The most common measure of error in experimental values is the standard deviation of a set of data. The deviation of a given point is the difference between it and the average or mean of the set of data.
It is calculated as the average squared deviation of each number from the mean. For example, for the numbers 1, 2, and 3 the mean is 2 and the variance or square of the standard deviation is:
(1-2)2 + (2-2)2 + (3-2)2
When the standard deviation for a sample of numbers is calculated, the formula is: Standard Deviation = The square root of (the average squared deviation of each number from the mean).
The mathematical formula is:
The above formula is a biased estimate as it assumes the number of data values are very large. However, when the number of values are small (N is small) then (N -1) is used as the denominator and the standard deviation is only an estimate as there are not enough points to obtain a precise answer.
This formula (see below) is then used for laboratory calculations to calculate the standard deviation or spread of the numbers being investigated.
It is useful to make up a table to help with the calculation of S.
Column 1: Data Value (X)
Column 2: Mean or Average (M)
Column 3: Square of the Data Value minus the Mean
Column 4: Sum of the squared Data Values minus the Means
Column 5: Column 4 value divided by the (number of values minus 1)