## Statisical 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 15-20) 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

$$r_s = 1-\frac{(6∑D^2)}{n(n^2-1)}$$

$$r_s$$ = 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.

$$\chi^2 = ∑ \frac{(O-E)^2}{E}$$

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

$$\chi^2 = ∑ \frac{(O-E)^2}{E}$$

$$\chi^2$$ = 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.

$$\mathsf{Degrees\;of\;freedom = (number\;of\;rows\;– 1) \times (number\;of\;columns\;– 1)}$$

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

$$U_1= n_1 \times n_2 + 0.5 n_2 (n_2 + 1)\;- ∑ R_2$$ $$U_2 = n_1 \times n_2 + 0.5 n_1 (n_1 + 1)\;- ∑ R_1$$

$$n_1$$ 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.

## Analysing qualitative data

Qualitative information (such as interview transcripts, photographs, creative writing and film) can be analysed objectively.

#### Frequency analysis

Frequency analysis identifies how often a particular theme occurs. For example, in a series of open-ended interviews about a city centre, 45% of the interviewees mentioned crime.

#### Thematic analysis

Thematic analysis helps you to look for recurring themes within your data. Coding is a useful technique of thematic analysis

• Descriptive coding Read the response several times. Try to summarise each line with a label consisting of a single word or short phrase
• Analytic coding Read your labels again. Try to group labels together into a few concepts

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