Question

Suppose the mean score on the first exam is a 60 out of 100. I would like to curve the exam to force the mean score to be an 80 out of 100. If I added 20 to everyone’s score, would the SD of the exam scores change? How about if I multiplied 4/3 to everyone’s score? Explain.

Part 2: Continuing from Part 1, fill out the formulas put below that will always be true. Let a = constant (- or +) and X = random variable.

EV (a + X) = SD (a + X) = SD (aX) =

EV (aX) = Variance (a + X) = Variance (a + X) =

Answer 1

let scores of student be x before adding 20 to each

now after adding 20 it it become X+20 let Y=X+20

we know that Variance(aX+b)=Var(aX)+Var(b)+2Cov(aX,b)

but a and b are constants or -1,+1 hence Var(b)=0 and Cov(aX,b)=0 and Var(aX)=(a^2)*Var(X)

hence shift of origin doesnt affect on Variance and change of scale changes variance by square of multiplier

i.e. Var(ax+b)=a^2 Var(x)

standard deviation(X)=(Var(X))^0.5

hence standard deviation also remains unchanged for shift of origin

but sd(aX+b)=|a|*sd(X) where |a|=absolute value of a

sd(Y)=sd(X+20)=sd(X)

hence after adding 20 marks to each score standard deviation remains unchanged

while, sd(4/3*X)=4/3*sd(x)

after multiplying each score by 4/3 standard deviation will be 4/3 times of earlier standrd deviation

##Part 2

E(a+X)=a+E(X)

Var(a+X)=Var(X)

sd(a+X)=sd(X)

E(aX)=aE(X)

Var(aX)=a^2 Var(X)

sd(aX)=|a|*sd(X)

E(aX+b)=aE(X)+b

Var(aX+b)=a^2 Var(X)

sd(aX+b)=|a|*sd(X)