F- Test, Z – Test

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An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled. Exact “F-tests” mainly arise when the models have been fitted to the data using least squares. The name was coined by George W. Snedecor, in honour of Sir Ronald A. Fisher. Fisher initially developed the statistic as the variance ratio in the 1920s

Assumptions of F- Test

Several assumptions are made for the test. Your population must be approximately normally distributed (i.e. fit the shape of a bell curve) in order to use the test. Plus, the samples must be independent events. In addition, you’ll want to bear in mind a few important points:-

  • The larger variance should always go in the numerator (the top number) to force the test into a right-tailed test. Right-tailed tests are easier to calculate.
  • For two-tailed tests, divide alpha by 2 before finding the right critical value.
  • If you are given standard deviations, they must be squared to get the variances.
  • If your degrees of freedom aren’t listed in the F Table, use the larger critical value. This helps to avoid the possibility of Type I errors.

Common examples

Common examples of the use of F-tests include the study of the following cases:

  • The hypothesis that the means of a given set of normally distributed populations, all having the same standard deviation, are equal. This is perhaps the best-known F-test, and plays an important role in the analysis of variance (ANOVA).
  • The hypothesis that a proposed regression model fits the data well. See Lack-of-fit sum of squares.
  • The hypothesis that a data set in a regression analysis follows the simpler of two proposed linear models that are nested within each other.

F Test to compare two variances by hand: Steps

Warning: F tests can get really tedious to calculate by hand, especially if you have to calculate the variances. You’re much better off using technology (like Excel — see below).

These are the general steps to follow. Scroll down for a specific example (watch the video underneath the steps).

Step 1: If you are given standard deviations, go to Step 2. If you are given variances to compare, go to Step 3.

Step 2: Square both standard deviations to get the variances. For example, if σ1 = 9.6 and σ2 = 10.9, then the variances (s1 and s2) would be 9.62 = 92.16 and 10.92 = 118.81.

Step 3: Take the largest variance, and divide it by the smallest variance to get the f-value. For example, if your two variances were s1 = 2.5 and s2 = 9.4, divide 9.4 / 2.5 = 3.76.

Why? Placing the largest variance on top will force the F-test into a right tailed test, which is much easier to calculate than a left-tailed test.

Step 4: Find your degrees of freedom. Degrees of freedom is your sample size minus 1. As you have two samples (variance 1 and variance 2), you’ll have two degrees of freedom: one for the numerator and one for the denominator.

Step 5: Look at the f-value you calculated in Step 3 in the f-table. Note that there are several tables, so you’ll need to locate the right table for your alpha level. Unsure how to read an f-table? Read What is an f-table?.

Step 6: Compare your calculated value (Step 3) with the table f-value in Step 5. If the f-table value is smaller than the calculated value, you can reject the null hypothesis.


A Z-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Because of the central limit theorem, many test statistics are approximately normally distributed for large samples. For each significance level, the Z-test has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the Student’s t-test which has separate critical values for each sample size. Therefore, many statistical tests can be conveniently performed as approximate Z-tests if the sample size is large or the population variance is known. If the population variance is unknown (and therefore has to be estimated from the sample itself) and the sample size is not large (n < 30), the Student’s t-test may be more appropriate.

A one-sample location test, two-sample location test, paired difference test and maximum likelihood estimate are examples of tests that can be conducted as z-tests. Z-tests are closely related to t-tests, but t-tests are best performed when an experiment has a small sample size. Also, t-tests assume the standard deviation is unknown, while z-tests assume it is known. If the standard deviation of the population is unknown, the assumption of the sample variance equaling the population variance is made.

One-Sample Z-Test Example

For example, assume an investor wishes to test whether the average daily return of a stock is greater than 1%. A simple random sample of 50 returns is calculated and has an average of 2%. Assume the standard deviation of the returns is 2.50%. Therefore, the null hypothesis is when the average, or mean, is equal to 3%. Conversely, the alternative hypothesis is whether the mean return is greater than 3%. Assume an alpha of 0.05% is selected with a two-tailed test. Consequently, there is 0.025% of the samples in each tail, and the alpha has a critical value of 1.96 or -1.96. If the value of z is greater than 1.96 or less than -1.96, the null hypothesis is rejected.

The value for z is calculated by subtracting the value of the average daily return selected for the test, or 1% in this case, from the observed average of the samples. Next, divide the resulting value by the standard deviation divided by the square root of the number of observed values. Therefore, the test statistic is calculated to be 2.83, or (0.02 – 0.01) / (0.025 / (50)^(1/2)). The investor rejects the null hypothesis since z is greater than 1.96, and concludes that the average daily return is greater than 1%.

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