Science Fair Guides · Analysis
Statistics for science fair projects,
without the maths degree.
Nobody loses a science fair because their statistics were insufficiently advanced. People lose because a bar chart hid the thing that mattered, because a p-value got described wrongly at the table, or because n = 3 could never have answered the question. Here is the small set of ideas that decides how your data reads.
← Back to the Science Fair Idea LabRule Zero
Plot it before you
calculate anything.
Before any average, any test, any p-value: put every measurement on a chart. Not a bar of the average with a whisker on top — one dot per run, conditions along the bottom. Four minutes in a spreadsheet, and the highest-value four minutes in your analysis.
Summary statistics are lossy by design. In 1973 Francis Anscombe built four datasets sharing the same means, variance, correlation and fitted line. Plotted, they are unmistakably different: a straight relationship, a clean curve, a line dragged off course by one stray point, and a vertical stack with a single outlier doing all the work. Every number you would normally report was identical. Only the picture told the truth.
What a plot catches and a test never will
- ·A ceiling. Your filter removes 99, 99, 100, 100, 99% — the mean looks brilliant, but the instrument has run out of room and good is indistinguishable from excellent.
- ·Two hidden populations. One condition splits into a cluster near 20 and one near 60, and the mean of 40 describes nothing that ever happened.
- ·Drift. Plot measurement against the clock too. If values creep upward all afternoon, your real variable may be room temperature.
- ·A decimal-point typo. 4.7 entered as 47 vanishes into a mean of thirty points and screams from a dot plot.
- ·A relationship that is real but bent. A straight-line correlation of 0.2 can sit on a clear curve.
Centre And Spread
The average is the
least interesting number.
Two sets of five measurements: minutes of runtime from a small motor driven by two different rechargeable cells. Both mean exactly 50 minutes, so a report that stopped at the mean would call them identical.
Cell A
48, 49, 50, 51, 52
Mean 50.0 min, standard deviation 1.6, range 4 minutes.
Cell B
20, 35, 50, 65, 80
Mean 50.0 min, standard deviation 23.7, range 60 minutes.
A t-test on those two sets returns t = 0 and p = 1.00 — no difference whatsoever. It is correct and useless, because the mean was never the question. If you need a cell for a device that must run forty minutes, A works every time and B fails one attempt in five. Opposite recommendations, none of it in the average.
Mean or median? Mean when the data are roughly symmetrical, median when they are skewed. Five dissolution trials give 42, 44, 45, 46 and 210 seconds, the last because the tablet lodged against the beaker wall. The mean is 77.4 seconds, describing no trial you ran. The median is 45, which is what you saw four times out of five — it stops one point dominating the summary while you decide what to do about it.
Error Bars
What the little
whiskers mean.
Standard deviation answers one question: typically, how far from the average does a single measurement land? For roughly bell-shaped data about two-thirds sit within one standard deviation of the mean and about 95% within two. Cell A had 1.6 minutes; cell B had 23.7.
Standard error is a different animal, and confusing the two is everywhere on fair floors. It is the standard deviation divided by the square root of the sample size, and it describes how precisely you have located the mean — not how variable your measurements were. For cell A that is 1.6 ÷ √5 = 0.7. Take more measurements and the standard error shrinks; the standard deviation does not, because the process is as variable as it always was.
So: standard deviation when your point is how variable your measurements are — almost always the right default — and standard error, or a 95% confidence interval, when your point is how well you know the mean. Every caption then states which, and what n was, because an unlabelled error bar is decoration. Never pick standard error because the bars come out shorter: judges know exactly what that is.
P-Values
What a p-value
actually says.
Start from a deliberately boring assumption: there is no real difference between your groups and everything you saw is noise. That is the null hypothesis. The p-value answers one narrow question — if that were true, how often would a difference this large turn up anyway? p = 0.03 means about three times in a hundred. That is all it means. The 0.05 threshold is a convention chosen in the 1920s; nothing magical happens between 0.049 and 0.051, and saying so at a judging table marks you out immediately. Decide which test you are running before you look at the data.
