# Statistical Significance Calculator

#### Measuring the Statistical Significance of Employment Decisions

While proof of discriminatory motive is central to disparate treatment cases, the U.S. Supreme Court in __Teamsters v. U.S.__ stated that, "[p]roof of discriminatory motive is critical, although it can in some situations be inferred from the mere fact of differences." __Teamsters v. United States__ 431 U.S. 324, 14 FEP Cases 1514 at 1519 (1977). In cases concerning the disparate treatment of protected employees, hiring, promotion, or termination rates for the protected class are compared to the rates for other employees to determine if the difference is statistically significant.

The level of statistical significance is the same as the probability that the event would occur by chance in a nondiscriminatory setting. A disparity is considered statistically significant if it would occur so rarely in a nondiscriminatory situation that we can rule out that it occurred by chance. Following the lead of social scientists, the courts typically require a demonstration of statistical significance at the 0.05 or 5 percent level, to permit an inference of discrimination. This is equivalent to requiring that the disparity is large enough that it would occur by chance less than 5 percent of the time, or in less than one in 20 nondiscriminatory events.

The chi-square test is one of the most commonly used and widely accepted statistical tests of bias in employment decisions. To assess the evidence of bias, employees are classified by two criteria, whether they are a member of the protected class or not, and whether or not they were subject to the employment decision in question (termination, promotion, hiring, etc.). The chi-square test compares the observed number of affected employees with the expected number, and provides a measure of the likelihood that the employment decisions were independent of membership in the protected class.

For example, if the proportion of employees 40 and over is 50 percent, we would expect the proportion of terminated employees who are 40 and over to be about 50 percent, give or take random variation in the termination process. If the proportion of older terminations is high enough, this provides evidence that promotions are not independent of age. The chi-square test gives us a handle on what proportion is high enough to conclude that promotions are age related.

The chi-square significance calculator provided here (without express or implied warranty for accuracy or applicability) uses a one-tailed Fisher exact test. This produces the most accurate result, and is reliable even with small sample sizes. It is our opinion that a one-tailed test is appropriate, since only rates which are disadvantageous for the protected class are relevant. In the table below, enter the number of protected class employees in the **Protected** row, with affected (e.g., terminated, promoted, hired) employees in the **Affected** column, and the number not affected in the **Not Affected** column. Do the same of the non-protected employees, by entering their information in the **Non-Protected** row.

### Chi Square Test

**In the table below, fill in the values for:**

For example, for terminations of employees 40 and over, the cells should be:

Protected & Affected | = | Older employees terminated | Protected & Not Affected | = | Older employees retained | |

Non-Protected & Affected | = | Younger employees terminated | Non-Protected & Not Affected | = | Younger employees retained. |

In this example, the resulting significance indicates the probability that there would be as many older employees terminated, if terminations were assigned independent age. The commonly accepted threshold for prima facie evidence is a significance level of 0.05. The 0.05 significance level is equivalent to requiring that the disparity would occur in no more than 1 in 20 age-neutral terminations.

In general, the significance is the probabilty that there would be as large a disparity, if employment decisions were independent of membership in the protected class. For terminations, large disparities occur in tables with more **Protected** employees **Affected**; for promotions or hiring, large disparities occur in tables with fewer **Protected** employees **Affected**.

Thanks to Tom Kirkman for the original version of the Fisher Exact test program.