NORMSDST and NORMSINV: Calculating Cumulative Normal Distribution

The NORMSDST and NORMSINV functions perform calculations on a standard normal distribution curve:

The results of NORMSDST and NORMSINV are returned as double-precision and are accurate to 6 significant digits.

A standard normal distribution curve is a normal distribution that has a mean of 0 and a standard deviation of 1. The total area under this curve is 1. A point on the X-axis of the standard normal distribution is called a normalized value. Assuming that your data is normally distributed, you can convert a data point to a normalized value to find the percentage of scores that are less than or equal to the raw score.

You can convert a value (raw score) from your normally distributed data to the equivalent normalized value (z-score) as follows:

The following diagram illustrates the results of the NORMSDST and NORMSINV functions:

Reference: Characteristics of the Normal Distribution

Many common measurements are normally distributed. A plot of normally distributed data values approximates a bell-shaped curve. The two measures required to describe any normal distribution are the mean and the standard deviation:

The mean is the point at the center of the curve.
The standard deviation describes the spread of the curve. It is the distance from the mean to the point of inflection (where the curve changes direction).

Syntax: How to Calculate the Cumulative Standard Normal Distribution Function

NORMSDST(value, 'D8');

where:

value: Is a normalized value.
D8: Is the required format for the result. The value returned by the function is double-precision. You can assign it to a field with any valid numeric format.

Syntax: How to Calculate the Inverse Cumulative Standard Normal Distribution Function

NORMSINV(value, 'D8');

where:

value: Is a number between 0 and 1 which represents the a percentile in a standard normal distribution.
D8: Is the required format for the result. The value returned by the function is double-precision. You can assign it to a field with any valid numeric format.

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Example: Using the NORMSDST and NORMSINV Functions

NORMSDST finds the percentile for the Z field. NORMSINV then returns this percentile to a normalized value.

DEFINE FILE GGPRODS
-* CONVERT SIZE FIELD TO DOUBLE PRECISION
X/D12.5 = SIZE;
END

TABLE FILE GGPRODS
SUM X NOPRINT CNT.X NOPRINT
-* CALCULATE MEAN AND STANDARD DEVIATION
COMPUTE NUM/D12.5 = CNT.X; NOPRINT
COMPUTE MEAN/D12.5 = AVE.X; NOPRINT
COMPUTE VARIANCE/D12.5 = ((NUM*ASQ.X) - (X*X/NUM))/(NUM-1); NOPRINT
COMPUTE STDEV/D12.5 = SQRT(VARIANCE); NOPRINT

PRINT SIZE X NOPRINT
-* COMPUTE NORMALIZED VALUES AND USE AS INPUT TO NORMSDST FUNCTION
-* THEN USE RETURNED VALUES AS INPUT TO NORMSINV FUNCTION
-* AND CONVERT BACK TO DATA VALUES
COMPUTE Z/D12.5 = (X - MEAN)/STDEV;
COMPUTE NORMSD/D12.5 = NORMSDST(Z, 'D8');
COMPUTE NORMSI/D12.5 = NORMSINV(NORMSD, 'D8');
COMPUTE DSIZE/D12 = NORMSI * STDEV + MEAN;
BY PRODUCT_ID NOPRINT
END

The output is:

Size

NORMSD

NORMSI

DSIZE