Effects of Linear Transformations

This section covers the effects of linear transformations on measures of central tendency and variability.

Example

Table below shows the temperatures of 5 cities to see the Linear transformation: temperatures of cities.

City	Degrees Fahrenheit	Degrees Centigrade
Houston	54	12.22
Chicago	37	2.78
Minneapolis	31	-0.56
Miami	78	25.56
Phoenix	70	21.11
Mean	54.000	12.220
Median	54.000	12.220
Variance	330.00	101.852
SD	18.166	10.092

To transform the degrees Fahrenheit to degrees Centigrade, we use the formula

C = 0.556F - 17.778

To get the mean in Centigrade, you multiply the mean temperature in Fahrenheit by 0.556 and then subtract 17.778 .

(0.556)(54) - 17.778 = 12.22.

• The same is true for the median.
• This relationship holds even if the mean and median are not identical as they are in the table above.

• The formula for the standard deviation is just as simple: the standard deviation in degrees Centigrade is equal to the standard deviation in degrees Fahrenheit times 0.556.
• Since the variance is the standard deviation squared, the variance in degrees Centigrade is equal to 0.5562 times the variance in degrees Fahrenheit.

If a variable X has a mean of μ, a standard deviation of σ, and a variance of σ2, then a new variable Y created using the linear transformation

Y = bX + A
will have a mean of bμ+A, a standard deviation of bσ, and a variance of b2σ2.

Quiz

1 You have 10 numbers. The mean is 5. You multiply each number in the group by 3. What is the new mean?

Answer >>

15

5x3 is 15

2 You have 8 numbers. The mean is 6. You add 5 to each number in the group. What is the new mean?

Answer >>

6+5 is 11

3 You have 12 numbers. The mean is 3, and the variance is 1. You multiply each number by 4. What is the new variance?

Answer >>

1*(4²) is 16

4 You have 15 numbers. The mean is 10, and the variance is 4. You multiply each number by 3. What is the new standard deviation?

Answer >>

3 x sqrt(4) is 6

5 Your teacher decides that he will add 10 points to all of your test grades. Which statistic is not changed by this decision?

	mean
	median
	mode
	standard deviation

Answer >>

standard deviation

The mean, median, and mode will all increase by 10, but the standard deviation remains unchanged.