Statistical data analysis, statistics, and scientific research assistance

Introduction

Any measurement of a variable is of real use only when accompanied by a quantitative statement of its associated uncertainty. The uncertainty can be seen as the range of values that a measurement could actually have. Without statements of uncertainty, we have little idea as to the validity of measurements, we are unable to decide whether a measurement is different from other measurements, and we cannot effectively compare results from different studies. Moreover, we will have no idea as to the size of uncertainties of derived variables resulting from the propagation of uncertainties during subsequent data manipulation.

Calculation

For a single measurement, the associated uncertainty could be expressed as half of the smallest unit of measurement of the measuring device. For a measurement of the long-axis (length) of a pebble with a ruler scaled in mm, the result may be 74.0 ± 0.5 mm. However, if several repeat measurements were taken of the same pebble, the calculation of the sample standard deviation (s) of the mean of this series of observations allows a formal estimate of uncertainty expressed as a measure of dispersion of the distribution about the mean value. If many measurements were taken of different pebbles, the standard deviation would again be appropriate as the uncertainty around the mean value of the sample. Supposing that the quantity in question is reasonably modeled by a normal probability distribution, then about 68% of the values lie within one standard deviation of the mean (i.e., within x ± s). Approximately 95% of the values of a quantity will lie within x ± 2s and 99% within x ± 3s. Uncertainties should routinely be expressed at 95% levels of confidence, i.e., as x ± 2s.

The uncertainty of the mean value (‘standard error’) is a somewhat different statistic. Although the distribution of the sample mean has the same mean as that of single values, the variance is lower by a factor dependent on the sample size. The standard error (s ) is the standard deviation divided by the square root of the sample size. For a sample of 100 pebbles, the mean length may be 66 mm, two standard deviations may be ± 10 mm (2s), and the standard error would be ± 1 mm (2s ).

Propagation

Uncertainties are combined using standard rules.

1. Division or multiplication by another measurement: Take a measurement a ± s_a and divide it or multiply it by another measurement b ± s_b. The value is a/b (or a*b) and the uncertainty is given by the formula:

e.g. 250 ± 20 / 10.0 ± 0.5 = 25.0 ± 2.4 (1s)

or 25.0 ± 4.8 (2s)

e.g. 250 ± 20 * 10.0 ± 0.5 = 2500 ± 240 (1s)

or 2500 ± 480 (2s)

2. Addition or subtraction by another measurement: Take a measurement a ± s_a and subtract it from or add it to another measurement b ± s_b. The value is a - b (or a + b) and the uncertainty is given by the formula:

e.g. 15.0 ± 1.0 - 10.0 ± 0.5 = 5.0 ± 1.1 (1s)

or 5.0 ± 2.2 (2s)

e.g. 15.0 ± 1.0 + 10.0 ± 0.5 = 25.0 ± 1.1 (1s)

or 25.0 ± 2.2 (2s)

3. Combinations: For equations that contain several variables and operators, it is simply a matter of combining uncertainties of the various components of the equation using the above rules.