Misstatement 1
"p = 0.03 means there is a 3% chance my result was down to luck."
What is true
It is the probability of data this extreme IF there were genuinely no effect — not the probability that there is no effect. Swapping those two questions is the commonest statistical error in student reports.
Misstatement 2
"p = 0.06, so there is no difference between my groups."
What is true
It means you did not gather enough evidence to rule out chance, not that the effect is zero. With n = 4 you would fail to detect almost anything. Write "no significant difference was detected", then give the effect size and n.
Misstatement 3
"p = 0.0001, so the effect is huge and really matters."
What is true
A p-value mixes effect size with sample size. A trivial difference goes significant if you measure enough units; an enormous one stays non-significant on three trials. Report the difference in real units beside it.
Choosing A Test
Six situations,
six answers.
The test follows from the shape of your data, never from your subject. Two questions get you there: am I comparing measured numbers or counting categories, and are my groups independent or the same units measured twice?
Two separate groups, a number you measured
Unpaired (two-sample) t-test
Filter design A against design B, eight runs each, dye removal in per cent.
Report as: t(14) = 6.12, p < 0.001, n = 8 per group
The same subject or unit measured twice
Paired t-test
Reaction time for the same twelve people before and after a warm-up. Each is their own control.
Report as: t(11) = 4.4, p = 0.001, n = 12 pairs
Three or more groups, a number you measured
One-way ANOVA, then a post-hoc test
Dissolution time at 10, 20, 30, 40 and 50 °C. ANOVA asks whether the means differ at all; Tukey says which pairs do.
Report as: F(4, 35) = 96.4, p < 0.001, Tukey post-hoc
Counts falling into categories
Chi-squared test
120 seeds across three soil treatments, each scored germinated or not. Counting things, not averaging a measurement.
Report as: χ²(2) = 11.3, p = 0.004, n = 120 seeds
Two numbers that vary together
Correlation, then linear regression
Turbidity against distance downstream at 24 points. Pearson for a straight line, Spearman if it is bent.
Report as: r = -0.81, p < 0.001, n = 24
Small n, or data that are clearly not bell-shaped
Mann-Whitney U, Wilcoxon signed-rank, Kruskal-Wallis
Rank-based versions of the first three, in order. A long tail or a hard floor at zero does not break them.
Report as: U = 9, p = 0.02, n = 8 and 8
The paired case is the one students most often get wrong, and it costs real evidence. Measure reaction time in twelve people before and after a warm-up. People differ enormously — say 60 ms of standard deviation between individuals — while each improves by a consistent 15 ms with a spread of 18. As two independent groups, the between-person variation swamps the effect and the test finds nothing. As pairs, that variation cancels and the same numbers give a clear result.
For three or more groups, do not run t-tests between every pair: five groups means ten comparisons and a good chance one comes up significant by accident. Run the ANOVA first.
Sample Size
n = 3 is a floor,
not a target.
Three replicates became the school default because it is the smallest number that lets you notice one weird result. It is not a standard for detecting anything. What you need depends on two things you can estimate in week one: how big an effect you are after and how noisy your measurement is. The shortcut is 16 ÷ d² measurements per group, where d is the difference between groups divided by the standard deviation within a group.
A big effect
Filter B removes 78% of dye, A removes 55%, spread within each about 7.5 points. d = 23 ÷ 7.5 ≈ 3.1, so 16 ÷ 9.6 ≈ 2 runs per group. This is why n = 3 sometimes appears to work: the effect was enormous.
A modest effect
A 4-point difference with an 8-point spread gives d = 0.5, so 16 ÷ 0.25 = 64 runs per group. At n = 3 you had almost no chance of detecting it, and a non-significant result would have told you nothing.
So do this early. Run five or six pilot trials, calculate the standard deviation, decide what difference would matter, and put both into that fraction. Better to learn in week one than week eleven that you need 64 per group: either reduce your noise or change the question. And be honest about what a replicate is. Three separate runs is n = 3. Three readings from one run is n = 1 measured three times — the mistake below.
