Frequency distribution

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In statistics, a frequency distribution is a list of the values that a variable takes in a sample. It is usually a list, ordered by quantity, showing the number of times each value appears. For example, if 100 people rate a five-point Likert scale assessing their agreement with a statement on a scale on which 1 denotes strong agreement and 5 strong disagreement, the frequency distribution of their responses might look like:

Rank	Degree of agreement	Number
1	Strongly agree	25
2	Agree somewhat	35
3	Not sure	20
4	Disagree somewhat	15
5	Strongly disagree	30

This simple tabulation has two drawbacks. When a variable can take continuous values instead of discrete values or when the number of possible values is too large, the table construction is cumbersome, if it is not impossible. A slightly different tabulation scheme based on the range of values is used in such cases. For example, if we consider the heights of the students in a class, the frequency table might look like below.

Height range	Number of students	Cumulative Number
4.5 -5.0 feet	25	25
5.0-5.5 feet	35	60
5.5-6 feet	20	80
6.0-6.5 feet	20	100

Applications

Managing and operating on frequency tabulated data is much simpler than operation on raw data. There are simple algorithms to calculate median, mean, standard deviation etc. from these tables.

Statistical hypothesis testing is founded on the assessment of differences and similarities between frequency distributions. This assessment involves measures of central tendency or averages, such as the mean and median, and measures of variability or statistical dispersion, such as the standard deviation or variance.

A frequency distribution is said to be skewed when its mean and median are different. The kurtosis of a frequency distribution is the concentration of scores at the mean, or how peaked the distribution appears if depicted graphically—for example, in a histogram. If the distribution is more peaked than the normal distribution it is said to be leptokurtic; if less peaked it is said to be platykurtic.

Frequency distributions are also used in frequency analysis to crack codes and refer to the relative frequency of letters in different languages.