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Wize University Statistics Textbook > Discrete Random Variables

Linear Transformation of Data

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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 cc to every data value

3 → 3 +c3\ \rightarrow\ 3\ +c


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

3 → 3d3\ \rightarrow\ 3d


Shifting & Scaling: multiplying/dividing every data value by the same constant dd and adding/subtracting the same constant cc

3 → 3d+c3\ \rightarrow\ 3d+c


Example

25, 60, 65

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What happens to the data set?




Watch Out!
When we talk about transforming the spread, we are talking about range, IQR, and standard deviation, but NOT variance!
If dd is a negative value, multiply the spread by the absolute value ∣d∣\mid d\mid.

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Example


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





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





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





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You multiply all values in a data set by 55 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.



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Linear Transformation

Given a linear equation Y=c+dXY=c+dX where cc is the shifting constant and dd is the scaling constant, then:

μY=c+dμXσY2=d2σX2σY=∣d∣σX\mu_Y=c+d\mu_X \\\sigma^2_Y=d^2\sigma^2_X \\\sigma_Y=|d| \sigma_X

Wize Concept
The standard deviation σ\sigma is the square-root of the variance σ2\sigma^2.


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→YX\rightarrow Y
Let XX represent the first data set and let YY represent the new data set.
Then Y=4X−3Y=4X-3
  • New mean: Y‾=4(8)−3=29\overline{Y}=4(8)-3=29
  • New standard deviation: sY=4×3=12s_Y=4\times3=12


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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=profitY=profit
X=revenueX=revenue

c=−5000c=-5000 (fixed cost, subtracted from profit)
d=0.85d=0.85 (she keeps 85% of the revenue because the salon gets 15% of the revenue)

Profit=0.85(Revenue)−5000Profit = 0.85(Revenue) - 5000
Y=0.85X−5000Y=0.85X-5000

Mean(profit):
μY=c+dμXμY=−5000+0.85(60,000)=46,000\mu_Y=c+d\mu_X \\ \mu_Y=-5000+0.85(60,000)=46,000

SD(profit):
σY=∣d∣σXσY=0.85(8000)=6800 \\\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.