IB Biology SLSkill Set 3 — Maths for BiologyPaper 1 & 2Core skill~10 min read
Handling Uncertainties
No measurement is exact. Every reading you take is really a small range of possible values, and the honest thing to do is say how wide that range is. Once you can do that, you can also say something much more useful: whether the difference you found between two results is big enough to be worth believing.
📘 What you need to know
Uncertainty is the range around a measurement within which the true value is expected to lie, written as a ± range.
Uncertainty is not the same as error. Error is the gap between your reading and the true value.
Absolute uncertainty carries units; percentage uncertainty lets you compare measurements of different sizes.
Adding or subtracting readings? Add the absolute uncertainties. Multiplying or dividing? Add the percentage uncertainties.
Error bars show uncertainty on a graph. Overlapping bars mean the two results are not clearly different.
R2 measures how well a line or curve fits the points. r measures the strength and direction of a linear relationship.
Correlation is not causation, however convincing the graph looks.
What uncertainty actually means
When you write 25.0 cm3, you are not claiming the volume is exactly 25.0. You are claiming it is close to 25.0, and the uncertainty says how close.
Every value in a results table really carries a band like this. You quote it once, in the column heading, rather than beside every number.
Where the number comes from
Analogue scales (ruler, thermometer, measuring cylinder): uncertainty is half the smallest division. A ruler marked in millimetres gives ±0.5 mm.
A balance with 10 g graduations measures to the nearest 10 g, so the true value could be 5 g either side — ±5 g.
A pipette graduated every 0.1 cm3 gives ±0.05 cm3.
Two readings, two uncertainties. A burette volume needs a start and an end reading, so the uncertainties add.
Uncertainty is not error. Uncertainty is honest bookkeeping about the limits of your equipment, and it exists even when everything goes perfectly. Error is the difference between what you measured and the true value, caused by the equipment or your technique.
Absolute and percentage uncertainty
An absolute uncertainty of ±0.5 cm3 sounds small — but that depends entirely on what you are measuring. On 25 cm3 it is minor; on 2 cm3 it is a quarter of the reading. Percentage uncertainty makes that comparison possible.
A leaf is measured as 84 mm long with a ruler marked in millimetres. State the reading with its uncertainty and find the percentage uncertainty.
Step 1: half the smallest division
Smallest division is 1 mm, so the uncertainty is ±0.5 mm.
Step 2: write the reading properly84.0 ± 0.5 mmStep 3: percentage uncertainty(0.5 ÷ 84) × 100 = 0.595…0.6% (1 s.f.)Measure something small with the same ruler — a 5 mm bud, say — and that jumps to 10%. Same equipment, very different reliability.
Combining uncertainties
Two rules cover almost everything you will meet at SL.
Adding or subtracting values
add the absolute uncertainties
Multiplying or dividing values
add the percentage uncertainties
The surprising one is subtraction. Taking one reading away from another does not cancel the uncertainties out — it makes the total worse, because both readings could be off in opposite directions.
WORKED EXAMPLE
A potato chip has mass 4.20 ± 0.01 g before soaking and 5.10 ± 0.01 g after. Find the change in mass with its uncertainty, and the percentage uncertainty.
Step 1: the change in mass5.10 − 4.20 = 0.90 gStep 2: subtraction, so add the absolute uncertainties0.01 + 0.01 = ±0.02 gStep 3: percentage uncertainty of the change(0.02 ÷ 0.90) × 100 = 2.22%0.90 ± 0.02 g, so 2.2%Each mass was only 0.2% uncertain, but the difference between them is 2.2%. Small differences are always the least reliable numbers in a table.
WORKED EXAMPLE
A rate is found by dividing a gas volume (2.0% uncertain) by a time (1.0% uncertain). Find the percentage uncertainty in the rate.
Division, so add the percentage uncertainties2.0 + 1.0 = 3.0%3.0% uncertainty in the rateThis tells you where to spend your effort: improving the timing barely helps here, because most of the uncertainty is coming from the volume measurement.
