MATH 1107
Elementary Statistics
Lecture 5
The Standard Deviation
as a Ruler
MATH 1107 –
Using the STD DEV for Relative Standing
Following from the example in your book (page
100)…is it a more impressive performance to
win the 800 meters by 8 seconds or the long
jump by 60 cm?
Was Babe Ruth a better hitter than Barry
Bonds?
MATH 1107 –
Using the STD DEV for Relative Standing
The only way to compare values in different units is to
standardize the deviations from the means. In other
words, we first have to convert all of the values into
similar units – standard deviations from the respective
means. THEN, we can compare them directly. This is
done through the application of a Z-score:
Value of
interest
(y
–
y)
z=
s
Mean of
data
Std dev
of data
MATH 1107 –
Using the STD DEV for Relative Standing
Continuing with the example in the book (page 101):
The winning 800m time of
129 second was 8
seconds better than the
mean of 137 seconds.
The std dev of all times
was 5 seconds. So:
The winning long jump
was 60cm longer than
the average 6m jump.
The std dev of all jumps
was 30cm. So:
(129-137)/5 = -1.6
60/30 = 2
In other words, the
winning time was 1.6
standard deviations
below the mean.
In other words, the
winning jump was 2
standard deviations
above the mean.
MATH 1107 –
Using the STD DEV for Relative Standing
Example from page 105:
The SAT scores have a mean of 1000 and a std dev of 200.
The ACT scores have a mean of 20.8 and a std dev of 4.8.
What ACT score would be equivalent to an SAT score of
1220?
First, lets convert the SAT score of 1220 to a Z-score:
(1220-1000)/200 = 1.10
Next, lets solve for the ACT score that has a Z-score of 1.10:
(x-20.8)/4.8 = 1.10 X=26.08
MATH 1107 –
Using the STD DEV for Relative Standing
Spot the Jack Russell weighs 19 pounds. The mean weight
for a Jack Russell Terrier is 16 pounds with a std dev of 1.5
pounds. Desdi the Maine Coon cat weighs 18 pounds
and frequently kicks Spot’s but around the house. The
mean weight for a Maine Coon is 17 pounds with a std
dev of .75 pounds. Which animal is most in need of a
diet?
MATH 1107 –
Using the STD DEV for Relative Standing
The 68-95-99.7 Rule (page 107):
Many distributions in life are “normal” or bell shaped.
When data follows a normal curve, the following is true:
MATH 1107 –
Using the STD DEV for Relative Standing
This is an important concept, because it allows
us to determine the probabilities associated with
certain outcomes. This is true, because the Zscores can be converted into probabilities of
occurrence.
MATH 1107 –
Using the STD DEV for Relative Standing
Example from page 107:
Suppose it takes you 20 minutes to drive to
campus, with a standard deviation of 2 minutes.
1. How often will you arrive on campus in less than 22
minutes?
2. How often will it take you more than 24 minutes?
3. 75% of the time you will arrive in X minutes or less. Solve
for x.
MATH 1107 –
Using the STD DEV for Relative Standing
Fun EXCEL exercises!