How to estimate random error from data?

Compute the standard deviation of repeated measurements to estimate random error.

Think of taking the same measurement many times, like timing how long it takes a cookie to bake, and then finding how spread out those times are; the standard deviation is a number that tells you that spread. First you find the average by adding all the measurements and dividing by how many you took, then for each measurement you see how far it is from that average and square that distance so all numbers are positive, then you add those squared differences and divide by one less than the number of measurements to get the variance, and the square root of that variance is the standard deviation which has the same units as the original measurements. This standard deviation estimates the random error because it reflects the typical size of fluctuations caused by unpredictable factors. If you want the uncertainty in the average value itself, divide the standard deviation by the square root of how many measurements you made to get the standard error.

Common mistakes include dividing by the wrong number when calculating variance, which gives a biased estimate; always divide by one less than the number of measurements when you are estimating the population variance from a sample. Another mistake is forgetting to square the differences so positive and negative deviations cancel out; squaring fixes that and keeps all contributions positive. A third mistake is confusing the standard deviation with the standard error; the standard deviation describes spread of individual measurements while the standard error describes the uncertainty in the mean and is smaller when you have more measurements.

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Estimate Random Error Using Standard Deviation

1. Take repeated measurements
2. Find the average (mean)
3. Compute deviations from mean
4. Square each deviation
5. Sum squared deviations
6. Divide by n−1 for variance
7. Take square root for standard deviation

Common Errors Estimating Random Error

1. Dividing by wrong number for variance
2. Forgetting to square deviations
3. Confusing standard deviation with standard error

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