Graphs for the IB IA: choosing the type, error bars, trendlines and R²

For most IB science IAs the right graph is a scatter graph of your processed data, independent variable on the x-axis, with error bars on every point and a line or curve of best fit chosen because the science predicts that shape. Use a bar chart only when the independent variable is categorical (species, sites, brands). Label both axes with the quantity and unit, and write a caption that says what the error bars represent.

Which type of graph should I use?

Let the type of independent variable decide.

A point-to-point line between experimental values only makes sense when the intermediate values really exist, as in a continuous time series. Joining up the means of five separate concentrations implies you know what happens between them, which you don't. For those, fit a line or curve, or leave the points unjoined.

What should go on the axes, title and caption?

• Axis labels: the quantity and its unit, e.g. "Concentration of H₂O₂ / %" or "Mean rate / cm³ s⁻¹". Uncertainties belong in the data table headers, not on the axis.
• Scales: evenly spaced, filling most of the plot. A scatter graph doesn't have to start at zero. A bar chart does, because the length of the bar is what the eye reads.
• Caption: a figure number, what is plotted against what, the sample size, and what the error bars show.

Plot what answers the research question. Usually that's the processed data (means, rates, calculated values), not every raw reading, though plotting the raw points faintly behind the means shows the reader the real spread.

What should the error bars show?

An error bar is only useful if the reader knows what it means. Pick one of these and say which in the caption.

SD and SE answer different questions. SD says how variable the organisms or samples were; SE says how sure you are of the mean, and it shrinks as n grows. Neither is "better". Just don't switch between them across graphs without saying so.

What overlapping error bars mean

Overlap is a rough visual check, not a result:

• SD bars: overlap tells you very little either way about whether two means differ.
• SE bars: with reasonably large, similar samples, overlapping SE bars mean the difference is not significant at p = 0.05. Non-overlapping SE bars don't guarantee that it is.
• Whatever the bars show, if your conclusion depends on whether two conditions differ, run the test. The statistical tests guide shows which one.

When should I add a trendline, and should it go through the origin?

Add a trendline when the science gives you a reason to expect that shape, and say what the reason is.

• Linear: when theory predicts proportionality or a straight line. Absorbance against concentration (Beer–Lambert) is the textbook case.
• Curve: enzyme rate against substrate concentration levels off as active sites become saturated, so a straight line through those points is the wrong model, however good its R² looks over a narrow range.
• No line: when the IV is categorical, or when you have no model and the shape is the finding.

Avoid high-order polynomials. A degree-5 polynomial through six points fits perfectly and means nothing. A quadratic is defensible when you expect an optimum (enzyme activity against temperature or pH) and you say so.

Through the origin? Only force a line through (0, 0) if the theory demands it and you have checked the data agree. Better: fit the line freely and look at the intercept. If theory says zero and your intercept is clearly not zero, that is evidence of a systematic error, such as a spectrophotometer not zeroed with the correct blank. Forcing the line through the origin hides exactly the thing your evaluation should discuss.

Don't extrapolate beyond your range unless you flag it as an assumption.

What does R² mean, and how do students misuse it?

R² (the coefficient of determination) is the proportion of the variation in y that your fitted line or curve accounts for. R² = 0.92 means the model accounts for 92% of the variation in the plotted values; the rest is scatter around it.

Common misreadings:

• "R² = 0.98, so my hypothesis is proven." A high R² shows a good fit to these points. It doesn't show cause, and it doesn't tell you whether the relationship is statistically significant.
• Fitting to means only. Five means usually give a much higher R² than the 25 readings behind them. Say which you used.
• Choosing the model with the highest R². Adding terms to a polynomial never lowers R², so the most complicated curve always looks best. Choose the model from the science, then report how well it fits.
• Mixing up r and R². The correlation coefficient r runs from −1 to +1 and carries a direction; for a straight line, R² = r². R² for a curve isn't the square of anything you can test with a correlation table.

How do I get an uncertainty for a gradient?

When your result is a gradient (a rate constant, a calibration factor, an activation energy from an Arrhenius plot), draw two more lines: the steepest and the shallowest straight lines that still pass through all (or nearly all) of the error bars.

In Figure 1 the best-fit gradient is 2.85 dm³ mol⁻¹. The steepest line has a gradient of 3.10 and the shallowest 2.60, so:

uncertainty in gradient = (3.10 − 2.60) ÷ 2 = 0.25 dm³ mol⁻¹, giving 2.85 ± 0.25 dm³ mol⁻¹ (about 9%).

The same two lines give a range for the intercept: here from −0.010 to +0.020. Zero lies inside it, so there's no evidence of a systematic offset, and that's a sentence worth writing in your conclusion.

How do I do this in Excel, Google Sheets or Logger Pro?

Excel

• Select the x and y columns, then Insert → Scatter.
• Chart Elements (+) → Error Bars → More Options → Custom → Specify Value, and select your column of SDs or uncertainties for both positive and negative.
• Add Trendline → choose the type → tick "Display equation" and "Display R-squared".

Google Sheets

Insert → Chart → Scatter; then Customise → Series → Error bars. Sheets offers constant, percentage and "standard deviation" bars, and the last has the same problem as Excel's. Per-point custom error bars are awkward in Sheets, so many students do the final graph in Excel, LibreOffice Calc or Logger Pro.

Logger Pro (or Vernier Graphical Analysis)

Set error bars from a column in the column options, then use the linear or curve fit tool. Logger Pro reports the uncertainty in slope and intercept directly, which saves the max/min construction.

For a full report with graphs, captions and error bars in context, see the annotated science IA example; fieldwork data and kite diagrams are covered in the ecology field IA guide.