A Level

# Statistical tests

## Spearman’s Rank Correlation Test

Spearman’s Rank Correlation is a statistical test to test whether there is a significant relationship between two sets of data.

The Spearman’s Rank Correlation test can only be used if there are at least 10 (ideally at least 15-15) pairs of data.

There are 3 steps to take when using the Spearman’s Rank Correlation Test

### Step 1. State the null hypothesis

There is no significant relationship between _______ and _______

### Step 2. Calculate the Spearman’s Rank Correlation Coefficient

r_s = 1-(6∑D^2) / (n(n^2-1))

• r_s = Spearman's Rank correlation coefficient
• D = differences between ranks
• n = number of pairs of measurements

Step 3. Test the significance of the result

Compare the value of r_s that you have calculated against the critical value for r_s at a confidence level of 95% / significance value of p = 0.05.

If r_s is equal to or above the critical value (p=0.05) the REJECT the null hypothesis. There is a SIGNIFICANT relationship between the 2 variables.

A positive sign for r_s indicates a significant positive relationship and a negative sign indicates a significant negative relationship.

If r_s (ignoring any sign) is less than the critical value, ACCEPT the null hypothesis. There is NO SIGNIFICANT relationship between the 2 variables.

## Chi-squared test

Chi squared in a statistical test that is used either to test whether there is a significant difference, goodness of fit or an association between observed and expected values.

chi^2 = ∑ (O-E)^2 / E

The chi squared test can only be used if

• the data are in the form of frequencies in a number of categories (i.e. nominal data).
• there are more than 20 observations in total
• the observations are independent: one observation does not affect another

There are 3 steps to take when using the chi squared test

### Step 1. State the null hypothesis

There is no significant association between _______ and _______

### Step 2. Calculate the chi squared statistic

chi^2 = ∑ (O-E)^2 / E

chi^2 = chi squared statistic

O = Observed values

E = Expected values

### Step 3. Test the significance of the result

Compare your calculated value of chi^2 against the critical value for chi^2 at a confidence level of 95% / significance value of P = 0.05, and appropriate degrees of freedom.

"Degrees of freedom" = ("number of rows" – 1) xx ("number of columns" – 1)

If Chi Squared is equal to or greater than the critical value REJECT the null hypothesis. There is a SIGNIFICANT difference between the observed and expected values.

If Chi Squared is less than the critical value, ACCEPT the null hypothesis. There is NO SIGNIFICANT difference between the observed and expected values.

## Mann Whitney U test

Mann Whitney U is a statistical test that is used either to test whether there is a significant difference between the medians of two sets of data.

The Mann Whitney U test can only be used if there are at least 6 pairs of data. It does not require a normal distribution.

There are 3 steps to take when using the Mann Whitney U test

### Step 1. State the null hypothesis

There is no significant difference between _______ and _______

### Step 2. Calculate the Mann Whitney U statistic

U_1= n_1 xx n_2 + 0.5 n_2 (n_2 + 1) - ∑ R_2

U_2 = n_1 xx n_2 + 0.5 n_1 (n_1 + 1) - ∑ R_1

• n_1 is the number of values of x_1
• n_2 is the number of values of x_2
• R_1 is the ranks given to x_1
• R_2 is the ranks given to x_2

### Step 3. Test the significance of the result

Compare the value of U against the critical value for U at a confidence level of 95% / significance value of P = 0.05.

If U is equal to or smaller than the critical value (p=0.05) the REJECT the null hypothesis. There is a SIGNIFICANT difference between the 2 data sets.

If U is greater than the critical value, then ACCEPT the null hypothesis. There is NOT a significant difference between the 2 data sets.