Descriptive statistics: mean, standard deviation & standard error

Living things vary. Measure the height of 100 people, the length of 100 leaves or the mass of 100 woodlice and no two values will be identical. Descriptive statistics are the numbers that sum up that variation so you can see what your data are actually telling you: where the values cluster (the mean), and how spread out they are (the standard deviation).

You will not usually be asked to calculate a standard deviation by hand in a written exam — you use a calculator or spreadsheet for that. What you are examined on is understanding what these numbers mean, choosing the right one, and interpreting them, especially error bars. That is what this page focuses on.

There are three ways to describe the “middle” of a set of data. You need to know what each is and when it is the sensible choice.

• Mean (the average) — add up all the values and divide by how many there are. This is by far the most used measure in biology and the one that feeds into statistical tests.
  Reading the symbols:
  • x̄the mean (said “x-bar”) — the answer you are working out.
  • Σ“the sum of” — the Greek capital letter sigma. It just says “add up everything that follows”. (Handy memory: sigma and sum both start with s.)
  • xeach individual value (each measurement you took).
  • nthe number of values in your sample (how many measurements). n for number.
  So Σx ÷ n simply means “add up all your measurements, then divide by how many there were”.
• Median — put the values in order and take the middle one (or the average of the middle two). Useful when there are extreme values (outliers) that would drag the mean, because the median is barely affected by them.
• Mode — the most common value. The only measure you can use for data that are categories (e.g. eye colour). On a graph it is the tallest bar or peak.

Which one should you use? (a common exam question)

You are often asked which measure of central tendency is most appropriate for a given set of data, and why. There is no single “best” one — it depends on the data. Here is how to choose.

In a perfectly symmetrical (normal) distribution the mean, median and mode all sit at the same point, right at the peak. As soon as the distribution becomes skewed (lopsided, with a long tail on one side), the three measures pull apart — and the order they fall in tells you which way the data are skewed. The diagram below shows all three cases at a glance:

Positive skew (right-skewed)

A few unusually large values stretch the distribution out into a long tail on the right. Those extreme values pull the mean towards them, so the mean ends up the highest of the three:

The mode stays put at the peak (the most common value), the mean is dragged furthest towards the tail, and the median sits in between. Biological example: the number of offspring per individual, where most have few but a handful have very many.

Negative skew (left-skewed)

A few unusually small values stretch the distribution into a long tail on the left. Now the mean is pulled downwards, so the order reverses:

A useful memory hook: the mean always chases the tail. Whichever side the long tail is on, that is the side the mean shifts towards — because the mean is the only one of the three affected by how far the extreme values are, not just how many. The median is barely moved and the mode not at all, which is exactly why the median is the safer measure of centre when data are skewed or contain outliers.

The simplest measure of spread is the range (highest value − lowest value). It is quick, but it uses only two values, so a single freak result can distort it completely. The standard deviation (SD) is better because it uses every value.

Standard deviation answers one question: on average, how far is each value from the mean? A small SD means the data are packed tightly around the mean; a large SD means they are widely spread. You will be able to see this directly in the interactive explorer further down the page.

Why n − 1 and not n? First, what it means in practice: n − 1 is simply one less than the number of values you have — so 5 values means you divide by 4, 10 values means you divide by 9, 20 values means you divide by 19. It is not a fixed number; it changes with your sample size.

And why one less? Because in biology you almost always measure a sample, not the whole population. A sample tends to look slightly less spread out than the population it came from, so dividing by the smaller (n − 1) nudges the answer up a little and corrects for that. You do not have to prove this at A-level — just remember: for a sample, divide by (n − 1).

Most continuous biological variables (height, mass, leaf length) follow a normal distribution — a symmetrical, bell-shaped curve: most values sit near the middle, and fewer and fewer values appear as you move out towards the extremes. (You can see this shape in the interactive explorer below.) Once data are normal, the standard deviation carves the curve into predictable slices:

• About 68% of values lie within ±1 SD of the mean.
• About 95% of values lie within ±2 SD of the mean.
• About 99.7% of values lie within ±3 SD of the mean.

This is why you will see “mean ± 2 standard deviations” used as an approximate 95% range. A value out beyond 2 SD from the mean is in the rare 5% tail — it is unusual, and that “unusual” idea is exactly what every significance test (chi-squared, t-test) is built on.

What “2 standard deviations” actually means

Because “2 SD” is used so much, it is worth making it completely concrete. Suppose you measured 20 leaves with a mean of 50 mm and a standard deviation of 8 mm. Then:

• 1 SD = 8 mm, so 2 SD = 16 mm.
• Mean − 2 SD = 50 − 16 = 34 mm  (the bottom of the range).
• Mean + 2 SD = 50 + 16 = 66 mm  (the top of the range).

So the “±2 SD range” here is simply 34 mm to 66 mm. The 68–95–99.7 rule says that about 95% of the leaves should fall inside that range, and only about 1 in 20 should fall outside it. In the diagram below, 19 of the 20 leaves land inside the green band — you can count them — and just one (the unusually long leaf) falls outside in the rare 5%.

This is the most important idea on the whole page. Every statistical test — chi-squared, the t-test, correlation — and the null hypothesis are all built on it. We will go very slowly and assume you have never met any of it before.

