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Changing the unit of measurement

Changing the unit of measurement . AP Statistics. 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

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Changing the unit of measurement

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  1. Changing the unit of measurement AP Statistics

  2. 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

  3. 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

  4. Page 56 1.45A school system employs teachers at salaries between $30000 and $60000. 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

  5. Page 56 #1.46Suppose 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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