Rankit
In statistics, rankits of a set of data are the expected values of the order statistics of a sample from the standard normal distribution the same size as the data. They are primarily used in the normal probability plot, a graphical technique for normality testing.
Example
[edit]This is perhaps most readily understood by means of an example. If an i.i.d. sample of six items is taken from a normally distributed population with expected value 0 and variance 1 (the standard normal distribution) and then sorted into increasing order, the expected values of the resulting order statistics are:
- −1.2672, −0.6418, −0.2016, 0.2016, 0.6418, 1.2672.
Suppose the numbers in a data set are
- 65, 75, 16, 22, 43, 40.
Then one may sort these and line them up with the corresponding rankits; in order they are
- 16, 22, 40, 43, 65, 75,
which yields the points:
| data point | rankit |
|---|---|
| 16 | −1.2672 |
| 22 | −0.6418 |
| 40 | −0.2016 |
| 43 | 0.2016 |
| 65 | 0.6418 |
| 75 | 1.2672 |
These points are then plotted as the vertical and horizontal coordinates of a scatter plot.
Alternative method
[edit]Alternatively, rather than sort the data points, one may rank them, and rearrange the rankits accordingly. This yields the same pairs of numbers, but in a different order.
For:
- 65, 75, 16, 22, 43, 40,
the corresponding ranks are:
- 5, 6, 1, 2, 4, 3,
i.e., the number appearing first is the 5th-smallest, the number appearing second is 6th-smallest, the number appearing third is smallest, the number appearing fourth is 2nd-smallest, etc. One rearranges the expected normal order statistics accordingly, getting the rankits of this data set:
| data point | rank | rankit |
|---|---|---|
| 65 | 5 | 0.6418 |
| 75 | 6 | 1.2672 |
| 16 | 1 | −1.2672 |
| 22 | 2 | −0.6418 |
| 43 | 4 | 0.2016 |
| 40 | 3 | −0.2016 |
Rankit plot
[edit]A graph plotting the rankits on the horizontal axis and the data points on the vertical axis is called a rankit plot or a normal probability plot. Such a plot is necessarily nondecreasing. In large samples from a normally distributed population, such a plot will approximate a straight line. Substantial deviations from straightness are considered evidence against normality of the distribution.
Rankit plots are usually used to visually demonstrate whether data are from a specified probability distribution.
A rankit plot is a kind of Q–Q plot – it plots the order statistics (quantiles) of the sample against certain quantiles (the rankits) of the assumed normal distribution. Q–Q plots may use other quantiles for the normal distribution, however.
History
[edit]The rankit plot and the word rankit was introduced by the biologist and statistician Chester Ittner Bliss (1899–1979).
See also
[edit]- Probit analysis developed by C. I. Bliss in 1934.