Question

Calculate and interpret the z-scores using R. Using the mtcars dataset in R. To complete the assignment, follow the steps below:

1. In the R console, type mtcars. This shows the whole dataset.
2. Using R, calculate the mean and standard deviation for the variables: mpg, cyl, disp, hp, drat, wt, qsec, gear, carb.
3. Using R, calculate the maximum values for the same variables in (b).
4. Using R, calculate the z-scores for the maximum values.
5. Interpret each z-score in (d). Is the maximum value unusual? Explain why or why not.

Answer 1

Whole dataset for mtcars:

Calculate the mean and standard deviation for the variables : mpg,cyl,disp,hp,drat,wt,qsec,gear,carb

• First you need to attach all the variables to the mtcars dataset using attach function.
• To find mean use the function mean() and for standard deviation use sd().

Calculate the maximum values for the variables: mpg,cyl,disp,hp,drat,wt,qsec,gear,carb

• To find maximum value of a variable use max().

Calculate Z-score for the maximum values:

• Z-score is also called standard score. It is calculated from mean and standard deviation.
• To calculate Z-score of a particular value(X), the formula is

                                              

here is mean of the variable

is standard deviation of the variable.

• Z-score for maximum value of mpg:

                here z-score value of maximum value of mpg is stored in 'z'

• Z-score for maximum value of cyl:

                  Z-score value is stored in 'a'

• Z-score for maximum value of disp:

                   Z-score value is stored in 'b'

• Z-score for maximum value of hp:

                   Z-score value is stored in 'c'

• Z-score for maximum value of drat:

                      Z-score value is stored in 'd'

• Z-score for maximum value of wt:

                    Z-score value is stored in 'e'

• Z-score for maximum value of qsec:

                     Z-score value is stored in 'f'

• Z-score for maximum value of gear:

                      Z-score value is stored in 'g'

• Z-score for maximum value of carb:

                      Z-score value is stored in 'h'

• We have already calculated mean,standard deviation and maximum values so you can directly calculate z-score value. and store it in different variables.

Interpreting each Z-score for each variable:

• The z-score value for maximum value of mpg is 2.29402.

It means the car which is having maximum value of mpg are having nearly 2.30 standard deviations from the mean of mpg.

• The z-score value for maximum value of cyl is 1.022

It means the car which is having maximum value of cyl are having 1.022 standard deviations from the mean of cyl.

• The z-score value for maximum value of disp is 1.94.

It means the car which is having maximum value of disp are having nearly 2 standard deviations from the mean of disp.

• The z-score value for maximum value of hp is 2.74.

It means the car which is having maximum value of hp are having 2.74 standard deviations from the mean of hp.

• The z-score value for maximum value of drat is 2.52.

It means the car which is having maximum value of drat are having nearly 2.52 standard deviations from the mean of drat.

• The z-score value for maximum value of wt is 2.28

It means the car which is having maximum value of wt are having nearly 2.30 standard deviations from the mean of wt.

• The z-score value for maximum value of qsec is 2.84

It means the car which is having maximum value of qsec are having nearly 2.84 standard deviations from the mean of qsec.

• The z-score value for maximum value of gear is 1.80

It means the car which is having maximum value of gear are having nearly 1.80 standard deviations from the mean of gear.

• The z-score value for maximum value of carb is 3.22

It means the car which is having maximum value of carb are having nearly 3.22 standard deviations from the mean of carb.

Is the maximum value unusual?

Yes the maximum values are different because different cars will have different values based on quality and quantity.

For example, duster 360 is having maximum cyl value but its am is having minimum value.

Answer 2

Whole dataset for mtcars:

Calculate the mean and standard deviation for the variables : mpg,cyl,disp,hp,drat,wt,qsec,gear,carb

• First you need to attach all the variables to the mtcars dataset using attach function.
• To find mean use the function mean() and for standard deviation use sd().

Calculate the maximum values for the variables: mpg,cyl,disp,hp,drat,wt,qsec,gear,carb

• To find maximum value of a variable use max().

Calculate Z-score for the maximum values:

• Z-score is also called standard score. It is calculated from mean and standard deviation.
• To calculate Z-score of a particular value(X), the formula is

                                              

here is mean of the variable

is standard deviation of the variable.

• Z-score for maximum value of mpg:

                here z-score value of maximum value of mpg is stored in 'z'

• Z-score for maximum value of cyl:

                  Z-score value is stored in 'a'

• Z-score for maximum value of disp:

                   Z-score value is stored in 'b'

• Z-score for maximum value of hp:

                   Z-score value is stored in 'c'

• Z-score for maximum value of drat:

                      Z-score value is stored in 'd'

• Z-score for maximum value of wt:

                    Z-score value is stored in 'e'

• Z-score for maximum value of qsec:

                     Z-score value is stored in 'f'

• Z-score for maximum value of gear:

                      Z-score value is stored in 'g'

• Z-score for maximum value of carb:

                      Z-score value is stored in 'h'

• We have already calculated mean,standard deviation and maximum values so you can directly calculate z-score value. and store it in different variables.

