In statistics, rankits of a set of data are the expected values of the

order statistic In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Importa ...

s of a sample from the standard

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...

the same size as the data. They are primarily used in the

normal probability plot The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...

, a

graphical technique Statistical graphics, also known as statistical graphical techniques, are graphics used in the field of statistics for data visualization. Overview Whereas statistics and data analysis procedures generally yield their output in numeric or tab ...

for

normality test In statistics, normality tests are used to determine if a data set is well-modeled by a normal distribution and to compute how likely it is for a random variable underlying the data set to be normally distributed. More precisely, the tests are a f ...

ing.

Example

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 In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is ...

population with

expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

0 and

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...

1 (the

standard normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...

) and then sorted into increasing order, the expected values of the resulting

s 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: These points are then plotted as the vertical and horizontal coordinates of a

scatter plot A scatter plot (also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram) is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data. ...

Alternative method

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:

Rankit plot

{{main, Normal probability plot A graph plotting the rankits on the horizontal axis and the data points on the vertical axis is called a rankit plot or a

. 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 In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

. A rankit plot is a kind of

Q–Q plot In statistics, a Q–Q plot (quantile-quantile plot) is a probability plot, a graphical method for comparing two probability distributions by plotting their '' quantiles'' against each other. A point on the plot corresponds to one of the q ...

– 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

The rankit plot and the word ''rankit'' was introduced by the biologist and statistician

Chester Ittner Bliss Chester Ittner Bliss (February 1, 1899 – March 14, 1979) was primarily a biologist, who is best known for his contributions to statistics. He was born in Springfield, Ohio in 1899 and died in 1979. He was the first secretary of the Internationa ...

(1899–1979).

External links

Engineering Statistics Handbook
Statistical charts and diagrams Normal distribution

Example

Alternative method

Rankit plot

History

See also

External links