1. Calculating Residuals © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

2. Calculating Residuals "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Statistics 1 AQA EDEXCEL OCR

3. Calculating Residuals Foot length and height of UK children Height (cm) Foot length (cm) Once we have found a regression line, we may need to know how close any particular observation is to the line. To do this, we find a residual. For the height and foot length data... y on x regression line To find the residual for the point we find

4. Calculating Residuals e.g. The marks for 10 students in Maths and Physics are as follows: ABCDEFGHIJ Maths, x41373839474234354849 Physics, y36203124354226272937 The regression line for y on x is Residual of point A = ( The residual is negative if the point is below the line.) To find y, substitute the value of x at point A into the regression line:

5. Calculating Residuals x SUMMARY To find the residual for a particular observation, A, calculate the y -coordinate on the regression line corresponding to the x -value at A, find The residual is negative if the point is below the line Since, the residual at A is also given by

6. Calculating Residuals Outliers Outliers are points that lie well away from the regression line. Since a residual measures the distance of a point from a line, residuals are used to identify outliers. Outliers can have a considerable effect on a regression line and make it unreliable.

7. Calculating Residuals e.g. The diagram is a scatter diagram of the data shown in the table. 38 77 116 125 144 123 182 51 yx If we were to draw the line “by eye”, the 1 st point... would lie well away from the line we would want to draw. However, the calculation of the regression line includes the 1 st point and distorts the position of the line.

8. Calculating Residuals The diagram shows the y on x regression line for all the data. The residuals are shown by the red lines. 38 77 116 125 144 123 182 51 yx The left-hand end of the line is further down than it would be without the 1 st point. e.g. The diagram is a scatter diagram of the data shown in the table.

9. Calculating Residuals xy 15 218 312 414 512 611 77 83 Removing the 1 st point... e.g. The diagram is a scatter diagram of the data shown in the table.

10. Calculating Residuals xy 15 218 312 414 512 611 77 83 e.g. The diagram is a scatter diagram of the data shown in the table. Removing the 1 st point gives

11. Calculating Residuals The sum of the squares of the residuals, Without the 1 st point, we have a regression line that is a much better fit. e.g. The diagram is a scatter diagram of the data shown in the table. Removing the 1 st point gives

12. Calculating Residuals Exercise (a) Find the equation of the regression line of y on x (b) Estimate the percentage of accidents to children in an area with 10 % open space. (c) Find the residual for A. 1. The table shows the number of accidents to children as a percentage of those to adults, y, in 9 areas of London together with the percentage of open space in those areas, x. 17·123·830·833·63738·24042·946·3 Children’s Accidents (%) 14·814·66·36·35·25·24·54·571·41·41·35Open Spaces(%) IHGFEDCBA

13. Calculating Residuals Solutions (a) Find the equation of the regression line of y on x (b) Estimate the percentage of accidents to children in an area with 10% open space. (c) Find the residual for A. (a) The equation of the regression line for y on x is Solution: (b) Nearly 29% of accidents will involve children. (c) At, Residual =

15. Calculating Residuals The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

16. Calculating Residuals calculate the y -coordinate on the regression line corresponding to the x -value at A, x SUMMARY To find the residual for a particular observation, A, The residual is negative if the point is below the line find Since, the residual at A is also given by

17. Calculating Residuals e.g. The marks for 10 students in Maths and Physics are as follows: 36 41 A 20 37 B 31 38 C 24 39 D 35 47 E 42 F 26 34 G 27 35 H 29 48 I 37 49 J Physics, y Maths, x The regression line for y on x is Residual of point A = ( The residual is negative if the point is below the line.) To find y, substitute the value of x at point A into the regression line:

18. Calculating Residuals Outliers Outliers are points that lie well away from the regression line. Since a residual measures the distance of a point from a line, residuals are used to identify outliers. Outliers can have a considerable effect on a regression line and make it unreliable.

19. Calculating Residuals e.g. The diagram is a scatter diagram of the data shown in the table. 38 77 116 125 144 123 182 51 yx If we were to draw the line “by eye”, the 1 st point... would lie well away from the line we would want to draw. However, the calculation of the regression line includes the 1 st point and distorts the position of the line.

20. Calculating Residuals e.g. The diagram shows the y on x regression line for the data in the table. The residuals are shown by the lines parallel to the y -axis. The sum of the squares of the residuals, Without the 1 st point, we have a regression line that is a much better fit. The 1 st point has the largest residual.