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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 (π₯) (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, π₯Μ (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 (π₯Μ ) 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 © www.teachitgeography.co.uk 2016 IQR 27066 Page 1 of 5 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 (Ο) allows you to see how reliable the mean tidal height is, and if there were many anomalies. Complete for table below. The π₯ and π₯Μ values are from the previous table. The Greek symbol, Sigma (β) is always used to indicate a total. The number of stations is N. Location Mean tide (π₯) π₯ β π₯Μ (π₯ β π₯Μ )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 β (π₯ β π₯Μ )2 = 8. Calculate the variance Variance = β (π₯ β π₯Μ )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 (Ο) allows us to see the βspreadβ 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? © www.teachitgeography.co.uk 2016 27066 Page 2 of 5 Central tendency, dispersion diagrams and standard deviation Normal distribution If the data is normally distributed, 68% of the data should lie within +1 Ο and -1 Ο 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 +1Ο and -1 Ο range. 12. With the aid of an atlas, locate the recording stations not falling within +1Ο and -1 Ο 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. © www.teachitgeography.co.uk 2016 27066 Page 3 of 5 Central tendency, dispersion diagrams and standard deviation Teaching notes Location Spring tide (metres) Neap tide (metres) Mean tide (π₯) (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, π₯Μ (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. © www.teachitgeography.co.uk 2016 27066 Page 4 of 5 Central tendency, dispersion diagrams and standard deviation Calculation of variance Location Mean tide (x) x β xΜ (x β xΜ )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 β (x β xΜ )2 = 232.5764 Variance β (π₯ β π₯Μ )2 N 23.25764 Ο 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: ο· the shape of the coastline β 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. ο· the depth of sea bed and continental shelf β this will influence the height of the tidal water. ο· the weather - strong winds and changes in atmospheric pressure can lead to a piling up of water giving higher tides. © www.teachitgeography.co.uk 2016 27066 Page 5 of 5