# Frequency Distributions

## Introduction

A frequency distribution is one of the most common graphical tools used to describe a single population. It is a tabulation of the frequencies of each value (or range of values). There are a wide variety of ways to illustrate frequency distributions, including histograms, relative frequency histograms, density histograms, and cumulative frequency distributions. Histograms show the frequency of elements that occur within a certain range of values, while cumulative distributions show the frequency of elements that occur below a certain value.

## Histograms

Frequency Histogram

1. A graphical representation of a single dataset, tallied into classes.
2. Frequency defined as the number of values that fall into each class.
3. Histogram consists of a series of rectangles whose widths are defined by the limits of the classes, and whose heights are determined by the frequency in each interval.
4. Histogram depicts many attributes of the data, including location, spread, and symmetry.
1. No rigid set of rules that determine the number of classes or class interval.
2. Between 5 and 20 classes suitable for most datasets.
3. Equal sized class widths are found by dividing the range by the number of classes.
4. Formal guide by which class intervals can be derived is the formula: K = 1 + 3.3 * log n
where K is the number of classes and n is the number of variables.
Relative Frequency Histogram
1. Relative frequency defined as the fraction of times the value occurs, or the freuqency of value(s) ÷ number of observations in the set.
2. Relative frequencies usually of more interest than the absolute frequencies.
3. Relative frequency histogram constructed by assigning the relative frequencies as heights of the rectangles.
4. Sum of all relative frequencies in a dataset is 1.
Density Histogram
1. Similar to frequency histogram except heights of rectangles are calculated by dividing relative frequency by class width.
2. Resulting rectangle heights called densities, vertical scale called density scale.
3. Noteworthy property: (class width * density) = relative frequency.
4. Total area of all rectangles equals 1.
Histogram Shapes

Example: Construct a frequency, relative frequency, and density histogram of net heat flux data at 130° E, 20° N for January 1960 to March 1998.

## `* * * * * * * * * *` Cumulative Frequency Distributions

Cumulative Frequency Distributions

1. Cumulative frequency distributions contain all information present in histograms, plus the following:
1. Allows user to easily estimate frequencies over several class intervals.
2. Provides better estimates of probablities since there is no arbitrary division of data into classes.
2. Cumulative distribution function is plotted with cumulative probabilites on the vertical axis and data values on the horizontal axis.
3. Cumulative frequencies are obtained by the formula F = m / (n + 1) where m is the mth value in order of magnitude of the series and n is the number of terms in the series.
4. F gives the probability that a randomly chosen value will not exceed the data value by which F was calcuated.
5. Probability of any exact value occuring is zero.
6. Concave downward (upward) cumulative frequency distributions indicative of positively (negatively) skewed data.

Example: Construct a cumulative frequency distribution of net heat flux data at 130° E, 20° N for January 1960 to March 1998.