Frequency distribution

In statistics, a frequency distribution is an arrangement of the values that one or more variables take in a sample. Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval, and in this way, the table summarizes the distribution of values in the sample.

Univariate frequency tables

Univariate frequency distributions are often presented as lists 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	20
2	Agree somewhat	30
3	Not sure	20
4	Disagree somewhat	15
5	Strongly disagree	15

A different tabulation scheme aggregates values into bins such that each bin encompasses a range of values. For example, the heights of the students in a class could be organized into the following frequency table.

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

A frequency distribution shows us a summarized grouping of data divided into mutually exclusive classes and the number of occurrences in a class. It is a way of showing unorganized data e.g. to show results of an election, income of people for a certain region, sales of a product within a certain period, student loan amounts of graduates, etc. Some of the graphs that can be used with frequency distributions are histograms, line graphs, bar charts and pie charts. Frequency distributions are used for both qualitative and quantitative data.

Joint frequency distributions

Bivariate joint frequency distributions are often presented as (two-way) contingency tables:

Two-way contingency table with marginal frequencies

	Dance	Sports	TV	Total
Men	2	10	8	20
Women	16	6	8	30
Total	18	16	16	50

The total row and total column report the marginal frequencies or marginal distribution, while the body of the table reports the joint frequencies.^[1]

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.

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

Notes

1. ^ Stat Trek, Statistics and Probability Glossary, s.v. Joint frequency

See also

Statistics portal

• Cross tabulation
• Cumulative frequency