Frequency distribution

In statistics, a frequency distribution is a list of the values that a variable takes in a sample. It shows the number of times each value appears.

Example of an (absolute) frequency distribution. This is the population pyramid of Angola, for the year 2005.

This is the population pyramid of China for the year 2005.

For example, imagine that 100 people are asked whether they agree with a statement, and have to give an answer on a scale from 1 to 5, where 1 means they strongly agree and 5 means they strongly disagree. The frequency distribution of their responses might look like this:

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

A simple table like this might not work well in some cases. For example, if a variable can take any value within a range (for example, a measurement like height), or if the number of possible values is too large, making a simple table would be difficult or even impossible. In this case, we could instead use ranges of values. For example, the frequency table for the heights of the students in a class might look like the table below.

Height range	Number of students
4.5–5.0 feet	25
5.0–5.5 feet	35
5.5–6 feet	20
6.0–6.5 feet	20

Applications

Working with a frequency distribution of data is often much simpler than working with the raw data. There are simple algorithms to calculate median, mean (statistics), standard deviation and so on from these tables.

Statistical hypothesis testing is based 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.