When it goes wrong
The mistake I made with my own data.
Early in the microplastic work I made up one batch of contaminated water, ran a single extraction on it, then took eight subsamples and counted particles in each under a microscope. Eight numbers, low variation, a beautifully tight standard deviation. I wrote n = 8.
It was n = 1. Those eight counts measured how consistently I could sample one beaker, not how consistently the extraction worked. Everything that genuinely varies between runs — how well the ferrofluid mixed, the particle load, the magnet contact time — was identical across all eight, so my error bars described my pipetting. The name for it is pseudoreplication, and it is the commonest serious statistics error in student projects because it looks exactly like diligence.
The fix was not more subsamples. It was fresh water, a fresh extraction, eight times over — four evenings instead of one. The new standard deviation was three times larger and the result held anyway. The test: ask what varied between the things you call replicates. If the answer is only your measuring, you have one data point.
Correlation
Two things moving
together, and why.
Everyone can recite that correlation does not imply causation, then writes a discussion that quietly implies it anyway. Being specific about the alternatives stops that. A concrete case: you grow one variety of bean in twenty plots across a school garden, measure height at six weeks, and find plants nearer the shed grew taller — r = 0.74, p < 0.001. Tempting to conclude the shed shelters them from wind. But the shed roof drains onto that corner, so the soil there holds far more water. Distance and moisture move together and your measurement cannot separate them.
- ·A causes B — the explanation you were hoping for.
- ·B causes A — reverse causation, easy to miss when the story sounds right.
- ·Something else causes both — a confounder, like a shed roof draining onto the near plots.
- ·Coincidence — especially if you tested many pairs of variables and reported the one that lined up.
The escape route is a controlled experiment: change one thing yourself, at random, holding the rest fixed. In the garden that means a fixed distance with watering varied and randomly assigned to plots. Randomising breaks the link between your variable and every confounder you have not thought of — which is most of them. Where that is impossible, say so and call the relationship an association. Judges respect that far more than an overclaim.
Outliers
When you may delete
a data point.
An outlier is a flag, never a verdict. The 1.5 × interquartile range rule and the three-standard-deviations rule are good for saying “go and look at this one”, and that is the whole of their authority. The 210-second trial is a fair candidate for exclusion, but only because you wrote in your notebook at the time that the tablet lodged against the beaker wall. Had you merely noticed afterwards that one number was awkwardly large, it stays. Four conditions hold together.
- 01You wrote down a physical reason at the time — the tablet stuck to the wall, the probe had not warmed up, you knocked the bench.
- 02The reason has nothing to do with the value being inconvenient. A run you would have kept had the number gone the other way is a run you must keep.
- 03You state how many points were removed, from which conditions, and why.
- 04You report the analysis both ways. If the conclusion survives, say so; if not, the honest headline is that your result rests on one measurement.
Done properly it takes one sentence: “One trial at 20 °C was excluded because the tablet lodged against the beaker wall, noted at the time; 39 of 40 trials are analysed. Including it raises the 20 °C mean from 45 s to 78 s and changes neither the direction nor the significance of the effect.” That makes you look careful; silently dropping it does the opposite, and judges do count. One warning, though: sometimes the outlier is the discovery. Before deleting anything, ask what would have to be true for that number to be real.
The Result Sentence
One sentence a judge
will respect.
Everything above exists so you can write one sentence properly. Five parts: direction, size in real units, uncertainty, the test with its output, and the sample size.
Says almost nothing
“Filter B was significantly better than filter A, which proves the modified design works.”
Says everything
“Filter B removed a mean of 78% of the dye (SD 8) against 55% (SD 7) for filter A — a difference of 23 percentage points, 95% CI 15 to 31 (unpaired t-test, t(14) = 6.12, p < 0.001, n = 8 runs per design).”
