Averages and Anomalies in Lab Data

When you repeat an experiment, you almost never get exactly the same number twice. Five students timing the same reaction get five slightly different times; three titrations give three slightly different titres. So what’s “the” result? This article works through how to turn a set of repeat measurements into one sensible value, how to spot results that don’t belong, and how to report the answer honestly.

Why repeat at all?

Every measurement contains some random error: small, unpredictable variations from judging an end point, reading a scale or starting a stopwatch. Repeating a measurement and averaging reduces the effect of these errors. Repeats also let you:

• check that results are repeatable (you get similar values each time)
• spot anomalous results caused by a mistake

Repeating doesn’t remove systematic errors, such as a balance that always reads 0.05 g too high. See experimental errors.

The three averages

Mean

Add up all the values and divide by how many there are. This is the average used in almost all chemistry calculations.

Mean = sum of values ÷ number of values

Median

Put the values in order and take the middle one (or the mean of the two middle ones if there’s an even number). The median isn’t pulled much by one extreme value.

Mode

The value that occurs most often. It’s rarely useful for continuous measurements such as volumes and temperatures, where exact repeats are uncommon, but it can be useful for categories or counts.

Worked example 1: mean, median and mode

Five measurements of the time for a cross to disappear (s): 42, 45, 43, 45, 44.

• Mean: (42 + 45 + 43 + 45 + 44) ÷ 5 = 219 ÷ 5 = 43.8 s
• Median: in order: 42, 43, 44, 45, 45 → 44 s
• Mode: 45 appears twice → 45 s

All three are close because the data are consistent. For a chemistry report, you’d normally give the mean, rounded sensibly: 44 s (the measurements were to the nearest second, so a mean of 43.8 s suggests more precision than the data have; some teachers accept 43.8 s, so follow your course’s convention).

Spotting anomalous results

An anomaly (outlier) is a result that doesn’t fit the pattern of the others, usually because something went wrong: a misread scale, an overshot end point, a wrongly recorded number.

Worked example 2: identifying an anomaly

Four measurements of the volume of gas produced in 1 minute (cm³): 34, 36, 35, 48.

• Three values are close together (34–36); 48 is far away.
• 48 cm³ is anomalous.
• Mean excluding it: (34 + 36 + 35) ÷ 3 = 35 cm³
• Mean including it: (34 + 36 + 35 + 48) ÷ 4 = 38.25 cm³, noticeably pulled upwards.

Notice the median of all four values (35.5 cm³) is barely affected by the anomaly. That’s the median’s advantage, but in chemistry the usual practice is to identify the anomaly, exclude it and take the mean of the rest.

When is it acceptable to exclude a result?

• There’s a clear reason to think something went wrong (you noted “overshot end point” or “stopwatch started late”), or
• It is obviously inconsistent with the other repeats.

Always record the anomalous result, mark it clearly, and explain why it was left out. Never quietly delete data. If you’re unsure, repeat the measurement. More formal statistical tests exist for rejecting outliers, but at school level, clear inconsistency plus a sensible explanation is usually expected.

Worked example 3: concordant titres

In titrations, the rule is to use only concordant results: titres within 0.10 cm³ of each other (some courses use 0.20 cm³).

Titres (cm³): rough 25.2; 24.65; 24.85; 24.60; 24.70.

• The rough titre is always excluded; it was done quickly to find the approximate end point.
• Accurate titres: 24.65, 24.85, 24.60, 24.70.
• Concordant set: 24.65, 24.60 and 24.70 (all within 0.10 of each other). 24.85 is 0.25 above 24.60, so it’s excluded.
• Mean titre = (24.65 + 24.60 + 24.70) ÷ 3 = 73.95 ÷ 3 = 24.65 cm³

This mean is then used in the titration calculations. See also titration errors.

Measuring spread: the range

The range is the difference between the highest and lowest values. It gives a quick idea of how much the results vary.

Range = highest − lowest

Worked example 4: using the range

Group A measures a temperature rise three times: 12.5, 12.8, 12.6 °C. Group B: 11.2, 13.9, 12.7 °C.

• Group A: mean = 12.6 °C; range = 12.8 − 12.5 = 0.3 °C
• Group B: mean = 12.6 °C; range = 13.9 − 11.2 = 2.7 °C

Both groups have the same mean, but Group A’s results are much more precise (repeatable). Group B should look for sources of random error, such as inconsistent stirring or reading the thermometer at different times.

A rough estimate of the uncertainty in a mean is ± half the range: Group A would report 12.6 ± 0.2 °C (rounding 0.15 up); Group B 12.6 ± 1.4 °C. See calculating uncertainty.

Precision vs accuracy

• Precise results are close to each other (small range).
• Accurate results are close to the true value.

Results can be precise but inaccurate, if a systematic error shifts them all. For example, a set of titres of 24.10, 24.15 and 24.10 cm³ is very precise, but if the burette was read from above every time, all of them are wrong in the same direction.

Worked example 5: pooling class results

Eight groups measure the enthalpy of neutralisation (kJ/mol): −54.1, −55.3, −53.8, −42.0, −54.9, −56.2, −55.0, −54.4.

1. Spot the anomaly: −42.0 is far from the others (all between −53.8 and −56.2). It’s probably from a group whose cup lost a lot of heat, or who misread a temperature.
2. Mean of the other seven: sum = −383.7; mean = −383.7 ÷ 7 = −54.8 kJ/mol
3. Range of the seven: −53.8 to −56.2, so 2.4 kJ/mol; uncertainty ≈ ±1.2 kJ/mol.
4. Compare with the literature value of about −57 kJ/mol: the class mean is slightly less exothermic, consistent with heat loss, a systematic error. See enthalpy of neutralisation.

Reporting an average

• Give the mean to a sensible number of significant figures, usually matching the precision of the original measurements. See significant figures.
• State how many results were used and which (if any) were excluded, and why.
• Give an indication of spread (range or uncertainty).

Common mistakes

• Including the rough titre in the mean.
• Including an obvious anomaly without comment.
• Deleting an anomaly without recording it.
• Quoting a mean to far more decimal places than the measurements.
• Thinking repeats remove systematic errors.

Key takeaways

• Repeating measurements and averaging reduces random error, not systematic error.
• The mean is the standard average in chemistry; the median resists extreme values.
• Identify and exclude anomalies, but always record and explain them.
• For titrations, average only concordant titres (within 0.10 cm³), never the rough.
• Use the range to judge precision, and quote the mean to a sensible precision. For more advanced spread measures, see standard deviation in chemistry data.