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Frequency distribution
 
 
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In statistics
Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
, a frequency
Frequency (statistics)

In statistics the frequency of an Event i is the number ni of times the event occurred in the experiment or the study. These frequencies are often graphically represented in histograms....
 distribution is a tabulation of the values that one or more variables take in a sample
Sampling (statistics)

Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference....
.

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
Likert scale

A Likert scale is a psychometrics scale commonly used in questionnaires, and is the most widely used scale in survey research. When responding to a Likert questionnaire item, respondents specify their level of agreement to a statement....
 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:

This simple tabulation has two drawbacks.






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Encyclopedia


In statistics
Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
, a frequency
Frequency (statistics)

In statistics the frequency of an Event i is the number ni of times the event occurred in the experiment or the study. These frequencies are often graphically represented in histograms....
 distribution is a tabulation of the values that one or more variables take in a sample
Sampling (statistics)

Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference....
.

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
Likert scale

A Likert scale is a psychometrics scale commonly used in questionnaires, and is the most widely used scale in survey research. When responding to a Likert questionnaire item, respondents specify their level of agreement to a statement....
 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:

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.

Joint frequency distributions


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

 Two-way 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.

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
Statistical hypothesis testing

A statistical hypothesis test is a method of making statistical decisions using experimental data. It is sometimes called confirmatory data analysis, in contrast to exploratory data analysis....
 is founded on the assessment of differences and similarities between frequency distributions. This assessment involves measures of central tendency or average
Average

In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "Expected value" value of the data set....
s, such as the mean
Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
 and median
Median

In probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half....
, and measures of variability or statistical dispersion
Statistical dispersion

In statistics, statistical dispersion is variability or spread in a variable or a probability distribution. Common examples of measures of statistical dispersion are the variance, standard deviation and interquartile range....
, such as the standard deviation
Standard deviation

In statistics, standard deviation is a simple measure of the variability or statistical dispersion of a data set. A low standard deviation indicates that all of the data points are very close to the same value , while high standard deviation indicates that the data are ?spread out? over a large range of values....
 or variance
Variance

In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value ....
.

A frequency distribution is said to be skewed
Skewness

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real number-valued random variable....
 when its mean and median are different. The kurtosis
Kurtosis

In probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real number-valued random variable....
 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
Histogram

In statistics, a histogram is a graphical display of tabulated frequency , shown as bars. It shows what proportion of cases fall into each of several Categorization....
. If the distribution is more peaked than the normal distribution
Normal distribution

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields....
 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.

See also

  • Cross tabulation
    Cross tabulation

    A cross tabulation displays the joint distribution of two or more variables. They are usually presented as a contingency table in a matrix format....
  • Cumulative frequency
    Cumulative frequency

    Cumulative frequency is the frequency of a random variable below a particular level. It tells how often the value of the random variable is less than or equal to a particular reference value....