

• This is equal to the standard error $\left(s/\!\sqrt n\right)$ divided by the mean

The "Comparison to standard deviation" subsection of the Wikipedia article on the coefficient of variation lists some situations where these different quantities would be useful relative to others.

From this standard deviation, would this experiment's results be considered precise?

If not, why not, and how would I interpret it?

In my opinion: Yes, these results are precise. In this case, since the mean value is far from zero and there's only one dataset being examined, I would use the coefficient of variation (CV) as my metric of precision. With these values, I get a CV of $1.88\%$. On the whole, I think this is an entirely reasonable CV for a titration experiment. The data could exhibit more precision, obviously, but that low of a CV implies very good repeatability to me.

As porphyrin noted, that $\pu{17.1 mL}$ measurement you obtained is perfectly reasonable in a statistical sense, as it's $25\%$ of a small dataset, residing $1.7$ standard deviations away from the mean. This puts it well line with the expected $32\%$ occurrence of values $x$ where $|x-\bar x|/s > 1.0$.

In other words, I disagree with Mithoron: I don't think that value is an outlier at all. The usual metric I'm familiar with for identifying outliers is where $|x-\bar x|/s > 3.0$, which should occur only $\approx 0.34\%$ of the time for normally-distributed data.

The uncertainty of the burette is $\pm\,\pu{0.05 mL}$.

A brief note: I believe the small uncertainty on the burette as compared to the standard deviation in your data $(0.05 / 0.38 \approx 13\%)$ indicates that the majority of the spread in your measurements comes from elements of the experimental apparatus/procedure other than the burette.

Further reading: This article at Inorganic Ventures is a good writeup of the mean and standard deviation, and how they relate (or not) to accuracy and precision.