A Level

# Data analysis

## 4. Data analysis

Sophisticated data analysis will help you spot patterns, trends and relationships in your results. Data analysis can be qualitative and/or quantitative, and may include statistical tests. An example of a statistical test is outlined below.

## Calculating wave energy

To calculate the energy of the waves, you need to have measured two things:

• Wave period (T): the time taken (in seconds) for 2 successive waves to pass a fixed point
• Wave height (H): the height of the wave (in metres) from trough to peak

### Step 1. Calculate wave period

Wave period is expressed in seconds.

If you have counted how many wave crests pass a fixed point in 60 seconds, wave period is calculated as

"Wave period" = 60 / "Number of wave crests passing a fixed point"

### Step 2. Calculate the wavelength

Wavelength is expressed in metres.

"Wavelength" = 1.56 xx ("Wave period")^2

### Step 3. Calculate the wave energy

Wave energy is expressed in joules per metre per second.

Wave energy is directly proportional to the wavelength (L). Wave energy is directly proportional to the square of wave height (H^2).

"Wave energy" = 740 xx "Wavelength" xx "Wave height"^2

## 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.

### Worked example

A geographer was interested in whether there was a difference in cliff gradient between places with a beach and places with no beach. Here are the results.

Cliff gradient where there is no beach (°) Cliff gradient where there is a beach (°)
20 15
35 21
32 36
16 12
41 10
23 18

### Step 1. State the null hypothesis

There is no significant difference in cliff gradient between places with a beach and places with no beach.

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

(a) Give each result a rank. Calculate the sum of the ranks for the two columns.

No beach Beach
20 6 15 3
35 10 21 7
32 9 36 11
16 4 12 2
41 12 10 1
23 8 18 5
TOTAL 49 TOTAL 29

(b) Calculate ∑R_1and ∑R_2

∑R_1 is the sum of the ranks in the first column (no beach) = 49

∑R_2 is the sum of the ranks in the first column (beach) = 29

n_1 = 6 and n_2 = 6

(c) Calculate U_1 and U_2

U_1 = 6 xx 6 + 0.5 xx 6 (6 + 1) - 29 = 28

U_2 = 6 xx 6 + 0.5 xx 6 (6 + 1) - 49 = 8

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

In this example, U_1 = 28 and U_2 = 8

U is the smaller of the two values, so U=8

The critical value at p=0.05 significance level for n_1=6 and n_2=6 is 5. Since our calculated value of 8<6 the null hypothesis is not rejected.

In conclusion, there is no significant difference in cliff gradient between places with a beach and places with no beach.