NORMSDST and NORMSINV: Calculating Normal Distributions

NORMSDST: Calculating Standard Cumulative Normal Distribution

How to:

Calculate the Cumulative Standard Normal Distribution Function

Reference:

Characteristics of the Normal Distribution

The NORMSDST function performs calculations on a standard normal distribution curve, calculating the percentage of data values that are less than or equal to a normalized value. A normalized value is a point on the X-axis of a standard normal distribution curve in standard deviations from the mean. This is useful for determining percentiles in normally distributed data.

The NORMSINV function is the inverse of NORMSDST. For information about NORMSINV, see NORMSINV: Calculating Inverse Cumulative Normal Distribution.

The results of NORMSDST 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:

z = (raw_score - mean)/standard_deviation

To convert from a z-score back to a raw score, use the following formula:

raw_score = z * standard_deviation + mean

The mean of data points xi, where i is from 1 to n is:

The standard deviation of data points xi, where i is from 1 to n is:

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.

Example: Using the NORMSDST Function

NORMSDST calculates the Z value and finds its percentile:

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
COMPUTE Z/D12.5 = (X - MEAN)/STDEV;
COMPUTE NORMSD/D12.5 = NORMSDST(Z, 'D8');
BY PRODUCT_ID NOPRINT
END

The output is:

Size              Z         NORMSD
----              -         ------
  16        -.07298         .47091
  12        -.80273         .21106
  12        -.80273         .21106
  20         .65678         .74434
  24        1.38654         .91721
  20         .65678         .74434
  24        1.38654         .91721
  16        -.07298         .47091
  12        -.80273         .21106
   8       -1.53249         .06270

NORMSINV: Calculating Inverse Cumulative Normal Distribution

How to:

Calculate the Inverse Cumulative Standard Normal Distribution Function

The NORMSINV function performs calculations on a standard normal distribution curve, finding the normalized value that forms the upper boundary of a percentile in a standard normal distribution curve. This is the inverse of NORMSDST. For information about NORMSDST, see NORMSDST: Calculating Standard Cumulative Normal Distribution.

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

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 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.

Example: Using the NORMSINV Function

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 shows that NORMSINV is the inverse of NORMSDST and returns the original values:

Size              Z         NORMSD         NORMSI            DSIZE
----              -         ------         ------            -----
  16        -.07298         .47091        -.07298               16
  12        -.80273         .21106        -.80273               12
  12        -.80273         .21106        -.80273               12
  20         .65678         .74434         .65678               20
  24        1.38654         .91721        1.38654               24
  20         .65678         .74434         .65678               20
  24        1.38654         .91721        1.38654               24
  16        -.07298         .47091        -.07298               16
  12        -.80273         .21106        -.80273               12
   8       -1.53249         .06270       -1.53249                8