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Scatter Diagrams - I know how to interpret a scatter diagram - I know how to identify a correlation - I know the difference between cause and effect

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Investigation We need to measure the following for everyone: Foot length Arm Span Height Hand Span Circumference of Head Make sure you know each of these details for yourself, so write it down!

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Graphs we can plot Foot length v Height Arm Span v Height Hand Span v Height Circumference of Head v Hand Span Etc, etc, etc…

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© Boardworks Ltd 2005 4 of 44 Scatter graphs As height increases, weight increases. What does this scatter graph show about the relationship between the height and weight of twenty Grade 10 boys? This is called a positive correlation. 40 45 50 55 60 140140150150160160170170180180190190 Height (cm) Weight (kg)

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© Boardworks Ltd 2005 5 of 44 Scatter graphs What does this scatter graph show? 50 55 60 65 70 75 80 85 020406080100120 Number of cigarettes smoked in a week Life expectancy It shows that life expectancy decreases as the number of cigarettes smoked increases. This is called a negative correlation.

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© Boardworks Ltd 2005 6 of 44 Interpreting scatter graphs Scatter graphs can show a relationship between two variables. This relationship is called correlation. Correlation is a general trend. Some data items will not fit this trend, as there are often exceptions to a rule. They are called outliers. Scatter graphs can show: positive correlation: as one variable increases, so does the other variable negative correlation: as one variable increases, the other variable decreases zero correlation: no linear relationship between the variables. Correlation can be weak or strong.

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© Boardworks Ltd 2005 7 of 44 Identifying correlation from scatter graphs strong positive correlation strong negative correlation zero correlation weak positive correlation weak negative correlation. A BC D EF Decide whether each of the following graphs shows,

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Correlation Coefficient The correlation coefficient gives us a numerical description of how strong the relationship between two variables is. The range of the value of r (the correlation coefficient) is from -1 to +1. From 0 to 1, there is a positive relationship. The closer to 1, the stronger the relationship. For -1 to 0, there is a negative relationship. The closer to -1, the stronger the relationship.

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© Boardworks Ltd 2005 9 of 44 Relationships between two variables

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© Boardworks Ltd 2005 10 of 44 0 5 10 15 20 25 0510152025 Strong positive correlation 0 5 10 15 20 25 0510152025 Strong negative correlation Weak negative correlation The line of best fit Lines of best fit are drawn on graphs by eye so that there are a roughly equal number of points above and below the line. Look at these examples, Weak positive correlation Notice that the stronger the correlation, the closer the points are to the line. If the gradient is positive, the correlation is positive and if the gradient is negative, then the correlation is also negative.

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© Boardworks Ltd 2005 11 of 44 The line of best fit The line does not have to pass through the origin. When drawing the line of best fit remember the following points, For an accurate line of best fit, find the mean for each variable. This forms a coordinate, which can be plotted. The line of best fit should pass through this point. The line of best fit can be used to predict one variable from another. It should not be used for predictions outside the range of data used. The equation of the line of best fit can be found using the slope and the y-intercept.

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© Boardworks Ltd 2005 12 of 44 The line of best fit 50 55 60 65 70 75 80 85 020406080100 Number of cigarettes smoked in a week Life expectancy This graph shows the relationship between life expectancy and the number of cigarettes smoked in a week.

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© Boardworks Ltd 2005 13 of 44 Solving problems with lines of best fit Work out an estimate for the equation of the line of best fit using the slope and y-intercept. Use the equation to estimate the life expectancy for someone who smokes 10 cigarettes a day. Why would an estimate of the number of cigarettes smoked for a life expectancy of 40 years not be reliable? Can you explain why there are so many outliers for this data?

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© Boardworks Ltd 2005 14 of 44 Cause and effect A study finds a positive correlation between the number of cars in a town and the number of babies born. “Buying a new car can help you get pregnant!” The local newspaper reports, Does the study support this conclusion? What might this be in the example above? Correlation does not necessarily imply that there is a causal relationship between the two variables. There may be some other cause.

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© Boardworks Ltd 2005 15 of 44 Cause and effect A study finds a negative correlation between the number of sleds sold and the temperature. Explain. Does the study support this conclusion? “If you want it to snow, go out and buy a sled!” The local newspaper reports,

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© Boardworks Ltd 2005 16 of 44 Homework Page 39 #s 1 - 5

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