Calculating the Arithmetic Mean of Repeated Measurements

What does a mean represent?

Imagine measuring the same quantity several times. The readings may not be identical because it is difficult to position a ruler or stop a timer in exactly the same way every time. The arithmetic mean gives one value that represents the set of readings. It is often called the average, although mean is the precise term we will use here.

Watch the mean portion of Math Antics – Mean, Median and Mode by mathantics. The block towers show what a mean represents before the video explains how to calculate it. You do not need the later sections on median and mode.

To calculate an arithmetic mean, add all the measurements, then divide by how many measurements you added:

The answer has the same unit as the measurements. An average height is still a height; an average time is still a time.

Calculating a mean from repeated measurements

Suppose you drop a ball from the same height three times and measure how high it bounces. You record $42\mathrm{cm}$, $44\mathrm{cm}$, and $43\mathrm{cm}$. All three readings measure the same quantity under the same intended conditions, so they belong in one mean.

Notice the two separate jobs in that calculation: 129 cm is the total, while 43 cm is the mean. We divide by 3 because there were three readings, not because the readings happened to be heights. Even if two readings were identical, each would still count as one measurement.

A mean need not equal any individual reading, and it need not be a whole number. Suppose three stopwatch readings are $10.0\mathrm{s}$, $10.2\mathrm{s}$, and $10.3\mathrm{s}$:

The division gives approximately $10.1667\mathrm{s}$. Here it is reported to one decimal place, like the original readings. Do not mistake the extra calculator digits for extra accuracy in your stopwatch measurements.

Check that the readings belong together before averaging. For instance, bounce heights from three drops at one release height form a set of repeated measurements. A bounce height measured after changing the release height belongs to a different set. Likewise, do not add a reading in centimetres to one in metres without first putting them in the same unit.

Check the readings before using the mean

Repeated measurements help reduce the effect of small, unpredictable differences in how you measure. They do not automatically correct every problem. A ruler positioned incorrectly in the same way each time, for example, could give consistently misleading readings.

Read Evaluating the data from BBC Bitesize for its explanation of repeatability, unusual results, and a worked mean calculation.

Consider what would happen if the bounce readings were $42$, $43$, $44$, and $83\mathrm{cm}$. Using all four gives:

The $83\mathrm{cm}$ reading is very different from the others. It might be a recording mistake, or the ball might have been dropped differently. Do not quietly erase it just because it changes the mean. Check what happened and, if possible, repeat the measurement. If you have a good reason to reject it, record that reason. The mean of the three remaining readings would then be:

The divisor changed from 4 to 3 because the number of readings included in the calculation changed. If there is no evidence that the unusual reading was faulty, keep it visible and investigate further rather than treating $43\mathrm{cm}$ as an unquestionable result.

Key takeaways

For repeated measurements of the same quantity under the same conditions, add the readings and divide by the number included. Write the mean with the correct unit, and check any unusually different reading before deciding how to handle it. A mean can give a useful single estimate, but it cannot make faulty measurements reliable by itself.

Next, you will use these measurement skills to interpret results shown in tables, bar charts, and line graphs.