Psychological Statistics

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The answers are in the attached Excel File.
1. If an original set of measurements is made in inches and has a mean of 36 and a variance of 144, what will be the mean, variance, and standard deviation if:

If a constant is added to or subtracted from each measurement, there is no effect on resulting variance or standard deviation; the resulting mean is obtained by increasing or decreased the original mean by that constant
If each measurement is multiplied or divided by a constant, the resulting mean is obtained multiplying or dividing the original mean by the constant; the resulting variance is obtained by multiplying or dividing the original variance by the square of the constant; the resulting standard deviation is obtained by multiplying or dividing the original standard deviation by the absolute value (magnitude or number without sign) of the constant.

a) the unit is changed to feet?

Mean= 36 inches
Variance = 144 square inches
Standard deviation = square root of variance= 12 inches =√144

If we change the units to feet, we will have to divide the measurements by 12
Resulting mean and standard deviation are obtained by dividing the original values by 12
and resulting variance is obtained by dividing the original value by square of 12 i.e. 144 =12^2 (^2 means square of)

Resulting values

Mean= 3 feet
Variance = 1 square feet
Standard deviation = square root of variance= 1 feet =√1

b) 6 inches must be added to each measurement to correct an error?

Mean= 36 inches
Variance = 144 square inches
Standard deviation = square root of variance= 12 inches =√144

If we add a constant value, mean increases by the constant value, standard deviation and variance remain constant 12
Resulting mean is obtained by increasing the original value by 6

Resulting values

Mean= 42 inches
Variance = 144 square inches
Standard deviation = square root of variance= 12 inches =√144

c) The measurements are converted to centimeters (2.5 cm = 1 inch) and then 15 cm is subtracted from each measurement to correct an error?

Mean= 36 inches
Variance = 144 square inches
Standard deviation = square root of variance= 12 inches =√144

Mean is obtained by multiplying the original mean by 2.5 and subtracting 15
Standard deviation is obtained by multiplying the original standard deviation by 2.5
Variance is obtained by multiplying the original variance by 6.25 =2.5^2

Resulting values

Mean= 75 cm
Variance = 900 square cm
Standard deviation = square root of variance= 30 cm =√900

2. Calculate the standard scores for each of the members of the following distribution of shoe sizes:

a) 5, 7, 7, 8, 9, 12

To convert a measurement to a standard score, first subtract the mean score from it and then divide this remainder by the standard deviation.
z=(x-Mean)/Standard deviation

Therefore, first we calculate the mean and standard deviation of the shoe sizes

Mean and standard deviation

X= X 2 =
5 25
7 49
7 49
8 64
9 81
12 144
Total= 48 412
n=no of ...