Reporting a centre only provides partial information about a dataset. Different samples or populations may have identical measures of centre, and yet differ from one another in other significant ways. It is highly recommended to compute measures of variability in addition to measures of location, to determine whether the data values are tightly clustered or spread out. Our primary measures of variability involves deviations from the mean. For example, a division about the mean is obtained by this expression, which is given as xi minus x-bar, where xi represents our observations in our dataset. That is, we want to know how far each observation is from the mean, and we do this by subtracting the mean from each observation. For population, it is obtained by the expression xi minus µ, because µ denotes the population mean. A deviation will be positive if the observation is larger than the mean and negative if the observation is smaller than the mean. If all the divisions are small in magnitude, then all the observations are close to the mean and there is little variability. On the other hand, if some of the deviations are large in magnitude, then some of the observations lie far from the mean, which suggests a greater amount of variability. The average of deviations is always zero as the sum of the deviations is always zero. Mathematically, this is given by the following equation. We consider instead the squared deviations which is given by this expression. The best-known estimates for variability are variance and standard deviation, which are based on squared deviations. The variance is an average of the squared deviations while the sample variance is denoted by s squared and is given by this equation. The population variance is denoted by sigma squared. For a population of size N, it is given by the sum of the squared deviations over all observations, divided by the population size N. Mathematically, we can express this using the following equation. The standard deviation is the square root of the variance. The sample standard deviation is denoted by s and the population standard deviation is denoted by the Greek letter sigma. Thus, we have this expression. In most statistical applications, the data being analyzed are for a sample. When we compute the sample variance, we are interested in using it to estimate the population variance. In fact, the sample variance is said to be an unbiased estimate of the population variance. Now we'll consider a worked example.