I have written about Measures of Variance – Standard Deviation in my previous post. This is good. We came a long way from Measures of Central Tendency to variance. But this measure do not indicate us, whether the distribution is symmetric or not. We have seen Frequency Distribution that differ widely in their nature and composition. Following are the two popular measures those are used to indicate the shape of a distribution of an interval or ratio variable.
- Skewness
- Kurtosis
Shape of the distribution is the shape of data when we plot the graph.
What is a symmetrical distribution?
when the mean, median and mode are identical, we call the distribution as symmetrical.
When it is not symmetrical, we call it as asymmetrical or skewed. Following are some types of shape.
UniModal:
The distribution has a single value that occurs most frequently.
Symmetrical:
the left side distribution of values mirrors the right side.
Bell-Shaped
The frequencies of cases decline towards the extreme values in the right and left tails. the resultant graph will be in the shape of a bell.
Normal Distribution
If all the values in a data set, are equally distributed, the shape would be symmetrical. For this type of data set, mean, median and model would be equal. 50% of the cases will lie above or below the mid point of the mean.
Skew
Skewness measures the symmetry of a distribution. Calculating the skewness will indicate the position of lower and higher values in a data set, that will pull the shape the distribution towards lower or higher end.
When the data has lot of low values, the shape will be +vely skewed.
If it has more higher numbers, it would be -vely skewed.
Kurtosis
Distributions of data may not have same shape always. Some are asymmetric, skewed to the left or right. Otheer distributions are bimodal, multimodal as shown above. I have explain these above.
Another measure to consider is, the shape of the tails of the distribution, on the far left and right. Kurtosis is the measure of the thickness or heaviness of the tails of a distribution.
Three categories of kurtosis are –
- Mesokurtic – typically measures with respect to normal distribution. The tails are similar to ND. The kurtosis of a mesokurtic distribution is neither high or low, rather it is considered as a base line for the two other classifications.
- Leptokurtic – Kurtosis is greater than mesokurtic. Peaks are thin and tall. Tails are thick ad heavy.
- Platykurtic – Kurtosis is lesser than meso kurtic. This has slender tails. This possess low peak.
See you in another interesting post.
Corrections on “kurtosis”:
1. Your graphs do not show differences in kurtosis – they only show differences in variance.
2. Kurtosis is unrelated to the height of the distribution. It measures tails (outlying values or potential outlying values) only.
Here are the correct interpretations.
Three categories of kurtosis are –
Mesokurtic – has kurtosis, or outlier character, similar to the normal distribution. The kurtosis of a mesokurtic distribution is neither high or low, rather it is considered as a base line for the two other classifications.
Leptokurtic – Kurtosis is greater than mesokurtic. Tails are heavy, meaning that the outlier character is more extreme than that of the normal distribution.
Platykurtic – Kurtosis is lesser than mesokurtic. Tails are light, meaning that the outlier character is less extreme than that of the normal distribution.
Peter, If you consider the area under all of these normal curves to be 1, then if a distribution has heavier tails, doesn’t that infer that the peak has to be lower? So in that way, kurtosis does affect the height of a distribution.