Transcript Statistics: Part III

OBJECTIVES
• Review vocabulary
• Variance
• Normal distributions
• Applications of statistics
MEAN ABSOLUTE DEVIATION
Mean Absolute Deviation is a measure of the dispersion of
data (how spread out the numbers are from the mean)
Mean Absolute Deviation Formula:
n
Mean absolute deviation 
xi

i
1
n

STANDARD DEVIATION
Standard Deviation is a measure of the dispersion of data
(how spread out numbers are from the mean)
The larger the standard deviation is, the more spread out the
numbers tend to be from the mean.
n
Standard Deviation Formula:
 
x i

i
1
n
 
2
MEAN ABSOLUTE DEVIATION AND STANDARD
DEVIATION
• Mean Absolute Deviation (MAD) and Standard
Deviation are statistics that are used to measure the
dispersion (spread) of the data
• Standard Deviation is a more traditional way to
measure the spread of data.
• Mean Absolute Deviation may be a better way to
measure the dispersion of data when there are
outliers since it is less affected by outliers.
Z-SCORE
• A z-score (or standard score) indicates how many
standard deviations a data point is above or below the
mean of a data set.
• A positive z-score indicates that a data point is above
the mean.
• A negative z-score indicates that a data point is below
the mean.
• A z-score of zero indicates that a data point is equal to
the mean.
Z-SCORE
The Z-Score Formula is:
x 


VARIANCE
• Variance is a measure of the dispersion of the data.
• It is used to find the standard deviation and is equal
to the standard deviation squared.
• Variance formula:
n
 
2
x i

i
1
n
 
2
WHAT’S NORMAL?
A normal curve is symmetric about the mean and has a
bell shape.
WHAT’S NORMAL?

Mean (
-1 Standard
Deviation (
)
-2
-3


)
+1 Standard
Deviation ( )

+2

+3

EMPIRICAL RULE
In a normal distribution with mean μ and standard
deviation σ:
68% of the data fall within σ of the mean μ.
95% of the data fall within 2σ of the mean μ.
99.7% of the data fall within 3σ of the mean μ.
APPLICATION
#1 Mr. Smith is planning to purchase new light bulbs for his
art studio. He tested a sample of 10 Power-Up bulbs and
found they lasted 4,356 hours on the average (mean) with
a standard deviation of 211 hours. Then, he tested 10
Lights-A-Lot bulbs and found the following results.
5,066
4,130
4,568
4,884
4,730
5,122
4,910
4,866
4,779
4,721
APPLICATION #1, CONTINUED
A) Which brand of light bulb has the greater average life
span?
B) Mr. Smith bought 2 Lights-A-Lot bulbs. One bulb
lasted for 5,150 hours and the other bulb lasted for
4,700 hours. Give the z-score for each bulb. Explain
what each z-score represents.
C) Which brand of light bulb do you think that Mr. Smith
should choose. Explain.
APPLICATION
#2 The chart below lists the height (in inches) of 10
players from two NBA teams. Use the data to answer
the following questions:
Chicago
Bulls
77” 81” 79” 77”
81”
79”
76”
82”
83”
69”
Toronto
Raptors
79” 78” 84” 79”
79”
80”
84”
81”
83”
71”
APPLICATION QUESTION #2, CONTINUED
A) Find the following for each team:
Chicago Bulls
Toronto Raptors
 =____________
78.4

2

14.6
= ___________
3.83
= ____________


2
79.8
= ____________
12.96
= ____________
3.6
 = _____________
APPLICATION QUESTION #2, CONTINUED
B) Find the z-score for a height of 6’11” for each team:
1.201
Chicago Bulls: _____________
0.889
Toronto Raptors: ___________
C) Is it more likely that a player for the Bulls or the
Raptors would be 6’11”? Explain.
He is more likely to play for the Raptors. The zscore for that height for the Raptors is 0.889 and
for the Bulls is 1.201. Therefore, a player who is
6’11” tall is more unusual (farther away from the
mean) if he plays for the Bulls.
APPLICATION QUESTION #2, CONTINUED
D) A player who is drafted by the Chicago Bulls has a zscore of -1.671. How tall is the player? Round to the
nearest inch.
6’0”
E) A player who is 6’7” tall has a z-score of -0.222. For
which team does he play?
Toronto Raptors
F) How many players on the Raptors are a height that is
between -0.4 and 0.6 standard deviations from the
mean?
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