Variation in a group or population is usually described with reference to one or more basic statistical functions. The mean or average of a group or distribution is the most common statistic used. It is calculated simply by adding all of the observations and dividing by the number of observations. The mean is sometimes referred to as the expected value since it is a measure of central tendenancy and in one single oberservation, represents all of the different observations in the population. All of the observations, when taken together, will be closer to the mean than any other single number.
We know that averages can be deceiving however since the average of the two numbers 0 and 100 is the same as the average of the two numbers 49 and 51. This makes the average or mean by itself a potentially poor measure of the actual expected value of a population even though it is the best one.
This leads us to the second common statistical measure of a population, the variance. The variance, as the name implies, describes in a single number how observations are dispersed about the mean. The variance is kind of an average in itself.
To calculate the variance of a set of numbers, you subtract the mean from all of the obervations in the sample or population and record these "deviations". These deviations describe for each number, how far it is from the mean. Since the outcome of the subtraction will be negative for observations which have a value less than the mean, you square the outcome of the subtraction (this yields a positive number for each deviation from the mean) and add up the squared deviations. One you have a total, you divide by the number of observations and you have calculated the variance, the average of the squared deviations from the mean.
Since you have employed this little trick of squaring each deviation to eliminate negative numbers, you can take the square root of the variance and voila...you have the standard deviation. So all of this statistical stuff doesn't have to be a big mystery. The variance and standard deviation are simply measures of the average difference or deviation of the numbers in your group (or population or sample, etc.) from the mean.
There are some nice properties of the standard deviation which allow you to project the number or percent of observations in your population that will be within plus or minus one standard deviation from the mean. If the population is normally distributed, about 68% of all the values randomly selected from the distribution will fall within plus or minus one standard deviation from the mean. About 95% will fall within plus or minus two standard deviations from the mean.
If someone tells you that few things are normally distributed in biological production, fear not. A guy by the name of Chebyshev figured out that regardless of the distributions shape (of course there are some weird exceptions), you can count on about 75% of the observations being within two standard deviations from the mean and about 50% will be within 1.41 standard deviations from the mean. A nice rule of thumb from a Russian whose first name was Pafnuty.
