# Level 1 CFA® Exam:

Tests on Correlation

For normally distributed variables, we can use the sample correlation coefficient and t-test with n-2 degrees of freedom to test whether the two variables are linearly correlated.

We use a two-tailed test with:

- the null hypothesis stating that the correlation coefficient in the population is equal to 0, and
- the alternative hypothesis stating that the correlation coefficient is different than 0.

\(t_{n-2} = \frac{r\times\sqrt{n-2}}{\sqrt{1-r^2}}\)

- \(t_{n-2}\) - t-statistic with "n-2" degrees of freedom
- \(r\) - sample correlation
- \(n\) - number of observations

For normally distributed variables, we can use the sample correlation coefficient and t-test with n-2 degrees of freedom to test whether the two variables are linearly correlated.

We use a two-tailed test with: the null hypothesis stating that the correlation coefficient in the population is equal to 0 and the alternative hypothesis stating that the correlation coefficient is different than 0.

We reject the null hypothesis if the test statistic is either larger or smaller than the critical level we can find it the t-table, e.g. if \(t_c=2.678\), then we reject the null if the test statistic is larger than 2.678 or smaller than -2.678.

One of the examples of nonparametric test is a test based on the Spearman rank correlation coefficient.

It is one of the most frequently used tests to examine the correlation between two variables. We often employ it when we cannot use t-test because random variables don’t meet assumptions about the distribution.

To calculate the Spearman rank correlation coefficient, first we have to give number to each observation from a sample. 1 is for the largest observation, 2 is for the second largest one, 3 is for the third largest one, and so on. The process of giving each observation a number is carried out for both random variables.

We calculate the Spearman rank correlation coefficient in the following way:

\(r_{s} = 1-\frac{6\times \Sigma^{n}_{i=1}d^{2}_{i}}{n\times (n^{2}-1)}\)

- \(r_{s}\) - Spearman rank correlation coefficient
- \(d_{i}\) - difference between the ranks of each pair of observations on X and Y
- \(n\) - number of observations

A test based on the Spearman rank correlation coefficient is __an example of a nonparametric test__.

It is one of the most frequently used tests to examine the correlation between two variables. We often employ it when we cannot use a t-test because random variables don't meet assumptions about the distribution.

To calculate the Spearman rank correlation coefficient, **1)** first we have to give a number to each observation from a sample. 1 is for the largest observation, 2 is for the second largest one, 3 is for the third largest one, and so on. The process of giving each observation a number is carried out for both random variables. **2)** Then we calculate the Spearman rank correlation coefficient as given above. **3)** To decide whether to reject the null or not, we use a t-test (test statistic for tests concerning correlation coefficient) if the sample is large enough (more than 30). In other cases, we use special tables to find critical values.

To decide whether to reject the null or not, we use t-test if the sample is large enough. In other cases, we use special tables to find the critical values.

- For normally distributed variables, we can use the sample correlation coefficient and t-test with n-2 degrees of freedom to test whether the two variables are linearly correlated.
- One of the examples of nonparametric test is a test based on the Spearman rank correlation coefficient.