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Anscombe’s Quartet – the importance of graphs!
Anscombe’s Quartet was devised by the statistician Francis Anscombe to illustrate how important it was to not just rely on statistical measures when analyzing data. To do this he created 4 data sets which would produce nearly identical statistical measures. The scatter graphs above generated by the Python code here.
Statistical measures
1) Mean of x values in each data set = 9.00
2) Standard deviation of x values in each data set = 3.32
3) Mean of y values in each data set = 7.50
4) Standard deviation of x values in each data set = 2.03
5) Pearson’s Correlation coefficient for each paired data set = 0.82
6) Linear regression line for each paired data set: y = 0.500x + 3.00
When looking at this data we would be forgiven for concluding that these data sets must be very similar – but really they are quite different.
Data Set A:
x = [10,8,13,9,11,14,6,4,12,7,5]
y = [8.04, 6.95,7.58,8.81,8.33, 9.96,7.24,4.26,10.84,4.82,5.68]
Data Set A does indeed fit a linear regression – and so this would be appropriate to use the line of best fit for predictive purposes.
Data Set B:
x = [10,8,13,9,11,14,6,4,12,7,5]
y = [9.14,8.14,8.74,8.77,9.26,8.1,6.13,3.1,9.13,7.26,4.74]
You could fit a linear regression to Data Set B – but this is clearly not the most appropriate regression line for this data. Some quadratic or higher power polynomial would be better for predicting data here.
Data Set C:
x = [10,8,13,9,11,14,6,4,12,7,5]
y = [7.46,6.77,12.74,7.11,7.81,8.84,6.08,5.39,8.15,6.42,5.73]
In Data set C we can see the effect of a single outlier – we have 11 points in pretty much a perfect linear correlation, and then a single outlier. For predictive purposes we would be best investigating this outlier (checking that it does conform to the mathematical definition of an outlier), and then potentially doing our regression with this removed.
Data Set D:
x = [8,8,8,8,8,8,8,19,8,8,8]
y = [6.58,5.76,7.71,8.84,8.47,7.04,5.25,12.50,5.56,7.91,6.89]
In Data set D we can also see the effect of a single outlier – we have 11 points in a vertical line, and then a single outlier. Clearly here again drawing a line of best fit for this data is not appropriate – unless we remove this outlier first.
The moral of the story
So – the moral here is always use graphical analysis alongside statistical measures. A very common mistake for IB students is to rely on Pearson’s Product coefficient without really looking at the scatter graph to decide whether a linear fit is appropriate. If you do this then you could end up with a very low mark in the E category as you will not show good understanding of what you are doing. So always plot a graph first!
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Is there a correlation between Premier League wages and league position?
The Guardian has just released its 2012-13 Premier League season data analysis – which shows exactly how much each club in the Premier League spent on wages last year (see the bar chart above). This can be easily plotted on a scatter graph to test how strong the correlation is between spending and league position. (y axis is league position, x axis is wage bill in millions of pounds).
The mean spending on wages is 89 million pounds. Our regression line is y = -0.08x + 17.52. We can see some of the big outliers are QPR (with a big wage bill but low premier league position) and Everton (with a low wage bill relative to others who finished in a similar position).
The Pearson’s product moment correlation coefficient (r) is -0.73. This is negative because in our case league position is numerically lower the higher up the league you are. This shows a pretty strong correlation between league spending and league position. An r value of -1 would be a perfect correlation in our case, whereas 0 would be no correlation.
Is there a correlation between turnover and league position?
We can also see what the correlation is between league position and overall club turnover (see the bar chart above). Here we can see there is a huge gulf between the top few clubs and everyone else in the league. There’s only 40 million pounds difference between the bottom ranked club for revenue Wigan and Newcastle, with the 7th biggest revenue. But then a massive jump up to those with the top 6 revenues.
This time we have a mean turnover of 128 million pounds and a regression line of y = -0.05x + 16.89. The Pearson’s r value this time is r = -0.79, so there is a slightly stronger correlation than from wages – and this is a strong correlation overall. So, both wage bills and turnover provide a pretty good predictor of where a team will finish – and also a decent yardstick to measure how well a team has done relative to their resources.
If you like this post you might also like:
Do Championship Wages Predict League position? A comparison between the Premier League and the Championship (England’s second tier).
Does Sacking a Manager Improve Results? How an improvement in team results is often just down to a statistical result – regression to the mean.
Maths Studies IA Exploration Topics – A large number of examples of statistics investigations to explore.
Essential resources for IB students:
1) Exploration Guides and Paper 3 Resources
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IB Maths Resources from Intermathematics 