Wize University Statistics Textbook > Discrete Random Variables

Linear Transformation of Data

Transforming Data

The measure of central tendency and measure of spread could change if we shift or scale the values in a given data set.

Shifting: adding or subtracting the same constant $c$ to every data value

3\ \rightarrow\ 3\ +c

Scaling: multiplying or dividing every data value by the same constant $d$

3\ \rightarrow\ 3d

Shifting & Scaling: multiplying/dividing every data value by the same constant $d$ and adding/subtracting the same constant $c$

3\ \rightarrow\ 3d+c

Example

Image illustrating Transforming Data: Shifting & Scaling

25, 60, 65

What happens to the data set?

Example

$Mean(y+5) = 10+5 = 15 \\Mean(3y) = 3y = 3(10)=30 \\Mean(3y+5)=3(10)+5=35$

$Median(y+5) = 8+5 = 13 \\Median(3y) = 3y = 3(8)=24 \\Median(3y+5)=3(8)+5=29$

$SD(y+5) = 6.325 \\SD(3y) = 3y = 3(6.325)=18.975 \\SD(3y+5)=3(6.325)=18.975$

You multiply all values in a data set by $5$ and then add four. The resulting mean is 34 and the resulting standard deviation is 25.

Determine the mean and standard deviation of the original data.

Original mean was 6 and standard deviation was 5

Original mean was 174 and standard deviation was 125

Original mean was 6 and standard deviation was 1

Original mean was 6 and standard deviation was 9

Original mean was 2.8 and standard deviation was 5.2

I don't know

Linear Transformation

Given a linear equation $Y=c+dX$ where $c$ is the shifting constant and $d$ is the scaling constant, then:

\mu_Y=c+d\mu_X \\\sigma^2_Y=d^2\sigma^2_X \\\sigma_Y=|d| \sigma_X

Example

Suppose that a data set has a mean of 8 and a standard deviation of 3. You wish to multiply all of the numbers by 4 and then subtract 3.

Determine the new mean and standard deviation.

We are transforming $X\rightarrow Y$

Let $X$ represent the first data set and let $Y$ represent the new data set.

Then $Y=4X-3$

New mean: $\overline{Y}=4(8)-3=29$
New standard deviation: $s_Y=4\times3=12$

Example: Linear Transformation

Jenny is a hairdresser at Swish Salon. She pays the salon a fixed fee of $5,000 to rent a chair (this is subtracted from her profit). In addition, the salon gets 15% of her revenue (which means she keeps 85% of her revenue). Suppose Jenny makes an average of $60,000 in revenue with a standard deviation of $8,000. What is the mean and standard deviation of her profit?

Hint: Profit = Revenue - Costs

$Y=profit$

$X=revenue$

$c=-5000$ (fixed cost, subtracted from profit)

$d=0.85$ (she keeps 85% of the revenue because the salon gets 15% of the revenue)

Profit = 0.85(Revenue) - 5000

$Y=0.85X-5000$

Mean(profit):

\mu_Y=c+d\mu_X \\ \mu_Y=-5000+0.85(60,000)=46,000

SD(profit):

\\\sigma_Y=|d| \sigma_X \\ \sigma_Y=0.85(8000)=6800

You are working overseas in Tokyo in as a consultant. Your average annual earning is ¥5,000,000 with a standard deviation of ¥1,500,000. The exchange rate is ¥1 = $0.009 USD. Each year, you must pay your agency a fixed fee of $10,000 USD.

Find the mean and standard deviation of your annual net earnings in US dollars.

(a) Mean annual net earnings in US dollars. [Enter whole number without commas, e.g. 12000](b) Standard deviation of annual net earnings in US dollars. [Enter whole number without commas, e.g. 6000]

I don't know

Linear Transformation of Data

Create a free account to get started

Transforming Data

Create a free account to get started

Create a free account to get started

Linear Transformation

Create a free account to get started

Example: Linear Transformation

Create a free account to get started