Interpreting each Z-score for each variable:

• The z-score value for maximum value of mpg is 2.29402.

It means the car which is having maximum value of mpg are having nearly 2.30 standard deviations from the mean of mpg.

• The z-score value for maximum value of cyl is 1.022

It means the car which is having maximum value of cyl are having 1.022 standard deviations from the mean of cyl.

• The z-score value for maximum value of disp is 1.94.

It means the car which is having maximum value of disp are having nearly 2 standard deviations from the mean of disp.

• The z-score value for maximum value of hp is 2.74.

It means the car which is having maximum value of hp are having 2.74 standard deviations from the mean of hp.

• The z-score value for maximum value of drat is 2.52.

It means the car which is having maximum value of drat are having nearly 2.52 standard deviations from the mean of drat.

• The z-score value for maximum value of wt is 2.28

It means the car which is having maximum value of wt are having nearly 2.30 standard deviations from the mean of wt.

• The z-score value for maximum value of qsec is 2.84

It means the car which is having maximum value of qsec are having nearly 2.84 standard deviations from the mean of qsec.

• The z-score value for maximum value of gear is 1.80

It means the car which is having maximum value of gear are having nearly 1.80 standard deviations from the mean of gear.

• The z-score value for maximum value of carb is 3.22

It means the car which is having maximum value of carb are having nearly 3.22 standard deviations from the mean of carb.

Is the maximum value unusual?

Yes the maximum values are different because different cars will have different values based on quality and quantity.

For example, duster 360 is having maximum cyl value but its am is having minimum value.

Answer 3

Whole dataset for mtcars:

Calculate the mean and standard deviation for the variables : mpg,cyl,disp,hp,drat,wt,qsec,gear,carb

• First you need to attach all the variables to the mtcars dataset using attach function.
• To find mean use the function mean() and for standard deviation use sd().

Calculate the maximum values for the variables: mpg,cyl,disp,hp,drat,wt,qsec,gear,carb

• To find maximum value of a variable use max().

Calculate Z-score for the maximum values:

• Z-score is also called standard score. It is calculated from mean and standard deviation.
• To calculate Z-score of a particular value(X), the formula is

                                              

here is mean of the variable

is standard deviation of the variable.

• Z-score for maximum value of mpg:

                here z-score value of maximum value of mpg is stored in 'z'

• Z-score for maximum value of cyl:

                  Z-score value is stored in 'a'

• Z-score for maximum value of disp:

                   Z-score value is stored in 'b'

• Z-score for maximum value of hp:

                   Z-score value is stored in 'c'

• Z-score for maximum value of drat:

                      Z-score value is stored in 'd'

• Z-score for maximum value of wt:

                    Z-score value is stored in 'e'

• Z-score for maximum value of qsec:

                     Z-score value is stored in 'f'

• Z-score for maximum value of gear:

                      Z-score value is stored in 'g'

• Z-score for maximum value of carb:

                      Z-score value is stored in 'h'

• We have already calculated mean,standard deviation and maximum values so you can directly calculate z-score value. and store it in different variables.

Interpreting each Z-score for each variable:

• The z-score value for maximum value of mpg is 2.29402.

It means the car which is having maximum value of mpg are having nearly 2.30 standard deviations from the mean of mpg.

• The z-score value for maximum value of cyl is 1.022

It means the car which is having maximum value of cyl are having 1.022 standard deviations from the mean of cyl.

• The z-score value for maximum value of disp is 1.94.

It means the car which is having maximum value of disp are having nearly 2 standard deviations from the mean of disp.

• The z-score value for maximum value of hp is 2.74.

It means the car which is having maximum value of hp are having 2.74 standard deviations from the mean of hp.

• The z-score value for maximum value of drat is 2.52.

It means the car which is having maximum value of drat are having nearly 2.52 standard deviations from the mean of drat.

• The z-score value for maximum value of wt is 2.28

It means the car which is having maximum value of wt are having nearly 2.30 standard deviations from the mean of wt.

• The z-score value for maximum value of qsec is 2.84

It means the car which is having maximum value of qsec are having nearly 2.84 standard deviations from the mean of qsec.

• The z-score value for maximum value of gear is 1.80

It means the car which is having maximum value of gear are having nearly 1.80 standard deviations from the mean of gear.

• The z-score value for maximum value of carb is 3.22

It means the car which is having maximum value of carb are having nearly 3.22 standard deviations from the mean of carb.

Is the maximum value unusual?

Yes the maximum values are different because different cars will have different values based on quality and quantity.

For example, duster 360 is having maximum cyl value but its am is having minimum value.