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

## Sediment analysis

### (a) Size of coarse sediments: mean size

Coarse sediments are pebbles and cobbles. If you have measured the a, b and c axes using calipers or a ruler, you could calculate the mean pebble size for each sample site on the beach.

"Mean size" = (a + b + c)/3 for each pebble

### (b) Size of fine sediments: phi sizes

Fine sediments are clay, silt and sand. Once you have sieved the sediment, you can calculate phi sizes. Use the conversion table if you do not have the phi sizes already.

Sediment size
mm phi
1.00 0
0.50 1
0.25 2
0.13 3
0.06 4
0.03 5
0.01 6

Calculate the percentage mass of sediment in each phi size category. For example, if total mass=100g and the mass of material at 5-10mm = 20g, then 20% of the total mass of sediment is 5-10mm in diameter. This can be presented in a number of ways

• using a histogram with % mass on the y axis and sediment size on the x-axis
• pie charts to show changes along the transect, which might be overlaid on a map or aerial photograph
• plot a scattergraph to show how mean sediment size varies with distance along the beach (see below).

Alternatively, use semi-logarithmic graph paper to plot a cumulative frequency graph of phi against mass. Plot phi size on the linear x-axis. Plot the cumulative mass of sediment on the logarithmic y-axis.

On your finished graph, find the phi size values at 16% and 84% cumulative mass. Use these figures in the following formula

("phi at 84% mass" - "phi at 16% mass")/2

Use the following table to interpret the result

result interpretation
0.35 very well sorted
0.35 - 0.5 well sorted
0.5 - 0.7 moderately well sorted
0.7 - 1.0 moderately sorted
1.0 - 2.0 poorly sorted
2.0 - 4.0 very poorly sorted
4.0 extremely poorly sorted

### (c) Zingg's shape classification

The analysis of the shape of coarse sediments can be divided into 4 categories: shape, sphericity, flatness and roundness.

The raw data needed for each pebble are the lengths of the a, b and c axes.

Calculate the ratio b-:a

Calculate the ratio c-:b

Now classify each pebble into one of the four groups shown in the table

Type of pebble b-:a c-:b
Sphere > 0.67 > 0.67
Disc > 0.67 < 0.67
Rod < 0.67 > 0.67

### (d) Krumbein's Index of Sphericity

The raw data needed for each pebble are the lengths of the a, b and c axes.

For each stone, calculate Krumbein's Index as follows

"Krumbein's Index" = ((bc)/a^2)^(1/3)

Krumbein's Index (K) K must be between 0 and 1. K = 1 for a perfectly spherical pebble. The lower that K is, the less spherical the pebble.

### (e) Cailleux’s Flatness Index

The raw data needed for each pebble are the lengths of the a, b and c axes.

For each stone, calculate Cailleux’s Flatness Index as follows

"Flatness Index" = (a + b) / (2c) xx 100

A perfectly equidimensional particle will have a Flatness Index of 100 and will increase infinitely as it become flatter.

### (f) Cailleux’s Roundness Index

The raw data needed for each pebble are:

• the length of the longest axis (l)
• the radius of curvature of the sharpest angle (r)

For each stone, calculate Cailleux Index as follows

"Roundness Index" = (2r)/lxx1000

Roundness Index =1000 for a perfectly spherical pebble. The lower the Roundness Index is, the more angular the pebble.

Cailleux's Roundness Index may be presented using box and whisker plots.

## 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.

### Worked example

A geographer is interested in whether longshore drift is operating along a coastal spit at Abermenai Point in Anglesey. Random samples of pebbles were taken at the base of the storm beach at 15 sites every 50m along a spit. Here are the results.

Distance along spit (m) Mean pebble size (cm)
0 11.3
50 12.2
100 12.8
150 13.6
200 14.5
250 15.3
300 13.5
350 11.2
400 10.0
450 9.3
500 8.0
550 10.2
600 9.8
650 10.2
700 7.4

### Step 1. State the null hypothesis

There is no significant relationship between distance along the spit and mean pebble size at Abermenai Point.

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

(a) Rank the measurements

Distance along spit (m) Mean pebble size (cm)
Data Rank Data Rank
0 1 11.3 8
50 2 12.2 9
100 3 12.8 10
150 4 13.6 12
200 5 14.5 13
250 6 15.3 14
300 7 13.5 11
350 8 11.2 7
400 9 10.0 5
450 10 9.3 3
500 11 8.0 2
550 12 10.2 6.5
600 13 9.8 4
650 14 10.2 6.5
700 15 7.4 1

(b) Calculate D and D^2

Distance along spit (m) Mean pebble size (cm)
Data Rank Data Rank D D^2
0 1 11.3 8 7 49
50 2 12.2 9 7 49
100 3 12.8 10 7 49
150 4 13.6 12 8 64
200 5 14.5 13 8 64
250 6 15.3 14 8 64
300 7 13.5 11 4 16
350 8 11.2 7 -1 1
400 9 10.0 5 -4 32
450 10 9.3 3 -7 49
500 11 8.0 2 -9 81
550 12 10.2 6.5 -5.5 30.25
600 13 9.8 4 -9 81
650 14 10.2 6.5 -7.5 56.25
700 15 7.4 1 -14 196

(c) Calculate ∑D^2i.e. the sum of the D^2column= 865.6

(d) Calculate r_s

r_s = 1-(6∑D^2) / (n(n^2-1))

r_s = 1-(6xx865.5) / (15xx(225-1))

r_s = -0.55

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

The critical value at p=0.05 significance level for 15 pairs of measurements is 0.521

Since our calculated value of 0.55> 0.521 (ignore the minus sign), the null hypothesis can be rejected.

In conclusion, there is a significant relationship between distance along the spit and mean pebble size at Abermenai Point. The negative sign for r_sindicates a significant negative relationship.