Chemistry · Lesson

Tables, graphs, calibration and honest data reporting

Make data visible without changing what they say. Use a calibration example to connect labelled observations, a model and the limits of an inferred result.

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By the end, you can

Label tables and graphs with quantities and units.
Distinguish calibration from adjusting an instrument.
Use a calibration model within its stated range and retain the original data.

A table should answer which quantity is which

An invented calibration record contains standards with known concentrations and measured instrument signals. A column headed “concentration / (mg/L)” tells you which quantity and unit the numbers describe. Here mg/L means milligrams of the specified material per litre of solution. “Signal / arbitrary unit” identifies a response whose scale belongs to the stated instrument. Sample labels and measurement conditions connect each row to its source.

Retain the original observations. A graph or fitted line is a presentation or analysis of those observations, not a replacement for them. State any averaging, correction, omission or transformation. A missing reading and a measured zero are different: write the missing status explicitly.

Calibration relates indications to reference information

Calibration establishes a relationship between reference quantities and instrument indications under stated conditions, including their uncertainties. It can help convert an indication into an estimated value of a quantity. Adjustment changes a measuring system so that its indications behave as intended. They are different operations.

A calibration that worked for one range, sample type or set of conditions need not work for all others. A chemical matrix—the other material present in a sample—can affect a signal. Record which standards, instrument, range and conditions support the relationship.

Follow a small linear model

For this teaching example, standards at 0, 1, 2 and 3 mg/L give signals 0.1, 2.1, 4.1 and 6.1 arbitrary units. A line through these invented values is S=2 C+0.1, where C is concentration in mg/L and S is the signal value on the named scale. The factor 2 has the units needed to convert concentration into signal.

An unknown with signal 3.1 gives C=(3.1−0.1)/2=1.5 mg/L in this model. That lies inside the standards’ 0–3 mg/L range. The exact-looking teaching line does not claim that real instruments have zero uncertainty, perfectly linear response or no interference.

Data points and a calibration model

Signal / arbitrary unit

Concentration / (mg/L)

Invented standards
— Model S = 2C + 0.1

Underlying data

Concentration / (mg/L)	Signal / arbitrary unit
0	0.1 · Invented standards
1	2.1 · Invented standards
2	4.1 · Invented standards
3	6.1 · Invented standards

Invented standards at 0, 1, 2, 3 mg/L yield 0.1, 2.1, 4.1, 6.1 arbitrary units. The line S=2C+0.1 models these values only within the stated scope. No measurement uncertainty is supplied by this teaching figure.

Show the relationship honestly

Put concentration on the horizontal axis and signal on the vertical axis for this example. Label both axes and use a scale that makes the four observations identifiable. The points are the data; the line is the model. A text table gives the same information to a reader who cannot use the graph.

A signal of 10.1 would give 5.0 mg/L by extending the same formula, but 5.0 is beyond the calibrated range. That is extrapolation. The calculation alone does not validate the extension. You would need suitable evidence or another measurement method. Even interpolation within the range requires the sample and conditions to match the model’s scope.

Worked examples

Estimate an unknown within the range

Use S=2 C+0.1 for an invented signal of 4.1 arbitrary units. What concentration follows?

Subtract the offset:4.1−0.1=4.0.
Divide by the slope:4.0/2=2.0 mg/L.
Check that 2.0 mg/L lies in the 0–3 mg/L calibration range.
Retain the matching-sample and condition assumptions.

The model gives 2.0 mg/L. A real result would also require uncertainty and evidence that the model applies to the sample.

Try it yourself

A reading was not taken. Should its table cell be entered as 0?
Show the answer and reasoning
No.
Zero is a measured or assigned numerical value. Missing data need an explicit missing status.
Are calibration and adjustment the same operation?
Show the answer and reasoning
No.
Calibration establishes a relationship with reference information. Adjustment changes the measuring system.
In the invented model S=2 C+0.1, what C follows from S=3.1?
Show the answer and reasoning
1.5 mg/L.
Subtract 0.1 and divide by 2; the result lies within the stated calibration range.
The model gives 5 mg/L for an indication beyond its calibrated range. Does applying the formula validate that result?
Show the answer and reasoning
No.
The extension is extrapolation. Its validity requires evidence beyond the arithmetic.

What to carry forward

Labels, units, samples and conditions give data their meaning.
Calibration and adjustment are different.
Separate observed points from the model fitted to them.
Retain raw data and explain processing.
A formula does not validate extrapolation.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.