The best way to reduce random errors is to repeat the measurement n times. Statistically the best estimate for these measurements is the average:
The uncertainty on this value is given by the standard deviation of the mean:
where S is the standard deviation defined as:
Excel Functions:
Use the following functions in Excel to compute the uncertainty on repeated measurements. In this example the values would be in the cells A1 to A10.
| Quantity | Function |
| average | =average(A1:A10) |
| standard deviation | =stdev(A1:A10) |
| standard deviation of the mean | =stdev(A1:A10)/(sqrt(count(A1:A10))) |
Example:
The period of a pendulum is measured 5 times.
| Period (s) | best estimate (average) | uncertainty (Sm) |
| 14.3 | 14.54 | 0.242074369 |
| 14.9 |
|
|
| 15.2 |
final result: 14.5 ± 0.2 s
|
|
| 14.5 | ||
| 13.8 |
Alternate simple method.
This method will approximate the uncertainty on an average when you have less than 10 values. Above 10, this method yields a result much smaller than the standard deviation of the mean and should not be used.
To approximate the uncertainty on an average, simply divide the range by the number of values: (Max - Min)/n
In the previous example, this method would give us: (15.2 - 13.8)/5 = 0.28 ~ 0.3.
The average period would be written as 14.5 ± 0.3 s.