Plotting data Representing data visually often helps people to see patterns or trends or to look for differences. It’s sometimes easier to look at a graph or chart than to look at a tables of values. On the following slides there are some interesting ways of representing data.

Ternary plot The first plot is called a Ternary plot. This plot is used when each item to be plotted has 3 pieces of connected information about it to represent. An example on the next slide is about soil types.

Ternary plot Can you work out how to read information from this plot about soil types?

Ternary plot The plot is built around 3 ‘axes’. What would a soil which is 25% sand, 30% clay and 45% silt be known as?

Ternary plot 25% sand 30% clay 45% silt

Ternary plot 4 soil types are plotted, what is the composition of each?

Ternary plot For each of the 4 soils, what’s the sum of the percentages? Will it always be this? Why? This is an important point about Ternary plots: they can only be used when the three categories are the only three possible ones, and so they always have to add up to 100%. There aren’t many situations where this is the case, which is perhaps one reason that we don’t often see these plots.

Food and calories Calories in food come from 3 main sources: carbohydrates, proteins and fats, so each food can be broken down according to the proportion of calories coming from these 3 sources. As an approximate guide: 1g of carbohydrate has 4 calories 1g of protein has 4 calories 1g of fat has 9 calories.

Food and calories A popular big burger contains 25g of protein, 46g of carbohydrate and 29g of fat. How many calories come from each source? What percentage of calories come from each source? On the next slide, the percentages have been calculated for some popular foods. Plot the values on a ternary plot.

Sources of calories in food: percentage values   Fat Carbs Protein Chocolate 49 45 6 Pizza 33 18 Chips 51 4 Apple 3 95 2 Cheese 74 1 25 Yoghurt 47 30 23 Pasta 79 15 Carrot 100 Crisps 57 38 5 Tuna 7 93

Sources of calories in food Who might be interested in this sort of information? Carbohydrate % Protein % Chocolate Yoghurt Pizza Pasta Chips Carrot Apple Crisps Cheese Tuna Fat %

Plotting very large data One of the issues with plotting data arises when we need to represent both relatively small and relatively large data at the same time. An example of this occurs if we want to look at the relative sizes of earthquakes and tremors. First we need to understand a little bit about how earthquakes are measured.

The Richter Scale The Richter scale is a term that many people will be familiar with – however, this is not strictly accurate as earthquakes are now measured using the Moment Magnitude Scale (MMS). The Richter scale was found to be less reliable for earthquakes measuring more than 7… … but what does a ‘7’ actually mean?

The MMS Scale The numbers refer to the amount of energy released by he earthquake as shown. How much more powerful is a MMS 7 earthquake than an MMS 5 one? MMS Approximate Energy (joules) 1 2 000 000 2 63 000 000 3 2 000 000 000 4 63 000 000 000 5 2 000 000 000 000 6 63 000 000 000 000 7 2 000 000 000 000 000 8 63 000 000 000 000 000 9 2 000 000 000 000 000 000 10 63 000 000 000 000 000 000

The MMS Scale It may be surprising that it is estimated there are over 500,000 earthquakes a year, of which only 100,000 are felt by humans since many of them are MMS 1 or 2. Seismologists record all activity and to help look for trends they sometimes plot the data too, hoping they will find ways of predicting events so that people can prepare. Last month there were over 500 which registered at MMS 4 or more.

The MMS Scale A selection of earthquake data from around the world during the last month are given. Plot them, putting the date on the horizontal axis and the joules released on the vertical axis. Date Approximate Energy (joules) 12/05 6.32 x 1010 13/05 2.52 x 1011 17/05 7.96 x 1012 18/05 19/05 7.01 x 1014 22/05 1.42 x 1015 25/05 8.93 x 1010 30/05 1.26 x 1014 02/06 1.26 x 1011 04/06 6.32 x 1013

The MMS Scale You will probably have found this a difficult task! If not an impossible one… The bigger numbers are so much larger than the smaller ones that it’s not easy to fit them on the same axes. If there are lots of ‘smaller’ values – as there are with earthquakes - the scale needs to allow people to see the differences, but then the large numbers are off the scale.

