Correlation

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 In this topic, we will look at patterns in data on a scatter graph.  We will see how to numerically measure the strength of correlation between two variables, using technology (MS Excel or a scientific calculator) to assist.  We will use the following vocabulary: correlation, line of best fit, dependent variable, independent variable. 3

Correlation Scatter graphs Scatter graphs are a way of representing two sets of data. It is then possible to see whether they are related. Positive correlation  As one variable increases, so does the other. Negative correlation  As one variable increases, the other decreases. No correlation  There seems to be no pattern linking the two variables. Positive Negative None

! Some scatter diagrams may appear to show correlation at first sight, but in fact do not. Watch out for these. The key is to consider if it is sensible to draw a line of best fit through the data.

Question 1 Scatter graphs In the study of a city, the population density (in people/hectare) and the distance from the city centre (in km) were investigated by choosing sample areas. The results are as follows: Plot a scatter graph and describe the correlation. Interpret what the correlation means. 6 AreaABCDE Distance Pop. density AreaFGHIJ Distance Pop. density Distance from centre (km) Pop. density (people/hectare) Dependent variable Independent variable

Correlation Scatter graphs In the study of a city, the population density (in people/hectare) and the distance from the city centre (in km) were investigated by choosing sample areas. The results are as follows: Plot a scatter graph and describe the correlation. Interpret what the correlation means. 7 AreaABCDE Distance Pop. density AreaFGHIJ Distance Pop. density Distance from centre (km) Pop. density (people/hectare) The correlation is negative, which means that as we get further from the city centre, the population density decreases.

Correlation Variability of bivariate data The following table shows babies heads’ circumferences (cm) and the gestation period (weeks) for six newborn babies. Calculate r, the Pearson product moment correlation coefficient. This can be done on calculators or in MS Excel. First Excel: 1.Copy the table into Excel, putting ‘Baby’ into cell A1. 2.In an empty cell, type ‘ =pearson(B2:G2,B3:G3)’. 3.Label this ‘PMCC’ BabyABCDEF Head size (x) Gestation period (y)

Correlation Variability of bivariate data The following table shows babies heads’ circumferences (cm) and the gestation period (weeks) for six newborn babies. Calculate r, the Pearson product moment correlation coefficient. BabyABCDEF Head size (x) Gestation period (y) Now with a calculator (Casio scientific): 1.Put your calculator into STAT mode 2.Select A+BX 3.Input the values into the table, pressing ‘ = ’ after each one and using the arrows to move into the y column once you have finished the x column. If you have a frequency column, just ignore it. When you have finished, press ‘AC’. 4.Press ‘shift’, ‘1’(STAT), select ‘reg’ then ‘r’ and press ‘= ’. This is the value of the PMCC.

Correlation Product moment correlation coefficient We can test the correlation of data by calculating the product moment correlation coefficient, r The value of this number tells you what the correlation is and how strong it is. The closer to 1, the stronger the positive correlation. The same applies for -1 and negative correlation. A value close to 0 implies no linear correlation. Positive correlation Negative correlation 10 No linear correlation

Question 1 Scatter graphs In the study of a city, the population density (in people/hectare) and the distance from the city centre (in km) were investigated by choosing sample areas. The results are as follows: Calculate the value of r, the product moment correlation coefficient. 11 AreaABCDE Distance Pop. density AreaFGHIJ Distance Pop. density Distance from centre (km) Pop. density (people/hectare) r =

Question 2 It is widely believed that those who are good at chess are good at bridge, and vice versa. A commentator decides to test this theory using as data the grades of a random sample of eight people who play both games. (i) Calculate the product moment correlation coefficient. I used Excel to do this (see file) and got r = PlayerABCDEFGH Chess grade Bridge grade

Question 2 It is widely believed that those who are good at chess are good at bridge, and vice versa. A commentator decides to test this theory using as data the grades of a random sample of eight people who play both games. (ii) Interpret the meaning of your result is a fairly strong positive correlation. This indicates that the commentator is correct, that those who are good at chess are good at bridge, and vice versa. 13 PlayerABCDEFGH Chess grade Bridge grade

Correlation Limitations of the product moment correlation coefficient Sometimes it may indicate correlation between unrelated variables:  Cars on a particular street have increased, as have the sales of DVDs in town – the PMCC would indicate positive correlation, when the two are most likely not linked.  The speed of computers has increased, as has life expectancy amongst people – these are not directly linked, although they are both due to scientific developments. 14

Summary We have looked at:  plotting scatter graphs  types of correlation and how to recognise them on a graph  using Excel or a calculator to calculate the product moment correlation coefficient, and interpreting it. It is a numerical measure of correlation.  using suitable vocabulary: correlation, line of best fit, dependent variable, independent variable. 15

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