The second can be checked, repeated and argued with. “Proves” is gone, because one experiment does not prove things, and the confidence interval tells the reader the true difference is unlikely to be below about 15 points. Then write it again in plain English for your board: “The modified filter removed about a fifth more dye, and that gap is far bigger than the run-to-run variation.” You need both, because a board is read in eight seconds and a report with a pencil.
Free Tools
You do not need
to buy anything.
All free, none needing a school licence. Pick the smallest tool that does your job.
Browser calculators for a t-test, a chi-squared test and confidence intervals. Nothing to install. Best for a one-off test when you know which one you need — you can paste in means and standard deviations rather than raw data.
Free, open-source, looks like SPSS and outputs tables you can drop straight into a report. Best when you need an ANOVA with a post-hoc test, or a regression.
Where your data should live from day one, one row per measurement. AVERAGE, MEDIAN, STDEV.S, T.TEST and CHISQ.TEST cover most school projects, and it charts instantly.
An interactive visual introduction to probability from Brown University. For intuition rather than running your numbers — the hypothesis-testing chapter makes p-values click in twenty minutes.
For the figures themselves, Datawrapper is free and produces clean charts with real axis labels. Whatever you use, keep your raw data in one flat sheet — one row per measurement, one column per variable, no meaning carried by cell colour — and never overwrite it. Build every summary, chart and test from that sheet, so an error found in week ten is one cell to fix.
Common Questions
What students ask
about the numbers.
What statistics do I need for a science fair project?
Far less than most students fear: a plot of your raw data, a mean or median with a stated measure of spread, error bars whose meaning you define in the caption, and one appropriate test reported with its output and sample size. Judges reward understanding, not the number of techniques used.
What does a p-value actually mean?
A p-value is the probability of seeing data at least as extreme as yours if there were genuinely no effect at all. It is not the probability that your hypothesis is true, and not the chance your result was a fluke. It says nothing about whether the difference is big enough to matter, which is why you report that difference in real units beside it.
Which statistical test should I use for my science fair project?
Follow the shape of your data, not your subject. Two separate groups with a measured number is an unpaired t-test; the same subject measured twice is a paired t-test; three or more groups is a one-way ANOVA with a post-hoc test; counts in categories is a chi-squared test. If the data are clearly not bell-shaped, use the rank-based equivalent: Mann-Whitney, Wilcoxon or Kruskal-Wallis.
How many trials do I need for a science fair project?
Three is a floor, not a target, and it only detects enormous effects. A workable rule for two groups is roughly 16 divided by the squared effect size per group, so a difference of one standard deviation needs about 17 per group and half a standard deviation about 64. Measure your own standard deviation in pilot trials and work the number out before you commit.
Can I remove an outlier from my data?
Only if you can point to a documented physical reason recorded at the time — contamination, a sensor not yet warmed up, a knocked bench — and never because the value is far from the others. If you do exclude a point, say how many and why, and report the analysis with and without it.
Should error bars show standard deviation or standard error?
Standard deviation describes how spread out your measurements were; standard error describes how precisely you have located the mean, and equals the standard deviation divided by the square root of the sample size. Standard error bars are always shorter, so they look tidier and mean something different. State which you used in the caption, with n.
Where To Go Next
Keep going.
The free tool that builds a project around your interests, equipment and time — or stress-tests the idea you have.
Where your statistics belong: results with no interpretation, and a discussion that sizes its own limitations.
How to put one chart on a board so a judge reads your result from two metres away.
If your sample involves people or animals, approval comes before you collect a single data point.
Statistics intimidated me for years, because every explanation started with formulas rather than with what the numbers were for. What made it click was realising the whole subject asks one question over and over: could this have happened by accident? You do not need to be good at maths — plot your data, know how variable it is, pick the test that matches its shape, and describe what you found without inflating it. That is a higher standard than most projects reach, and an afternoon’s work.
— Fionn
Google Science Fair Global Grand Prize, 2019
Science Fair Guides · Idea Lab · fionnferreira.com