Question

The Value of Variance

The Value of Variance, Subscribe More often than not, when we are presented with statistics we are given only a measure of central tendency (such as a mean). However, lots of useful information can be gleaned about a dataset if we examine the variance, skew, and the kurtosis of the data as well. Choose a statistic that recently came across your desk where you were just given a mean. If you can't think of one, come up with an example you might encounter in your life. How would knowing the variance, the skew, and/or the kurtosis of the data give you a better idea of the data? What could you do with that information?

Answer 1

Mean and variance is the two most commonly used method of data summarization. but if the data value is not normally distributed and it is more complex then other methods are needed to understand the data. So-

When you run the descriptive statistics, you get a lot of numbers – the sample size, average, standard deviation, range, maximum, minimum and a host of other numbers. You spy two numbers: the skewness and kurtosis. What do these two statistics tell you about your sample?

skewness and kurtosis are two commonly listed values when you run a software’s descriptive statistics function. Many books say that these two statistics give you insights into the shape of the distribution.

Skewness is a measure of the symmetry in distribution. The asymmetrical dataset will have a skewness equal to 0. So, a normal distribution will have a skewness of 0.   Skewness essentially measures the relative size of the two tails.

Kurtosis is a measure of the combined sizes of the two tails. It measures the amount of probability in the tails. The value is often compared to the kurtosis of the normal distribution, which is equal to 3. If the kurtosis is greater than 3, then the dataset has heavier tails than a normal distribution (more in the tails). If the kurtosis is less than 3, then the dataset has lighter tails than a normal distribution (less in the tails). Careful here. Kurtosis is sometimes reported as “excess kurtosis.” Excess kurtosis is determined by subtracting 3 forms the kurtosis. This makes the normal distribution kurtosis equal to 0. Kurtosis originally was thought to measure the peakedness of a distribution.   Though you will still see this as part of the definition in many places, this is a misconception

The mean and variance are raw moments, and the skewness and kurtosis are normalized/standardized moments (normalized with standard deviation). Unlike mean and variance, skewness and kurtosis are unit-free/dimensionless moments. For example, if our data is in inches (height), then mean and variance will be in inches, but skewness and kurtosis will be unit-free (not in inches). Therefore, skewness and kurtosis will not be affected by any linear change in the scale of the data (inches to the centimeter).