What is the difference between the formulas for the standard deviation of a population and the standard deviation for a sample?
Standard deviation of a population implies that the "mean" is a calculable value; standard deviation of a sample implies that the "mean" is an estimate (based on the average of the sample values).
The standard deviation (for both population and sample statistics) is based on the corresponding variance.
If all values of a population are known, then the average of these values is the true mean value.
The population variance can be calculated as the average of the squares of the differences between individual values and the true mean.
If only a sample of the entire population is known. We can not calculate the true mean and must settle for the sample mean. This means that besides the difference between the sample values and the true mean we have another source of deviation, the error between our estimated "sample mean** and the true mean.
As a result, in calculating the sample variance our deviation must be increased. This is done by dividing the sum of the squares of the individual deviations from the sample mean by one less than the number of sample values.