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 Equation: Cumulative Gaussian distribution

## Introduction

A frequency distribution plots the number of observations as a function of value. A cumulative frequency distribution plots the cumulative number of observations as a function of value. Each Y value is the number of observations in the original data set that have a value less than or equal to the X value.

The advantage of creating a cumulative distribution is that you don't have to make any choice regarding bin width.

If your data follow a Gaussian distribution, the cumulative distribution has a sigmoidal shape.

## Step-by-step

1.Create an XY table, and enter your X and Y values. The X values correspond to the value in the original data set, and the Y values are the number (or fraction or percent) of values in the original data set that are less than or equal to the Y value.

Alternatively, enter a stack of values onto a Column data table, and run the frequency distribution analysis choosing to create a cumulative frequency distribution with no bins.

2.From the cumulative frequency distribution, click Analyze, choose Nonlinear regression and then choose one of the Cumulative Gaussian distribution equations from the "Gaussian" group of equations.

3.If your data are entered as counts (rather than percentages or fractions) constrain N to a constant value equal to the number of observations.

## Models

The details of the model depend on whether the Y values are percentages, fractions or counts.

Here is the model if the data are percentages, so the last Y value equals 100.

Top=100

z=(X-Mean)/SD

Y=Top * zdist(z)

Here is the model if the data are fractions, so the first line of the model defines Top to equal 1.00.

Top=1.0

z=(X-Mean)/SD

Y=Top * zdist(z)

And finally, here is the model if the data are numbers of observations, so the largest value equals the number of observations (N). In this case, you should constrain N to be a constant value equal to the number of observations.

z=(X-Mean)/SD

Y=N * zdist(z)

Mean is the average of the original distribution, from which the frequency distribution was created.

SD is the standard deviation of the original distribution.

Both of these parameters are expressed in the same units as the X values plotted on the graph, which is the same as the Y values in the original distribution from which the frequency distribution was generated.