Two ordinary words are used in a special way in statistics: “by chance” and “significant”. We will build them from three everyday examples, then use them on a real experiment.

First, three examples of “chance”

1. A coin

A fair coin means an ordinary, honest coin — one that is equally likely to land on heads or tails, with nothing done to it to favour either side. (An unfair or “biased” coin would be weighted so it lands one way more often.)

Flip a fair coin 10 times. You expect 5 heads — but you will very often not get exactly 5. You might get 4, or 6, or 7. That does not mean the coin is faulty. It is just the ordinary random variation you get whenever chance is involved. That everyday variation is what statisticians mean by “by chance”.

2. A medicine

Two people have a cold. One takes a vitamin and feels better after 5 days; the other takes nothing and feels better after 6 days. Did the vitamin work? You cannot tell — people shake off colds in different times anyway, so a one-day difference could easily happen by chance, with the vitamin doing nothing at all. This is the exact problem every experiment faces: is the difference I see real, or just natural variation?

3. Two dice — and this one explains the bell shape

Here is the example that shows why random results form that famous bell shape. Roll two dice and add them. The total can be anywhere from 2 to 12 — but the totals are not equally likely. Why not? Because some totals can be made in many different ways and others in only one:

• A total of 2 can only happen one way: 1 + 1.
• A total of 12 can only happen one way: 6 + 6.
• But a total of 7 can happen six ways: 1+6, 2+5, 3+4, 4+3, 5+2, 6+1.

So a 7 comes up far more often than a 2 or a 12 — simply because there are more ways to make it. Count the ways for every total and you get this:

Watch it happen: the Galton board

A Galton board makes this idea physical. Drop a bead in at the top and it hits a peg. At every peg it bounces left or right at random (a 50–50 choice, like flipping a coin). After bouncing through all the rows, it drops into a bin at the bottom. Drop hundreds of beads and watch what shape they build.

How much variation is “normal”? Back to the coin

The coin behaves just like the dice. There are many ways to end up with about 5 heads, but very few ways to get an extreme result like 10 heads — so middle results are common and extreme ones are rare. The chart below shows how often each number of heads comes up, which lets us answer the key question: where does an “ordinary” result end and a “rare” one begin?

The suspicion ladder (this is the key idea)

Forget formulas for a moment. Imagine a friend keeps flipping a coin and telling you how many heads they got out of 10. At what point would you start to suspect the coin is rigged? Your gut already knows the answer — let’s follow it down this ladder:

Look at what happened as you went down the ladder: the more heads they got, the less likely luck could explain it (the % on the right shrinks), and the more suspicious you became. Those two things always move together — and this is the bit that stops the logic feeling backwards:

Scientists just draw a line on this ladder and agree: “once luck’s chance drops below 5%, I’m suspicious enough to stop believing it was luck.” On the coin that line falls at about 8 heads. Getting 8, 9 or 10 heads is where you cross from “just chance” into “something real”. Making that switch is the entire job of a statistical test.

Now the biology: the fertiliser experiment

You test a fertiliser by growing two groups of plants and measuring their heights:

The fertilised plants are taller, so the fertiliser works — right? Not necessarily. Plants vary in height even when treated identically (just like the coin does not give exactly 5, and the two cold-sufferers recovered in different times). If you had taken untreated plants and simply split them into two groups at random, the two means would still differ by a centimetre or two — purely by chance. So the real question is:

What “significant” means — and the 5% line

A result is called significant when it is so unlikely to have happened by chance that “chance” is no longer a believable explanation, so you conclude something real is going on. That is the whole idea — it is the coin’s “9 heads out of 10” moment, applied to an experiment.

We just need an agreed cut-off for “so unlikely”. Scientists use 5%, written p < 0.05:

The direction of this trips everyone up, so hold onto it:

What “significant” does NOT mean

Standard deviation, which you met above, tells you how spread out your individual measurements are. But there is a second, different question that matters just as much: how much can you trust the mean itself?

Here is the idea. You measured one sample of leaves and got a mean. If you went out and collected a completely new sample of the same kind of leaves, you would get a slightly different mean, just by chance. Collect a third sample — a slightly different mean again. The standard error of the mean (SE) is a number that estimates how much your mean would jump around if you kept repeating the experiment like this. A small standard error means your mean is reliable; a large one means it could easily have come out quite different.

What is an error bar?

When you draw a bar chart or point graph of your means, each mean is a single dot or the top of a bar. An error bar is a small vertical line drawn through that point: it goes up a certain amount and down the same amount, with a little cap at each end, to show how much uncertainty or variation there is around the mean. It is a way of saying “the mean is here, but the true value is somewhere in this range.”

You get to choose how long to make the error bars, and you must state which you have used in the graph title or legend, because the two choices mean different things:

• Error bars of ±1 standard deviation — use these when you want to show how variable the raw measurements are.
• Error bars of ±2 standard error — use these when you want to compare two means and see whether they are really different. Because ±2 SE marks out the ~95% range (see the box above), this is the version that lets you judge overlap, and the one that matters most at A-level. (You will sometimes also see ±1 SE bars, but the overlap rule below is most reliable with ±2 SE.)