Level of precision
Your processed answers must not be more precise than the data they came from. Readings to one decimal place cannot give a mean to four.
Every value in a raw data column should be written to the same level of precision: 4.20, not 4.2, if the balance reads to two decimal places.
The apparatus decides the precision, not the calculator.
Error bars on a graph
An error bar is the uncertainty drawn onto the plot. Usually vertical, for the dependent variable, though you can also draw horizontal bars for the independent variable. Error bars can show the range, the standard deviation or the standard error, so you must state which.
The last point has a horizontal bar too, because the independent variable was uncertain as well — common when concentrations are made by dilution.
The sentence worth memorising: “the error bars overlap, so the difference between these means may be due to chance rather than the variable I changed.” It fits a huge number of data questions.
How well does the line fit? R2
The coefficient of determination, written R2, measures how closely a trend line or curve matches the points it was drawn through.
R2 = 0 — the line tells you nothing; you cannot predict the dependent variable from the independent one.
Between 0 and 1 — you can predict, and the closer to 1, the better the fit.
R2 = 1 — a perfect fit, with every point sitting on the line.
A high R2 does not prove the line is the right model. A straight line can fit a curve reasonably well over a short stretch and still be the wrong shape.
Correlation, and what it does not prove
A correlation is an association between two variables. The correlation coefficient, r, tells you whether a linear relationship exists and how strong it is.
r = +1 — perfect positive: every point on a straight line, and as one variable rises so does the other.
r = −1 — perfect negative: as one rises, the other falls.
r = 0 — no linear relationship at all.
The closer r sits to +1 or −1, the stronger the correlation.
All four sets have the same number of points. Only the tightness of the pattern changes, and that is exactly what r is measuring.
Correlation is not causation
This is the single most repeated warning in the whole course, and it is worth understanding rather than reciting. Two variables can move together for three quite different reasons:
One really does cause the other.
Something else causes both. Ice cream sales and drowning deaths rise together, because hot weather drives both.
Coincidence, especially in a small sample.
Causation means one variable genuinely influences the other, and showing it needs more than a graph — usually a controlled experiment where you change one variable and hold the rest steady, plus a plausible biological mechanism.
Significance: the null hypothesis
Statistical tests exist to answer one question: is the pattern I found bigger than what chance alone would produce?
The null hypothesis says there is no significant difference or association between the variables.
The alternative hypothesis says there is one.
The test result lets you accept or reject the null hypothesis.
If the test finds no significant difference, then any difference you can see in the data is put down to chance alone.
Notice that you never “prove” the alternative hypothesis. You reject the null one, which is a more modest claim, and that modesty is the whole point of doing statistics.
💡 Exam tip
Half the smallest division for analogue scales; two readings means double the uncertainty.
Absolute uncertainties add for + and −; percentage uncertainties add for × and ÷.
Quote the uncertainty to the same precision as the measurement: 84.0 ± 0.5 mm.
Always state what your error bars represent — range, standard deviation or standard error.
Use the overlap sentence when comparing two means with error bars.
Write “correlation does not show causation” and then say what else would be needed to test it.
In the IA, put the uncertainty in each column heading of the raw data table.
⚠ Common mix-up
Confusing uncertainty with error. Uncertainty is a range you expect; error is a gap you did not intend.
Thinking uncertainties cancel out on subtraction. They add.
Adding percentage uncertainties when adding values, or absolute ones when multiplying. Match the rule to the operation.
Quoting a processed value more precisely than the raw data allows.
Treating a high R2 as proof that the line is the correct model.
Reading causation into a correlation, especially where a third variable could explain both.
Saying the null hypothesis was proved. You accept or reject it; you never prove it.
Up next: Graphing Skills — picking the right graph, drawing a proper line of best fit, taking a tangent for an initial rate, and knowing where the line stops being trustworthy.
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