Mean, Range and Repeatability

Introduction

A table of ten repeated readings is hard to take in at a glance. Scientists summarise such data with two simple numbers: the mean , which gives a single best estimate, and the range , which shows how much the readings vary. Together they tell you both what the answer is and how much you can trust it. This page also explains two words that describe trustworthy data: repeatable and reproducible .

Core explanation

The mean. The mean (arithmetic average) is the sum of the readings divided by the number of readings:

mean = sum of readings ÷ number of readings

It is the best single estimate of the true value when errors are random, because random errors above and below partly cancel. Anomalous readings should be excluded before calculating the mean.

Precision of the mean. Quote the mean to the same resolution as the readings — usually the same number of decimal places. If burette readings are recorded to 0.05 cm³, a mean such as 23.4666… cm³ should be rounded to two decimal places, 23.47 cm³; never copy every digit from the calculator.

The range. The range is the largest reading minus the smallest reading:

range = maximum − minimum

It is a quick measure of spread . A small range means the readings are close together, so the data are precise. A large range means considerable scatter.

Estimating uncertainty from the range. A simple estimate of the uncertainty in the mean is half the range. For titres of 24.10, 24.20 and 24.15 cm³, the range is 0.10 cm³, so the mean could be written as 24.15 ± 0.05 cm³.

Repeatability. Results are repeatable if the same experimenter, using the same method and equipment, obtains similar results when repeating the measurement. Repeatability checks that random errors are small.

Reproducibility. Results are reproducible if a different person, or the same person using different equipment or a different method, obtains similar results. Reproducibility is a stronger test, because it can reveal systematic errors linked to one particular instrument or technique. When other laboratories can reproduce a result, the scientific community becomes much more confident in it.

Concordant results. In titrations, results within 0.10 cm³ of each other are often called concordant . Only concordant titres are used to calculate the mean titre; a rough first titre is usually excluded.

Step-by-step reasoning

To summarise a set of repeated readings:

1. Look for any anomalous reading and decide whether to exclude it, giving a reason. 2. Add the remaining readings. 3. Divide by the number of readings used to find the mean. 4. Round the mean to the resolution of the readings. 5. Subtract the smallest reading from the largest to find the range, and comment on the spread.

Visual explanation

Plot each reading as a dot on a vertical scale. Draw a horizontal line through the middle of the cluster at the mean. Then draw a bar from the lowest dot to the highest dot: the length of that bar is the range. Short bars mean precise data; long bars mean scattered data.

Real-world analogy

A cricket batter's average tells you their typical score, while the gap between their best and worst innings tells you how consistent they are. Two players with the same average may differ greatly in reliability — exactly what mean and range capture together.

Real-world example

Drug manufacturers test the mass of tablets taken from each batch. They report the mean mass to check the dose is correct and the range to check the tablets are consistent. A batch with the right mean but a large range could contain tablets with dangerously high or low doses.

Why?

Why is the mean more reliable than a single reading? Each reading carries its own random error. When several readings are averaged, errors in opposite directions partly cancel, so the mean is usually closer to the true value than an individual reading.

Common misconception

"The range tells you how accurate the result is." The range measures spread, which describes precision. A set of readings could have a tiny range and still all be wrong because of a systematic error, so a small range alone does not show accuracy.

Worked example

Question: Four titres are recorded: 25.60 (rough), 24.35, 24.40 and 24.30 cm³. Calculate the mean titre and range of the concordant results.

Reasoning: Exclude the rough titre, 25.60 cm³. Sum = 24.35 + 24.40 + 24.30 = 73.05 cm³. Mean = 73.05 ÷ 3 = 24.35 cm³. Range = 24.40 − 24.30 = 0.10 cm³.

Answer: Mean titre = 24.35 cm³; range = 0.10 cm³, so the results are concordant and precise.

Quick check

1. Readings of 18.2, 18.6 and 18.1 °C are recorded. What is the range? Answer: 18.6 − 18.1 = 0.5 °C.

Exam focus

Show the sum and the division when calculating a mean, and state which readings you excluded and why. Give the mean to an appropriate number of decimal places. Remember that "repeatable" refers to the same person and method, while "reproducible" involves a different person, method or equipment.

Advanced insight

The range depends heavily on the two most extreme readings, so a single unusual value can inflate it. For larger data sets, scientists prefer the standard deviation , which uses every reading. The standard deviation of the mean, s ÷ √n, becomes the usual uncertainty in more advanced work.

Summary

The mean, found by dividing the sum of readings by their number, is the best estimate of the true value. The range, maximum minus minimum, measures spread and so indicates precision; half the range gives a rough uncertainty. Repeatable results agree when the same person repeats the experiment; reproducible results agree when others or different methods are used.

Practice questions

1. Calculate the mean of 3.42 g, 3.46 g and 3.44 g. Answer: (3.42 + 3.46 + 3.44) ÷ 3 = 10.32 ÷ 3 = 3.44 g. 2. State the range of the readings in question 1 and use it to estimate the uncertainty in the mean. Answer: Range = 3.46 − 3.42 = 0.04 g; uncertainty ≈ half the range = ±0.02 g. 3. What is the difference between repeatable and reproducible results? Answer: Repeatable results agree when the same person repeats the experiment with the same method and equipment; reproducible results agree when a different person, method or set of equipment is used. 4. Why is the rough titre usually left out of the mean titre? Answer: It is done quickly to find the approximate end point, so it usually overshoots and is not as precise as the careful titres.