How do I estimate random error from repeated trials?

Calculate the standard deviation of your repeated measurements to estimate random error; if you only have a few trials you can use the range as a simple estimate.

When you take the same measurement many times you will get slightly different answers because of small random variations. To find the standard deviation you first find the average of all the measurements, then for each measurement you see how far it is from that average, square those differences, average the squared differences, and finally take the square root of that average. The result is a single number that tells you how spread out the measurements are around the average, which is the random uncertainty. If you only have a very small number of trials, calculating the range by subtracting the smallest value from the largest gives a quick, rough idea of the spread.

Students often think that taking more trials will always remove bias, but more trials only reduce random error not systematic error, so check your method if all values are shifted one way; another common mistake is forgetting to square the differences when calculating standard deviation, which would cancel out positive and negative deviations and give a misleading zero, so follow each step carefully; some people try to use the range as a precise substitute for standard deviation with many measurements, which underestimates or overestimates uncertainty, so use range only for quick checks and standard deviation for proper quantification.

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Estimate Random Error from Repeated Trials

1. Find the average of measurements
2. Compute each deviation from average
3. Square each deviation
4. Average the squared deviations
5. Take square root of average
6. Use range for few trials

Common Errors Estimating Random Error

1. Confusing random error with systematic error
2. Forgetting to square deviations
3. Using range as precise substitute

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