A mean is the sum of the readings divided by how many readings you used. In measurement and quantities you meet it whenever an experiment is repeated to give a more reliable value.
The skill has two parts: deciding which readings belong in the average, and dividing by the correct number.
How do I calculate a mean properly?
- Look at all the readings first. Do they cluster? Is one far from the others?
- Decide about any anomalous result and say why it is excluded.
- Add the readings you are keeping.
- Divide by the number you kept, not the number you started with.
- Give the mean to the same number of decimal places as the readings, with the unit.
Worked example
A student times a trolley rolling down a ramp five times: 2.31 s, 2.28 s, 2.35 s, 3.05 s, 2.30 s. (Invented example data.)
Step 1, look at the set: four readings are between 2.28 s and 2.35 s. The 3.05 s reading is about 0.7 s away from all of them.
Step 2, decide: 3.05 s is anomalous, possibly a late stop. It is left out, and the reason is stated.
Step 3, add the rest: 2.31 + 2.28 + 2.35 + 2.30 = 9.24 s.
Step 4, divide by four readings: 9.24 ÷ 4 = 2.31 s.
Answer: mean time = 2.31 s (anomalous 3.05 s omitted).
Another everyday use is finding a period. If 10 oscillations of a pendulum take 18.5 s (itself a mean of repeated timings), one oscillation takes 18.5 ÷ 10 = 1.85 s.
The mistake to watch for
There are two common slips. The first is to keep the anomalous reading in the sum and average everything.
Mistaken answer: (9.24 + 3.05) ÷ 5 = 12.29 ÷ 5 = 2.458 s, about 2.46 s
This value is higher than four of the five readings and close to none of them.
The second is to remove the anomalous reading but still divide by five:
Mistaken answer: 9.24 ÷ 5 = 1.848 s
The correction is to divide by the number of readings actually used. Always ask whether your mean lies inside the cluster of readings. A mean far from every reading is a warning sign.
Check yourself
1. Find the mean of 15.2 cm, 15.6 cm and 15.4 cm.
Show answer
15.2 + 15.6 + 15.4 = 46.2. 46.2 ÷ 3 = 15.4 cm.
2. Twenty oscillations of a pendulum take 34.0 s. Find the period.
Show answer
34.0 ÷ 20 = 1.70 s.
3. A wire’s diameter is measured as 4.8 mm, 4.9 mm, 4.7 mm, 6.1 mm and 4.8 mm. Identify the anomalous reading and find the mean.
Show answer
6.1 mm is far from the cluster, so it is anomalous. Remaining: 4.8 + 4.9 + 4.7 + 4.8 = 19.2. 19.2 ÷ 4 = 4.8 mm.
Where this leads next
After calculating a mean, the natural question is how much to trust it, which is the focus of stating a realistic uncertainty from supplied instrument data. You may also want to revisit reading an analogue scale, and then try the measurement and quantities practice set. The bounds and rounding explainer shows how many decimal places a mean can honestly carry.
If experiment questions feel like guessing which readings to keep, our teachers can work through examples with you in online one-to-one Physics tuition.