The most common measure of the error in an experimental quantity is the standard deviation of a set of data. The deviation of a given data point is just the difference between it and the average of the set of data, X_i - X_ave.

The proper standard deviation for a set of data is s,



however this only holds when we have a lot of points. Since we typically don't have enough data to compute a proper standard deviation, we use the estimate of the standard deviation, S

Note that the only difference is the N vs. N-1 in the denominator. If N is large, S and s become almost identical.

Statistically, for a large set of measurements about 68% will lie within one standard deviation of the average value, 95% will lie within 2 and 99.7% will lie within 3. For example, if you have a set of data reported as 1.00 +- 0.1, 68% of the data will lie within the range 0.9 to 1.1, 95% within the range 0.8 to 1.2, and 99.7% within the range 0.7 to 1.3. See the help file on the normal curve for more information.

Standard deviations are typically reported to only one significant figure.

Example: What is the estimated standard deviation of the following set of numbers?

2.45, 2.56, 2.19, 2.00, 2.22

Solution: The estimated standard deviation (S) is given by the second equation above. First we need to compute the average of the data:

X_ave = (2.45+2.56+2.19+2.00+2.22)/5

X_ave = 2.28

0.029 + 0.078 + 0.008 + 0.078 + 0.004 = 0.197

S = sqrt(0.197/(5-1)) = 0.2

Information provided by: http://learn.chem.vt.edu