# ZSTAT( ) function

Returns the standard Z-statistic.

## Syntax

`ZSTAT(actual, expected, population)`

## Parameters

actual

Numeric. When specifying parameters as numbers, the actual count, such as a leading digit or a leading digit combination. When specifying parameters as proportions, the expected proportion of the value being test and must be between 0 and 1 inclusive (i.e., greater than or equal to 0 and less than or equal to 1).

expected

Numeric. When specifying parameters as numbers, the expected count, such as a leading digit or a leading digit combination. When specifying parameters as proportions, the expected proportion of the value being test and must be between 0 and 1 exclusive (i.e., greater than 0 and less than 1).

population

Numeric.The total number of items being tested. This parameter must be a positive whole number greater than 0.

Numeric.

## Remarks

This function calculates the standard Z-statistic for use in many problem-solving tasks, including digital analysis. It outputs the result with a precision of three decimal places.

You can use this function to evaluate the likely frequency of occurrence of a given result in a specified period or category. The larger the resulting Z-statistic, the more unlikely the occurrence. For example, a Z-statistic of 1.96 has a significance of 0.05, representing the likelihood of a one time in 20 occurrence, whereas a Z-statistic of 2.57 has a significance of 0.01, representing the likelihood of a one time in 100 occurrence. For information on the Z-statistic, consult a statistics textbook.

You can specify the parameters for ZSTAT( ) as either numbers or proportions:

• When you specify both parameters as numbers, the function computes the Z-statistic using floating-point arithmetic.

• When you specify both parameters as proportions, the function computes the Z-statistic using fixed-point arithmetic, and you need to use a decimal multiplier to control rounding.

When using an expression within an expression to calculate the actual or expected parameter value, you must specify the level of precision you want in the result by using a decimal multiplier. ACL has a precision of 8 digits, therefore a multiplier of 1.00000000 will return the greatest precision attainable.

## Examples

a. Parameters expressed as numbers

Based on 10 years of previous data, you know that the distribution of worker disability claims per month is normally highly uniform. In April, May, and June of this year, claims were higher by about 10 percent, averaging 220 per month instead of 200. Claims in July and August were slightly low, at 193 and 197. The total claims for the year were 2,450. To test whether these high and low results were significant, use the Z-statistic.

The actual number of claims for April to June is higher than expected at 660. The expected number of claims for this period is 25 percent of the 2,450 annual claims, or 612.5. The Z-statistic for these counts is calculated as 2.193:

ZSTAT(660, 612.5, 2450)

A Z-statistic of 1.96 has a significance of 0.05, and 2.57 has a significance of 0.01. Thus, the probability that the higher rates of claims are due to chance is between 1:20 and 1:100.

The actual number of claims for July and August is lower than expected at 390. The expected number of claims for this period is one sixth of the 2,450 annual claims, or 408.33. The Z-statistic for these proportions is calculated as 0.967:

ZSTAT(390, 408.33, 2450)

This is not a very significant result. Z-statistics of 1.000 and less are very common and can typically be ignored.

b. Parameters expressed as proportions

Based on 10 years of previous data, you know that the distribution of worker disability claims per month is normally highly uniform. In April, May, and June of this year, claims were higher by about 10 percent, averaging 220 per month instead of 200. Claims in July and August were slightly low, at 193 and 197. The total claims for the year were 2,450. To test whether these high and low results were significant, use the Z-statistic.

The actual number of claims for April to June is represented by the proportion 660/2450, which is higher than expected. The expected number of claims for this period should be 25 percent of the 2,450 annual claims. The Z-statistic for these proportions is 2.193:

ZSTAT((1.00000000 * 660 / 2450), 0.25, 2450)

A Z-statistic of 1.96 has a significance of 0.05, and 2.57 has a significance of 0.01. Thus, the probability that the higher rates of claims are due to chance is between 1:20 and 1:100.

The actual number of claims for July and August is low at 390. The expected number of claims for this period should be one sixth, or 16.6667 percent of the 2,450 annual claims. The Z-statistic for these proportions is 0.967:

ZSTAT((1.00000000 * 390 / 2450), 0.16667, 2450)

This is not a very significant result. Z-statistics of 1.000 and less are very common and can typically be ignored.