Changing the unit of
measurement
AP Statistics
Linear transformation
A linear transformation changes the
original variable x into a new variable
given by x
an
newequation of the form
 a  bx a shifts all
Adding thexnew
constant
values of x upward or downward by
the same amount
Multiplying by the positive constant b
changes the size of the unit of
measurement
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
Page 56 1.45
A 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
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 xnew  1.05x
, where x = the old salary and xnew = the
new salary. Both IQR and s will increase
by a factor of 1.05