Download Central tendency, dispersion diagrams and standard deviation

Central tendency, dispersion diagrams and standard deviation
Measures of central tendency - the mean, median and mode are used to provide the average or
normal values of a data set.
Measures of dispersion - the range and standard deviation are used to help us calculate if these
figures are very similar to, or different from each other. They provide an indication of the
spread of the data.
Student tasks
1. Look at the data set below of the heights of tides around the coast of the UK.
Location
Spring tide
(metres)
Neap tide
(metres)
Mean tide (𝑥)
(metres)
Aberdeen
4.9
3.7
(4.9 + 3.7) / 2 = 4.3
Cromer
7.4
4.6
Dover
8.5
6.3
Felixstowe
5.0
2.4
Ilfracombe
27.7
1.0
Liverpool
9.3
7.63
Milford Haven
31.75
6.1
Portsmouth
5.2
3.2
Weymouth
3.7
3.3
Ullapool
14.01
4.4
Total mean tide height (metres)
Mean tidal height, 𝑥̅ (metres)
Data used courtesy of the UK Tide Gauge Network noc.ac.uk/data/british-oceanographic-data-centre
2.
Calculate the height of the mean tide for each of the recording stations. Aberdeen has
been completed as an example.
3.
Calculate a single mean height (𝑥̅ ) for the ten recording stations, i.e. add your ten mean
tide heights and divide by ten.
4.
Plot a dispersion diagram for your mean tidal range values
5.
Calculate the mode and median using the ten mean tide calculations.
Mode
6.
Median
Calculate the range and the interquartile range (IQR). The IQR is the difference between
the upper quartile and the lower quartile using the dispersion diagram or the data listed in
ascending order.
Range
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IQR
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Central tendency, dispersion diagrams and standard deviation
7.
How are the mean, mode and median different from each other? Which is more useful?
Standard deviation
A calculation of the standard deviation (σ) allows you to see how reliable the mean tidal height
is, and if there were many anomalies.
Complete for table below. The 𝑥 and 𝑥̅ values are from the previous table. The Greek symbol,
Sigma (∑) is always used to indicate a total. The number of stations is N.
Location
Mean tide (𝑥)
𝑥 − 𝑥̅
(𝑥 − 𝑥̅ )2
Aberdeen
4.3
4.3 - 8.0 = -3.7
13.69
Dover
7.4
-0.6
0.36
Felixstowe
3.7
-4.3
18.49
8.47
0.47
0.216225
4.2
-3.8
14.44
9.21
1.21
1.452025
Cromer
Ilfracombe
Liverpool
Milford Haven
Portsmouth
Weymouth
Ullapool
∑ (𝑥 − 𝑥̅ )2 =
8.
Calculate the variance
Variance = ∑ (𝑥 − 𝑥̅ )2
N
9.
The standard deviation is the square root of the variance. Round the square root of the
variance to two decimal places.
The standard deviation (σ) allows us to see the ‘spread’ of the data, i.e. is it clustered around
the mean or spread out over a large range. A large standard deviation will indicate a large
spread and a small standard deviation will indicate a small spread in the data set.
The Mediterranean
The range in a data set for ten recording stations in the Mediterranean is 0.8 metres with a
standard deviation of 10.
10.
What do a range of 0.8 metres and a standard deviation of 10 indicate?
11.
How does your standard deviation for the UK compare with the Mediterranean recording
stations?
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Central tendency, dispersion diagrams and standard deviation
Normal distribution
If the data is normally distributed, 68% of the data should lie within +1 σ and -1 σ of the mean,
i.e. with symmetry about the mean.
By Dan Kernler (Own work) [CC BY-SA 4.0 (http://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia Commons
With 68% of the data in the central region of the normal bell shaped distribution curve, y 32% of
the data is outside of the +1σ and -1 σ range.
12.
With the aid of an atlas, locate the recording stations not falling within +1σ and -1 σ of the
mean.
13.
Examine the two maps of tidal speeds around the UK at
www.greenrhinoenergy.com/renewable/marine/tidal_stream.php. The yellow shading
shows largest tidal range, which then decreases through shades of orange/pink. The
purple has lowest tidal range.
14.
Describe and suggest possible explanations for the variations in tidal heights around the UK
coast.
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Central tendency, dispersion diagrams and standard deviation
Teaching notes
Location
Spring tide
(metres)
Neap tide
(metres)
Mean tide (𝑥)
(metres)
Aberdeen
4.9
3.7
4.3
Cromer
7.4
4.6
6
Dover
8.5
6.3
7.4
Felixstowe
5.0
2.4
3.7
Ilfracombe
27.7
1.0
14.35
Liverpool
9.3
7.63
8.47
Milford Haven
31.75
6.1
18.93
Portsmouth
5.2
3.2
4.2
Weymouth
3.7
3.3
3.5
Ullapool
14.01
4.4
9.21
Total mean tide height (metres)
80.05
Mean tidal height, 𝑥̅ (metres)
8.0
Mode
This difficult with many decimal places. One could argue there is no mode, or
that when decimal places are rounded to single figures it would be 4.
Median
6.7
Ordered data
3.5, 3.7, 4.2, 4.3, 6 , 7.4, 8.47, 9.21, 14.35, 18.93
Range
15.43
IQR
5.01 (in this case the difference between data point 3 and data point 8)
Mean, mode and median comparison
In combination the mean, mode and median are useful for giving a summary statement about
sets of data. The mean is most useful on small sets of data as it can be influenced by extreme
values. The mode is not always useful in data sets with no repeated values as it is entirely based
on frequency. The median is not as arithmetically sound as the mean as it is based on rankings
however it is not influenced by extreme values.
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Central tendency, dispersion diagrams and standard deviation
Calculation of variance
Location
Mean tide (x)
x − x̅
(x − x̅)2
Aberdeen
4.3
4.3 - 8.0 = -3.7
13.69
Cromer
6
-2
4
Dover
7.4
-0.6
0.36
Felixstowe
3.7
-4.3
18.49
Ilfracombe
14.35
6.35
40.3225
Liverpool
8.47
0.47
0.216225
Milford Haven
18.93
10.93
119.3556
Portsmouth
4.2
-3.8
14.44
Weymouth
3.5
-4.5
20.25
Ullapool
9.21
1.21
1.452025
∑ (x − x̅)2 =
232.5764
Variance
∑ (𝑥 − 𝑥̅ )2
N
23.25764
σ
4.822617
Tidal speeds around the UK
Using www.greenrhinoenergy.com/renewable/marine/tidal_stream.php.
The tidal speeds are highest around the coasts of Norfolk, Isle of Wight, Bristol Channel and the
northern/southern extremities of Wales. The lowest speeds on located in inlets around Scotland
and bays of south Cornwall and Devon. The northeast coast of England has a low range.
Relevant factors include:

the shape of the coastline – the narrow channel between the UK and France leads to
funnelling effect of water in a small space. The effect is similar/identical in other funnel
shaped estuaries and inlets, e.g. the Bristol Channel.

the depth of sea bed and continental shelf – this will influence the height of the tidal
water.

the weather - strong winds and changes in atmospheric pressure can lead to a piling up of
water giving higher tides.
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