IB Biology HLSkill Set 3 — Maths for BiologyPaper 1, 2 & IAPractical skill~12 min read
Handling Uncertainties
No measurement you ever take is exactly right. That is not carelessness, it is physics — every instrument has a limit to how finely it can read. Uncertainty is how scientists own up to that limit honestly, and it is why error bars exist.
📚 What you need to know
Uncertainty is the range of values around a measurement within which the true value is expected to lie. It is written as a range with a ± sign.
For a scale, the uncertainty is usually half of the smallest division.
Uncertainty is not the same as error. Error is the gap between your measured value and the true value.
Error bars on a graph can show range, degree of precision, standard error or standard deviation — you must say which.
If error bars showing standard deviation overlap, the two sets of data are not significantly different.
Level of precision: never write more decimal places than your equipment can actually measure.
R2, the coefficient of determination, measures how well a line or curve fits the data. It runs from 0 to 1.
The correlation coefficient (r) measures the strength and direction of a linear relationship, from −1 to +1.
A null hypothesis says there is no significant difference or association. Statistics let you accept or reject it.
What uncertainty actually is
Definition
Uncertainty is the range of values around a measurement within which the true value is expected to lie.
Say you measure a leaf with a ruler marked in millimetres and it looks like 74 mm. The leaf is not exactly 74.000 mm. It is somewhere close to 74, and the honest way to say so is 74 ± 0.5 mm.
Where does the 0.5 come from? The smallest division on the ruler is 1 mm, and the true value could be up to half a division either side of the mark you read. That gives the rule you will use again and again.
The rule for a scale
uncertainty = ± half of the smallest division
Finer graduations mean a narrower band. A syringe marked every 0.1 cm3 would give ±0.05 cm3 instead.
Two more examples. A balance with 10 g graduations reads to the nearest 10 g, so the uncertainty is ±5 g. A pipette marked every 0.1 cm3 gives ±0.05 cm3. Same rule every time: halve the smallest division.
Uncertainty is not error
These two words get swapped around constantly, and they mean different things.
Uncertainty
Error
What it is
The range in which the true value is expected to lie
The difference between a measured value and the true value
Where it comes from
The limits of the equipment you are using
Equipment faults or practical technique that push readings away from the truth
Can you remove it?
No. You can only reduce it by using finer equipment
Often yes, by improving the method or calibrating the equipment
How you report it
As a range with a ± sign
You discuss it in the evaluation
A worn-out thermometer that reads 2 °C too warm every single time gives you an error, not an uncertainty — and no amount of repeating will fix it. Uncertainty is built into the instrument’s scale; error is something that went wrong.
Error bars
An error bar is the uncertainty of a measurement drawn onto the graph. It sits above and below the point, or from side to side, and it turns a single dot into an honest statement about a range.
Error bars are usually vertical, for y-values, but they can be plotted horizontally for x-values.
They can be used to show the range, the degree of precision, the standard error or the standard deviation.
Because they can show any of those, you must always state what your bars represent. In an exam the question will tell you.
What the bar shows
What it tells the reader
Range
The gap between the lowest and highest value recorded
Degree of precision
How close the repeat readings sit to each other
Standard error
How reliable the mean is as an estimate of the true population mean
Standard deviation
The spread of the data around the mean
Reading overlap
This is the bit that carries marks. If two sets of error bars showing standard deviation overlap, the two data sets are not significantly different — the difference you can see between the means could easily have come about by chance.
Overlap is a warning sign, not a verdict. Non-overlapping bars only tell you the difference is worth testing.
Level of precision
Precision is about how finely you write your numbers down, and there is a simple honesty rule behind it.
The precision you write with must not exceed the precision of the equipment. A balance reading to 0.1 g cannot give you an answer of 4.283 g.
Every value in one raw data set should be written to the same number of decimal places. Writing 4.0, 4.25 and 4 in the same column looks careless and loses marks.
Processed values, such as means, follow the same rule — a mean of three readings taken to one decimal place should be given to one decimal place.
Writing extra decimal places does not make you look more scientific — it makes you look like you trust equipment further than it deserves. Match your data table to your instrument and stop there.
The coefficient of determination, R2
When you draw a line or curve of best fit, R2 tells you how well that line actually describes the points.
Value of R2
What it means
R2 = 0
The dependent variable cannot be predicted from the independent variable at all. R2 is normally zero or above
Between 0 and 1
The dependent variable can be predicted, but how well depends on the value. The closer to 1, the better the fit
R2 = 1
A perfect fit — every point sits on the line
A good fit is not the same as a good explanation. A high R2 only says the line matches your points. It does not promise the line is the right model for what is going on biologically.
