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**Changing the unit of measurement**

AP Statistics

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**Linear transformation**

A linear transformation changes the original variable x into a new variable given by an equation of the form Adding the constant a shifts all values of x upward or downward by the same amount Multiplying by the positive constant b changes the size of the unit of measurement

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**Effect of a linear transformation**

Multiplying each observation by a positive number b multiplies both measures of center (mean and median) and measures of spread (standard deviation and IQR) by b Adding the same number a (either positive or negative) to each observation adds a to measures of center and to quartiles but does not change measures of spread

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**How much will the mean salary increase? The median salary? **

Page A school system employs teachers at salaries between $30000 and $ The teacher’s union and the school board are negotiating the form of next year’s increase in the salary schedule. Suppose that every teacher is given a flat $1000 raise. How much will the mean salary increase? The median salary? The mean and the median will both increase by 1000 Will a flat $1000 raise increase the spread as measured by the distance between the quartiles? No. Each Quartile will increase by 1000 therefore the difference will remain the same Will a flat $1000 raise increase the spread as measured by the standard deviation of the salaries? No, the standard deviation remains unchanged when the same amount is added to each data value

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Page 56 #1.46 Suppose that the teachers in the previous exercise each receive a 5% raise. The amount of the raise will vary from $1500 to $3000, depending on present salary. Will a 5% raise across-the-board increase the spread of the distribution as measured by the distance between the quartiles? Do you think it will increase the standard deviation? A 5% across-the –board raise will cause both the IQR and s to increase. The transformation being applied here is , where x = the old salary and = the new salary. Both IQR and s will increase by a factor of 1.05

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Looking at data: distributions - Describing distributions with numbers IPS chapter 1.2 © 2006 W.H. Freeman and Company.

Looking at data: distributions - Describing distributions with numbers IPS chapter 1.2 © 2006 W.H. Freeman and Company.

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