The MMS Scale One possibility is to use semi-logarithmic graph paper. This has a linear scale on the horizontal axis. On the vertical axis each ‘cycle’ of numbers represents 10 times the value of the previous cycle.

The MMS Scale It could be: 1 2 3 4 5 6 7 8 9 10

The MMS Scale It could be: 1 2 3 4 5 6 7 8 9 10 then 20 30 40 50 60 70 80 90 100

The MMS Scale It could be: 1 2 3 4 5 6 7 8 9 10 then 20 30 40 50 60 70 80 90 100 then 200 300 400…

The MMS Scale It could be: 1 2 3 4 5 6 7 8 9 10 then 20 30 40 50 60 70 80 90 100 then 200 300 400… Note the ‘overlap’ at 10, 100, 1000 etc. and note that the lines get closer together.

The MMS Scale For our purposes, the first cycle will be: x1010 and then x1011 and then x1012 Label the axes and plot the values.

Plotting Large Data This type of graph has uses in science, for those looking at bacterial growth, the spread of infection, charting planets and distances and many others. Using a scale which gets ten times bigger is common, however, sometimes a different scale is used, such as on the following slide. It’s just important to remember that it’s not a linear scale…

Plotting Large Data From the graph, can you determine: Which country has the biggest increase in cases in any three day period? For Chile (the pink line) when is the biggest increase in cases?

Teacher notes: Plotting Data This edition looks at a range of visual representations of data. The focus for these activities is on making sense of graphics, and although this doesn’t necessarily relate to any particular curriculum content objective, the process of understanding information presented in novel ways and interpreting the data is a useful real-life skill as well as involving many aspects of problem solving.

Teacher notes: Ternary Plots This activity could be used with a wide range of students as the only previous knowledge required is expressing one value as a percentage of another. Slide 4 Give students plenty of time to look at this to try to make sense of it. Slide 7 Make it ‘girl friendly’: Ask students to discuss it with a friend before giving an answer Dot colour % Sand % Clay % Silt Blue 12 64 24 Pink 41 35 Yellow 75 15 10 Black 85

Teacher notes: Ternary Plots Slide 10 Fat: 29 x 9 = 261 Carbohydrates: 46 x 4 = 184 Protein: 25 x 4 = 100 Total calories: 545 Fat: 48% Carbs: 34% Protein: 18% Slide 11 A blank ternary plot is available as a separate download.

Teacher notes: Plotting large data For this activity students will ideally be familiar with standard index notation. This activity involves quite challenging concepts. Through trying to plot the raw data, students should come to the realisation that ‘something different’ is needed. Slide 18 Students should attempt to plot the data and should soon realise that it is very difficult to do. Frustration and ‘failure’ can be key to genuinely appreciating the issues and understanding why an alternative is helpful. If students do manage to plot the data ask them how accurate it is for the lower values – are they able to distinguish between 6.32 x 1010 and 2.52 x 1011 ?

Teacher notes: Plotting large data Slide 21 to 24 It is worth spending some time looking at the paper, perhaps asking students to describe what they notice about it before showing them the slides. Slide 25 It should be relatively straight-forward to plot the data. No pattern emerges, but that is usual – particularly since these earthquakes are not from a single region. A separate file of a semi-logarithmic blank graph with 6 cycles is available to download. Slide 28 The USA has an increase in the last few days of approximately 8000 cases. Chile has the steepest line between June 8th and 11th, but this represents approximately 1500 cases, whereas between 23rd and 26th June there are about 2000 more cases.

Acknowledgements Ternary soil plot from: http://stackoverflow.com/questions/12520003/representing-ternary-plot-data-for-lookups Calories in food data from: http://www.calorieking.com/foods/ Earthquake data from: http://ds.iris.edu/seismon/eventlist/index.phtml Semi-logarithmic paper from: http://incompetech.com/graphpaper/logarithmic/