Correlation and the correlation coefficient
Correlation is an association between two variables. It says they move together; it does not say one causes the other.
Positive correlation: as variable A goes up, variable B goes up.
Negative correlation: as variable A goes up, variable B goes down.
Causation is different — that is when one variable genuinely influences the other.
The correlation coefficient, r, puts a number on it. It tells you whether a linear relationship exists and how strong it is.
Perfect correlation, where every point sits on one straight line, gives r of exactly +1 or −1. Real biological data almost never does.
🧠
The sign and the size mean different things
The sign of r tells you the direction — up or down. The size of r tells you the strength. So r = −0.9 is a much stronger relationship than r = +0.3, even though it is negative.
Statistical tests and hypotheses
A statistical test is what turns “these look different to me” into a defensible claim. Every test starts from two hypotheses.
Hypothesis
What it claims
Null hypothesis
There is no significant difference, or no significant association, between the two variables
Alternative hypothesis
There is a significant difference, or a significant association, between the two variables
The test lets you accept or reject the null hypothesis. If the test shows no significant difference, then any difference you can see in the data is put down to chance alone.
Test
Use it to…
t-test
Decide whether the means of two data sets differ significantly
Correlation test
Find whether a correlation exists and how strong it is
Chi-squared test
Decide whether the gap between observed and expected values is significant
You are given the formulas in the exam. What you are marked on is choosing the right test, stating the null hypothesis properly, and saying what your result means about the biology — not just quoting a number.
WE 1
Stating a measurement with its uncertainty
A student measures the diameter of a fungal colony with a ruler graduated in millimetres and records 46 mm. State the reading with its uncertainty, and explain your answer. (3 marks)
Step 1: find the smallest division
The ruler is marked every 1 mm.
Step 2: halve it
The true value could be up to half a division either side of the mark, so the uncertainty is ±0.5 mm.
Step 3: write the reading as a range
The diameter lies somewhere between 45.5 mm and 46.5 mm.
46 ± 0.5 mmwriting 46.0 mm with no ± claims more precision than the ruler can give
WE 2
Interpreting error bars
A graph shows the mean height of seedlings grown in two light conditions. The means are 8.2 cm and 9.4 cm, and the standard deviation error bars overlap. Evaluate what can be concluded. (3 marks)
Point 1: describe what you see
The mean height in the second condition is higher, by 1.2 cm.
Point 2: read the bars
The error bars overlap, so the two sets of data are not significantly different.
Point 3: give the conclusion carefully
The difference between the means could have arisen by chance, so there is no evidence that light condition affected seedling height.
No significant difference — the overlap means chance cannot be ruled outsay what the bars represent, and run a t-test if you want to be sure
WE 3
Reading R2 and r together
A scatter graph of stomatal density against light intensity has a line of best fit with R2 = 0.91 and a correlation coefficient of r = +0.95. Explain what these two values tell you. (3 marks)
Point 1: what R2 says
R2 = 0.91 is close to 1, so the line of best fit is a good fit to the data points.
Point 2: what r says
r = +0.95 is close to +1, so there is a strong positive linear relationship: as light intensity rises, stomatal density rises.
Point 3: the limit of both
Neither value shows that light intensity causes the change in stomatal density. Correlation is not causation.
A strong positive relationship that the line describes well — but not proof of causethe “not causation” sentence is nearly always the third mark in questions like this
💡 Exam tips
Halve the smallest division. That single rule handles rulers, syringes, thermometers and balances.
Always write the unit after a ± value: ±0.5 mm, not just ±0.5.
When you describe error bars, say what they represent before you interpret them.
Overlapping standard deviation bars means “not significantly different” — use that exact phrase.
Keep every value in a column to the same number of decimal places.
Whenever you mention a correlation, add the line about causation. It is a free mark.
⚠ Common mistakes
Treating uncertainty and error as the same thing. One comes from the equipment’s scale, the other from something going wrong.
Saying a difference is significant just because the means look different. Check the bars, then test.
Not saying whether bars show SD, SE or range. Without that, the bars mean nothing.
Writing more decimal places than the instrument allows. It looks precise and is actually dishonest.
Reading a high R2 as proof the model is right. It only says the line matches the points.
Turning a correlation into a cause. Two variables can rise together because a third one drives both.
Up next: Graphing Skills — picking the right graph, drawing a line of best fit that an examiner will accept, and pulling rates out of a curve with a tangent.
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