Warm-up
O Make sure to use a ruler and proper scaling
for the box-plots.
O This will be taken up for a grade!
O Today we start the last chapter before our
big test!
O Start thinking about what you want to put on
your index card for the test.
Objectives
O Learn how shifting and rescaling affect the
measures of center and spread.
O Describe a Normal model and discuss when
it is appropriate to use.
O Learn how to standardize a Normal model by
calculating z-scores.
O Describe why standardizing the model is
critical when comparing data sets.
I. Shifting & Rescaling Data
A. Shifting data
o Moves the distribution to the R or L (translation) by
adding or subtracting a constant value from each
data point.
Example 1
O A summary of what I pay my hourly workers
at my company are in the table below. What
Pay
Pay + raise
happens to this summary if I decide to give
Min
12
everyone a $10/hr raise?
Q1
15
Med
22
Mean
30
Q3
55
Max
95
IQR
Sx
15
What does shifting do to…
O Measures of spread?
O Measures of position including center?
O Shape?
I. Shifting & Rescaling Data
B. Rescaling data
o Dilates the distribution by multiplying or
dividing by a common factor.
Example 2
O A summary of what I pay my hourly workers at my
company are in the table below. What happens to
this summary if I decide to give everyone a 20%
raise?
Pay
Pay + raise
Min
12
Q1
15
Med
22
Mean
30
Q3
55
Max
95
IQR
Sx
15
What does rescaling do to…
O Measures of spread?
O Measures of position including center?
O Shape?
You try!!
O Suppose the class took a 40 point quiz.
Results show a mean score of 30, median
32, IQR 8, SD 6, min 12, and Q1 27.
(Suppose YOU got a 35.) What happens to
each of the statistics if…..
O I decide to weight the quiz 50 points, and will
add 10 points to every score.
O I decide to weight the quiz as 80 points, and
double each score.
O I decide to count the quiz as 100 points; so
the scores are doubled and 20 points are
added.
I. Normal Distribution
A. Characteristics
o Unimodal, symmetric, bell-shaped.
o The mean and median are the same and
located in the middle of the peak.
o Are described by the mean and standard
deviation N(µ, σ) : where µ = the mean and
σ = std dev
II. Normal Distribution
B. Importance
o Are good descriptions for some distributions of real
data. (IQ, SAT’s, characteristics in biological
populations)
o Are good approximations to the results of many
kinds of chance outcomes.
o Many statistical inferences are based on the
normal curve model.
II. Normal Distribution
C. 68-95-99.7 RULE
Example 3
O If µ = 72 and σ = 4, What percent of the
class scored between 68 and 76?
Example 3
O What percent of students failed?
O What percent of students scored above an 80?
O Is it likely that someone got an ‘A’ on the test?
III. Standardizing Normal
Distribution
O Has a mean of 0 and a standard deviation of 1.
O To standardize data, convert each to a z-score
x x

z
s
x = value (data point)
𝑥= mean
s = standard deviation
III. Standardizing Normal
Distribution
Example 4
O The Virginia Cooperative Extension reports that
the mean weight of yearling Angus steers is 1152
pounds. Suppose the weights of all such animals
can be described by a Normal model with a
standard deviation of 84 pounds.
O a) How many standard deviations from the mean
would a steer weighing 1000 pounds be?
O b) Which would be more unusual, a steer
weighing 1000 pounds, or one weighing 1250
pounds?