The Median of Grouped Data © Christine Crisp “Teach A Level Maths” Statistics 1.
Linear Scaling of Regression Data © Christine Crisp Teach A Level Maths Statistics 1.
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Transcript of Linear Scaling of Regression Data © Christine Crisp Teach A Level Maths Statistics 1.
Linear Scaling of Linear Scaling of Regression DataRegression Data
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”Statistics 1Statistics 1
Linear Scaling of Regression Data
"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
EDEXCELOCR
Linear Scaling of Regression Data
We may want to change the units that have been used when collecting data.
For example, we may want kilometres instead of miles or kilograms instead of pounds.
Sometimes we may simply want to reduce the size of the numbers in data items.
In both these cases we talk about scaling or coding the data.
When dealing with regression lines, we can alter a regression line to different units without converting the original data.
Linear Scaling of Regression DataA researcher has the following data giving the average daily maximum temperature for each month in 1980 and the corresponding figures for the sales of milk in a supermarket.
J F M A M J J A S O N D
Temp (°F), F 40 38 46 55 60 70 65 69 64 58 47 49
Sales ( thousands of
pints ), y8 9 8 5 6 3 4 2 3 7 8 10
The y on F regression line is
Fy 213408417 The correlation coefficient is 0·89. Recent data are measured in degrees Celsius and thousands of litres and the researcher wants to compare the sets of results by converting the older ones.
Linear Scaling of Regression Data
761
yY
To convert from pints to litres, we must divide by 1·76 so, if Y is the new variable,
The conversion from degrees Fahrenheit to degrees Celsius is
)32(9
5 FC
As the conversions are both linear, instead of converting all the data we can simply substitute into the regression line.We first need to rearrange both conversion equations.
Linear Scaling of Regression Data
761
yY
)32(9
5FC
Yy 761
325
9FC 32
5
9 CF
As we have only a small amount of data, we can check the effect of the scaling by converting the data and drawing both regression lines.
Substituting in
Fy 213408417
32
5
9213408417761 CY
CY 384100111761 CY 21830266
836384108417761 CYSimplifying:
Linear Scaling of Regression Data
The correlation coefficient is a measure of the spread of the data so is not altered by linear scaling.( Although the scales are different on the
diagrams, we can see that the scatter of the points is unchanged. )
CY 21830266 Fy 213408417 Product Moment Correlation Coefficient
(p.m.c.c.)
Graphs showing milk sales against temperature
Linear Scaling of Regression Data
e.g.2 A set of data connects two variables, p and t. However, in order to calculate a regression line, the data has been coded using the formulae
x 60t and y p 1000
If the regression line of y on x is y = 3·29 + 4·15xfind the equation of the regression line for p
on t.
Solution:
Substitute for y and x:
xy 154293 )60(1542931000 tp
tp 249291003
Linear Scaling of Regression Data
SUMMARY
To convert a scaled or coded regression equation, substitute for the variables using the conversion formulae.
The product moment correlation coefficient (p.m.c.c.) is not changed by linear scaling.
Linear Scaling of Regression DataExercise1. Data for 2 variables, v and z have been scaled
using the formulae
1000zy,100
vx
If the equation of the resulting y on x regression line is xy 162440 find the equation of the regression line of v
on z.
Solution:
1001624401000
vz
vz 02160441000
xy 162440 becomes
Linear Scaling of Regression DataExercise
(b) Use your answer to (a) to find the equation of the regression line of t on m.
(a) Find the equation of the y on x regression line, giving the values of the constants correct to 2 d.p.
(c) What effect would the conversion have on the product moment correlation coefficient?
2. A company rep. records the distance travelled, m miles, and time taken, t minutes, for 5 journeys. The data were converted to kilometres and hours and are summarised below. The formulae for the conversions are
60
ty
5
8mx
1422 x 432 y 10326 yx4624642 x
5n
and
Linear Scaling of Regression Data
1422 x 432 y 10326 yx4624642 x
(b) The equation of the regression line of t on m.
(a) The y on x regression line:
(c) The conversion has no effect on the p.m.c.c.
xy 020031
n
yxyxS xy
n
xxS xx
22
xx
xy
S
Sb xbya
bxay
441111 258047
01910 03451
60
ty ,
5
8mx
5
8020031
60
mt
mt 921861
xy 020031 becomes
so
Solution:
Linear Scaling of Regression Data
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.
Linear Scaling of Regression Data
We may want to change the units that have been used when collecting data.
For example, we may want kilometres instead of miles or kilograms instead of pounds.
Sometimes we may simply want to reduce the size of the numbers in data items.
In both these cases we talk about scaling or coding the data.
When dealing with regression lines, we can alter a regression line to different units without converting the original data.
Linear Scaling of Regression Data
A researcher has the following data from 1980 giving the monthly average daily maximum temperature and sales of milk in a supermarket.
1087324365898Sales
( thousands of pints ), y
494758646965706055463840Temp (°F), F
NM J J A S O DAMFJ
The y on F regression line is
Fy 213408417 The correlation coefficient is 0·89. Recent data are measured in degrees Celsius and thousands of litres and the researcher wants to compare the sets of results by converting the older ones.
Linear Scaling of Regression Data
761
yY
To convert from pints to litres, we must divide by 1·76 so, if Y is the new variable,
The conversion from degrees Fahrenheit to degrees Celsius is
)32(9
5 FC
As the conversions are both linear, instead of converting all the data we can simply substitute into the regression line.We first need to rearrange both conversion equations.
Linear Scaling of Regression Data
761
yY
)32(9
5FC
Yy 761
325
9FC 32
5
9 CF
As we have only a small amount of data, we can check the effect of the scaling by converting the data and drawing both regression lines.
Substituting in
Fy 213408417
32
5
9213408417761 CY
CY 384100111761 CY 21830266
836384108417761 CYSimplifying:
Linear Scaling of Regression Data
The correlation coefficient is a measure of the spread of the data so is not altered by linear scaling.( Although the scales are different on the
diagrams, we can see that the scatter of the points is unchanged. )
CY 21830266 Fy 213408417
Product Moment Correlation Coefficient (p.m.c.c.)
Graphs showing milk sales against temperature
Linear Scaling of Regression Data
SUMMARY
To convert a scaled or coded regression equation, substitute for the variables using the conversion formulae.
The product moment correlation coefficient (p.m.c.c.) is not changed by linear scaling.
