Link to Data Set: www.stat.berkeley.edu/~statlab...
Here I show you how to understand numbers for skewness and kurtosis, with some example data and histograms on newborns. With Excel formulas for skewness and kurtosis. Roughly speaking, Skewness measures whether data stretch out farther in one tail than another, and Kurtosis measures whether the data has heavy tails (higher probability of outliers), or whether data is more concentrated in the center.
Definition: Deviation = xi-mean
As for intuition WHY it makes sense that the 3rd power indicates which side outliers might be on: An odd power preserves the sign. Numbers below the mean will have a negative deviation. LARGE negative deviations will become VERY LARGE negative numbers when cubed. If there are a preponderance of numbers farther away below the mean than above, you'll get a negative skewness measure. The opposite if they are on the positive side. If symmetric, they will effectively cancel each other out.
As for intuition why it makes sense that the 4th power indicates the presence of many "outliers" on both sides, or the lack of them (here by "outliers" I just mean numbers pretty far away from average, think more than one standard deviation)... By "many", we mean compared to a normal distribution. Picture if you will, two distributions. Distribution A is a perfect semicircle lying with its flat side down (See blue curve here: upload.wikimedia.org/wikipedi...) Distribution A will have negative kurtosis. Distribution B is shaped something like the Eiffel Tower, but with skinny tails (yet fatter farther out than a normal would have) stretching off on both sides (see Red image). It will have positive Kurtosis.
When raising things to the 4th power, numbers more than 1 will get pretty big, numbers less than 1 will get pretty small. If all of the numbers in a distribution are within one standard deviation, kurtosis will be a very small number. The more numbers with z scores of 2, 3, and 4 that there are, the more 2^4 and 4^4 we have to add in. This makes a very large kurtosis number. At the end of calculating kurtosis we normally subtract 3 since that is what a normal distribution's kurtosis is. So, negative sort of indicates "more closely packed" than a normal, negative means more with z scores farther away.
23 июл 2024
