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Standard Deviation
The problem: given a set of data, we can estimate the mean value, or the average, of the items in the set simply enough by adding up all the items and dividing by the number of them. But how can we estimate the variation between the data items and their mean?
The standard deviation is a measure of the scatter between a set of measurements. It helps to answer questions like, "How much variability is there in my results?"
One solution is to find the largest and smallest values. This gives the range in which the values lie, but this measure is very sensitive to the occurrence of extreme cases.
A better approach is to calculate the sum of the deviations from the mean, and divide the sum by the number of items to give a figure for the average deviation. This is simple enough to do by hand, but this method is inconvenient from a mathematical point of view, due to needing to take the absolute value of each deviation.
Standard Deviation Definition
The preferred method avoids the mathematical difficulties by squaring each deviation. Then the average of the squared deviations is calculated, to give a result caused the variance and standard deviation is the square root of this variance.
Thus we come to the standard deviation formula, the basic method for calculating standard deviation for a set of items: calculate the square root of the average value of the squares of the distances of each item from the mean for the whole set.
The standard deviation equation is expressed mathematically as:
| std deviation = sqrt(sum((value - mean) 2 ) / N) or |  |
where N is the number of items, and the mean is calculated first as:
| mean = sum(value) / N or |  |
How can we make use of standard deviation? I said at the start that it is a measure of variability. It can be shown that 68% (say, two thirds) of all items in a reasonably large sample will be within one standard deviation of the mean, 95.4% will be within two std deviations, and more than 99.7% will be within three std deviations of the mean.
Population Standard Deviation
The discussion so far has focussed on how to calculate standard deviation for a sample of items. Can we estimate the standard deviation of the values found in the population from which our sample was taken? Well yes: it can be shown that the population standard deviation is found by doing a similar calculation to that for the sample standard deviation, but instead of dividing by the number of items in the sample, we divide by one less than this number.
Standard Errors
Now that we have an estimate for the variation of the sample items from their mean value, can we estimate the standard error of this mean value? In other words, given our sample, how confident can we be in our estimate of the mean for the population as a whole? It can be shown that the standard error of the mean is equal to the standard deviation of the sample divided by the square root of the number of items in that sample.
For example, if we measure the heights of 100 young adults and find the mean height to be 178 centimetres with a standard deviation of 7 cm, then the standard error of the mean is 7/sqrt(100)=0.7cm. We can be very confident that the average height of young adults generally is within 3 x 0.7 = 2.1cm of 178cm.
How accurate is our calculated standard deviation? It can be shown that the degree of uncertainty in this value, or the standard error of the standard deviation, is given by dividing the standard deviation by the square root of double the number of items. So for our 100 young adults, the standard error of the standard deviation is 7/sqrt(2x100) or near enough 0.5cm. We can therefore be very confident that the true standard deviation lies between 5.5cm and 8.5cm.
How to Calculate Standard Deviation
Unfortunately, the standard deviation formula is not so easy to calculate by hand. Finding the standard deviation is not as easy as finding the average deviation.
UltimaCalc has a window which acts as a standard deviation calculator. To perform a standard deviation calculation, you simply enter all the values in turn. The mean and standard deviation are calculated automatically. Apart from calculating standard deviation, UltimaCalc will calculate a large number of other properties of the data. Look at the standard deviation window help page for more information.