Variance and standard deviation

Variance and standard deviation

 

Measures of central tendency (mean, median, and mode) provide information on the data values at the center of the data set. Measures of dispersion (quartiles, percentiles, ranges) provide information on the spread of the data around the center. In this section, we will look at two more measures of dispersion called the variance and the standard deviation.

               

Variance:

The variance of the data is the average squared distance between the mean and each data value.

Variance tells us that how our entire dataset is going to vary across the mean.

If the variance of our dataset is high which means that our data is too far from its average.

If the variance is low which means that our dataset is concentrated towards its mean.

The variance has the following properties.

·                  It is never negative since every term in the variance sum is squared and therefore either positive or zero.

·                  It has squared units. For example, the variance of a set of heights measured in centimeters will be given in centimeters squared. Since the population variance is squared, it is not directly comparable with the mean or the data themselves. In the next section, we will describe a different measure of dispersion, the standard deviation, which has the same units as the data.

Standard Deviation:

the variance is a squared quantity, it cannot be directly compared to the data values or the mean value of a data set. It is therefore more useful to have a quantity that is the square root of the variance. This quantity is known as the standard deviation.

In statistics, the standard deviation is a very common measure of dispersion. Standard deviation measures how to spread out the values in a data set are around the mean. More precisely, it is a measure of the average distance between the values of the data in the set and the mean. If the data values are all similar, then the standard deviation will be low (closer to zero). If the data values are highly variable, then the standard variation is high (further from zero).

The standard deviation is always a positive number and is always measured in the same units as the original data. For example, if the data are distance measurements in kilograms, the standard deviation will also be measured in kilograms.

The mean and the standard deviation of a set of data are usually reported together. In a certain sense, the standard deviation is a natural measure of dispersion if the center of the data is taken as the mean.