Describe data


A type of average, calculated by dividing the sum of a set of values by the number of values.

mean = Sum(Xi)/N


A type of average, found by arranging the values in order and then selecting the one in the middle. If the total number of values in the sample is even, then the median is the mean of the two middle numbers.


The most frequent value in a group of values.


The average of the square of the distance of each data point from the mean

variance = Sum((Xi-mean)^2))/N

Standard Deviation

The standard deviation is the square root of the variance.

sd = sqrt(variance)


Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.

skewness = (Sum((Xi-mean)^3))/((N-1)*SD^3)

Skewness of symmetric data is zero


Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case.

kurtosis = (Sum((Xi-mean)^4))/((N-1)*SD^4) -3

The standard normal distribution has a kurtosis of zero. Positive kurtosis indicates a "peaked" distribution and negative kurtosis indicates a "flat" distribution.