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Stage 4: Data analysis

Stage 4: Glacial deposition: calculations | data presentation | statistics | Drumlins

Glacial deposition: calculations

Calculating the Cailleux 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

Ci = (2r/l)x1000.

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

Calculating 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

K = cube root of bc/a2

K = 1 for a perfectly spherical pebble. K must be between 0 and 1. The lower K is, the less spherical the pebble.

Zingg's shape classification

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
Blade	< 0.67	< 0.67

Analysing the extent of sediment sorting

Depending on the sieves that you used to sort sediment, you may have results for sediment size in millimetres or as 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

Use semi-logarithmic graph paper to plot a cumulative frequency graph of phi against mass. Plot phi 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

Glacial deposition: data presentation

Constructing a rose diagram for stone orientation

Sort the results into a tally chart showing the frequency (i.e. number) of measurements in each 15 interval (0-15, 16-30, 31-45, etc). Since each orientation will have two values 180° apart, the rose diagram will be a mirror image of itself across the 180° line. The map right shows how this has been done for glacial deposits exposed by coastal erosion in Somerset.

Plotting a roundness frequency histogram

Plot a frequency distribution of the Ci. Sort the results into a tally chart showing the frequency (i.e. number) of measurements in each 5 interval (0-5, 5.1-10, 10.1-15, etc).

Plotting a sediment size histogram

Calculate the percentage of the total mass at each sediment size. 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.

Plot a histogram with % mass on the y axis and sediment size on the x-axis.

Glacial deposition: statistics

Statistical analysis of stone orientation

The orientation of stones in two different deposits can be compared statistically using the chi-squared test. Need more information about this test?

Sort the data for each site into 45° intervals (i.e. 0-45, 46-90, 91-135 and 135-180). These are the observed values. Calculate the expected values. E = row total X column total

Chi squared = (O - E)² / E

Calculate degrees of freedom = (no of rows - 1) x ( no of columns - 1)

Choose a significance level e.g. 1% ( This means that chance should only account for the results in up to 1% of occasions the field test is carried out )

Compare the result with the critical value in the table. If the calculated value is greater than the critical value in the table the null hypothesis must be rejected.

Drumlins

(i) Step 1 of 2: analysing drumlin distribution by nearest-neighbour analysis

Nearest-neighbour analysis can be used to describe the degree of clustering of drumlins. If you have collected data on the distances between each drumlin. Now calculate the mean distance between drumlins. The nearest-neighbour statistic is calculated as

2 x (mean distance between drumlins) x v (number of drumlins ÷ total area of drumlin field)

Make sure that you use the same units for distance throughout and take care calculating the area. For example, if you have measured distances in metres, calculate the area in square metres.

The result should be between 0 and 2.15. To interpret: clustered distribution = 0; random distribution = 1.0; regular distribution = 2.15.

(ii) Step 2 of 2: vector analysis of drumlin orientation

Sort the results into a tally chart showing the frequency (i.e. number) of measurements in each 40 interval (0-40, 41-80, 81-120, etc) - or use the tally chart you made when drawing a rose diagram in Stage 3. These intervals are called azimuths. Calculate the midpoint azimuth for each interval, i.e. the middle value. For example, for the 0-40 azimuth, the midpoint is 20. You should have a table which looks like this.

Midpoint azimuth (φ)	Frequency (f)
20	10
60	12
100	4
140	1
180	2
220	3
260	3
300	4
340	6

Now calculate the resultant vector (Φ). This is the central point of the distribution of drumlins.

Φ = tan ­¹ (A ÷ B)

where A = ∑ (f . sin φ) and B = ∑ (f . cos φ)

It's most straightforward to work this out in a table.

Midpoint azimuth (φ)	Frequency (f)	sin φ	f . sin φ	cos φ	f . cos φ
20	10	0.34	3.42	0.94	9.40
60	12	0.87	10.39	0.50	6.00
100	4	0.98	3.94	-0.17	-0.68
140	1	0.64	0.64	-0.77	-0.77
180	1	0.00	0.00	-1.00	-1.00
220	0	-0.64	0.00	-0.77	0.00
260	3	-0.98	-2.94	-0.17	-0.51
300	4	-0.87	-3.48	0.50	2.00
340	6	-0.34	-2.04	0.94	5.64
	f(n)=41	∑ (f . sin φ)	= 9.63	∑ (f . cos φ)	= 20.08

Since A = 9.63 and B = 20.08

Φ = tan ­¹ (A ÷ B) = tan ­¹ (9.63 ÷ 20.08) = 25.6

Therefore 25.6° is the central point of drumlin orientation.

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