Mean and Standard Deviation
If you make several measurements on samples that should be identical, such as the determination of the mass of some analgesic tablets, the results should express two things: the average of the measurements and the size of the uncertainty.
There are two common ways of expressing an average: the mean and the median. The mean (x) is the arithmetic average of the individual results (x1, x2, etc.), or
where the numerator is the sum of the values. The mean is equal to the sum of all the measurements divided by the number of measurements. For four tablets observed to have masses of 428 mg, 479 mg, 442 mg, and 435 mg, the mean is
The median is the value that lies in the middle among the results. Half of the measurements are above the median and half are below the median. For results of 465 mg, 485 mg, and 492 mg, the median is 485 mg. When there is an even number of results, the median is the average of the two middle results.
In addition to expressing a mean value for a series of results, we must also express the uncertainty. This usually means expressing either the precision of the measurements or the observed range of the measurements. The range of a series of measurements is defined by the smallest value and the largest value. For the masses of the four tablets, the range is from 428 mg to 479 mg. Using this range, we can express the results by saying that the true value lies between 428 mg and 479 mg.
However, the most common way to specify precision (agreement within the series) is by the standard deviation, s, which for a small number of measurements is given by the formula
where xi, is an individual result, x is the average (mean), and n is the total number of measurements. For the masses of the four tablets, we have
Thus we can say the mass of a typical tablet in the group is 446 mg with a sample standard deviation of 23 mg. Statistically this means that any additional measurement has a 68% probability (68 chances out of 100) of being between 423 mg (446 - 23) and 469 mg (446 + 23). That is, we can use the mean and standard deviation to express the mass of a typical tablet as 446 +/- 23 mg. Thus the standard deviation is a measure of the precision of a given type of determination.
Although the standard deviation is a good measure of the precision of a given set of data, it can be difficult to compare the standard deviation from two different types of measurements directly. You might need to do such a comparison to determine the largest source of uncertainty in an experimentally determined answer. In the example above, the standard deviation was 23 mg, but how would that compare to a standard deviation of 0.25 mL for set of volume measurements with a mean value of 35.49 mL? One way to do this comparison is with a relative standard deviation. A relative standard deviation, RSD, is simply the ratio of the standard deviation over the mean (typically multiplied by 100 to express as a percentage).
RSD = 100*(s/x)
For our two examples, (23 mg / 446 mg)*100 = 5.2 % and (0.25 mL / 35.49 mL)*100 = 0.70 %. Now that these two numbers are expressed as percentages it is clear that the precision of the volume measurement is better than the precision of the mass measurement.
Last update: Monday, August 25, 2003 at 6:12:49 PM
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