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AP STATISTICS
Section 2.2 The Normal Distribution
Objective: To be able to calculate percentiles using the
normal distribution.
Normal Distributions:
• Bell-shaped density curve.
• All basically the same shape.
• Identified by the mean μ and standard deviation σ.
SIDE:
• Use Greek letters to represent parameters. (population)
• Use standard letters for statistics. (sample)
Notation: X ~ N ( μ, σ)
Sketch:
Inflection point: a point on the graph where the curve
changes concavity.
68-95-99.7 Rule:
In a normal distribution, approximately
• 68% of all observations lie within _____ standard
deviation of the mean.
• 95% of all observations lie within _____ standard
deviations of the mean.
• 99.7% of all observations lie within _____ standard
deviations of the mean.
Ex. Adult males weights are normally distributed with a mean of
190 pounds and a standard deviation of 30 pounds. Find the
proportion of adult males whose weights fall in the following
regions.
a. X < 190
b. X < 160
c. X > 250
e.
160 < X < 250
g. 100 < X < 250
d. X = 250
f. X > 280
h. X < 130 or X > 250
Q: What does changing μ but not σ do to the distribution?
Q: What does changing σ but not μ do to the distribution?
Why do we use the Normal Distribution:
1. Good model for real world data.
2. Easy to approximate percentiles.
3. Many statistical inference procedures are based on
normality.
Equation for the Normal Distribution:
∞
1
𝐞
−∞ σ 2π
−(𝑥−μ)2
2σ2
𝑑𝑥
Standard Normal Distributions:
Diagram:
Z ~ N(0,1)
Using the z-table, find the proportion of observations such
that:
(Area to the left)
Z < -1
Z < 2.06
Z < .56
(Area to the right)
Z > 1.53
Z > -1.05
(Area in between two values)
-1 < Z < 1
-2.54 < Z < -.26
Working backwards with the table:
What z-score represents the 40th percentile?
What z-score represents the first quartile?
What z-score represent the upper 10 percent of the area?
Using X = weight of an adult male and X~N(190,30), find
the proportion of observations such that:
X < 145
X < 213
X > 245
X > 153
122 < X < 200
X < 132 or X > 205
Working backwards from a percentile to a value of X.
Q: What weight represents the 85th percentile?
Q: What body weight represents the heaviest 5% of adult
males?
**If a z-score falls outside the range on the z-table, then it
is approximately 0.
Example:
In 2013 Jose Reyes had a batting average of .337. During
that season X ~ N(.266,.028). In 2010 Chipper Jones had
a batting average of .364. During that season X ~ N(.280,
.037). Calculate the percentile for each player within their
season. Which player had the better overall season?
What batting average represents the 70th %-ile in 2013?
ASSESSING NORMALITY:
1. Construct a histogram or stem and leaf plot and look
for a bell-shaped pattern.
-Good for large data sets.
2. Compare relative frequencies to the 68-95-99.7 Rule.
-Mark the x-axis with 𝑥 ± 𝑠, 𝑥 ± 2𝑠, 𝑎𝑛𝑑 𝑥 ± 3𝑠 and observe how
closely the observations follow the 68-95-99.7 Rule.
Ex. Pulse Rate Data
3. Normal Probability Plot (Normal Quantile Plot) NPP
• Most common method for assessing normality.
• It is a plot of (𝑥 𝑖 , 𝑧 𝑖 )
• IF THE PLOT APPEARS FAIRLY LINEAR, THEN WE CAN
ASSUME THAT THE DATA FOLLOWS A NORMAL
DISTRIBUTION.
• If most of the points are above the reference line, then the data is
skewed right.
• If most of the points are below the reference line, then the data is
skewed left.
• This graph will be very important later in the course!
Ex. Pulse Rate Example
How a NPP works: (extra)
1. Rank the data from min to max.
2. Calculate the percentile for each point such that
ile is calculated for each 𝑥(𝑖) .
3. Calculate the z-score for each percentile.
4. Plot all ordered pairs (𝑥 𝑖 , 𝑧 𝑖 ).
𝑖−0.5
%𝑛