Transcript Normal Distribution and Z

WHAT IS NORMAL DISTRIBUTION?
• Mean (average) and the Median (middle #)
are the same number. Or at least very close!!
• Symmetric
• Can look tall and skinny, short and wide or
somewhere in the middle.
WHAT ARE STANDARD DEVIATIONS?
A measure of spread that expresses the amount
of variation in the data from the mean.
Symbolized by σ
Which graph has a larger
spread (larger standard
deviation)?
WHAT IS THE EMPIRICAL RULE ?
Shows where a certain percentage of the data is found in a normal
distribution.

68% of the data falls within one standard deviation of the mean.

95% of the data falls within 2 standard deviations of the mean.

99.7% (almost 100!) of the data falls within 3 standard deviations of
the mean.
µ
If µ = 70 and σ = 3, What percent of the class scored between 67 and 73?
HOW CAN I FIND STANDARD DEVIATION?
Use your calculator!!
Mean →
Standard deviation →
# of data points →
SCROLL DOWN TO FIND MORE DATA!!
Median →
Remember …. Range = Max- min
Z-SCORES
FORMULA
𝒙−𝝁
𝒛=
𝝈
x = data point
µ = mean
σ= standard deviation
DEFINITION
- The measure of how
many standard
deviations a point is
from the mean.
- Positive z-score is
above the mean
- Negative z-score is
below the mean
HOW DO I CALCULATE Z-SCORE (EXAMPLE)
Calculate the z-score for the following data points if
µ = 45 ft and σ = 10 ft.
X1 = 55 ft
X2 = 30.ft
X3 = 22.45 ft.
µ
# of Std. deviations = z-score
If a data point is 2.5σ above the mean, what is its z-score?
HOW DO I FIND THE DATA POINT IF I
KNOW THE Z-SCORE?
You have a Normal Distribution with mean µ = 235.7 and
Standard σ=41.58. Which data point has a z-score of
-3.45?
𝒙−𝝁
𝒛=
𝝈
𝒙 − 𝟐𝟑𝟓. 𝟕
−𝟑. 𝟒𝟓 =
𝟒𝟏. 𝟒𝟖
X = 92.6
Do the next one on your own!
GUIDED PRACTICE WORKSHEET PRACTICE
The distribution of weights of 9-ounce bags of a particular brand of potato chips is
approximately Normal with mean of 9.12 ounces and standard deviation of 0.05
ounces.
1. Label the model
2. What would be the z- score for a bag weighing 9.25 z?
3. What would be the z-score for a bag weighing 9.05 oz?
4. Which bag weight is more unlikely?
Suppose your friend receives an 80% on a test in AP
World History and a 90% on a test in Underwater
Basket Weaving. The average score on the AP World
history test was 72% with a standard deviation of
6.5. The average score on the Underwater Basket
Weaving test was 88% with a standard deviation of
3. Which student should be happier with their
score?
• Find the z-score of each one
• Compare the z-scores.
EXAMPLE
AP World
x = 80
µ = 72
σ= 6.5
𝟖𝟎 − 𝟕𝟐
𝒛=
𝟔. 𝟓
z = 1.23
UBW
x = 90
µ = 88
σ= 3
𝟗𝟎 − 𝟖𝟖
𝒛=
𝟑
z = .67
How likely is it that a bag weighs less than 9 oz?
What is the probability that a bag weighs between
9 and 9.1 oz?
What is the probability that a bag weighs more
than 9.2 oz?
HOW TO FIND PROBABILITY USING THE
CALCULATOR
2ND VARS (DIST), then select #2 normalcdf
Normalcdf(L,U, µ, σ)
L = lower data point
U = upper data point
µ= mean
σ = standard deviation
L = a and U = b
L = -1E99, U = x
L = x, U = 1E99
How likely is it that a bag weighs less than 9 oz?
What is the probability that a bag weighs between
9 and 9.1 oz?
What is the probability that a bag weighs more
than 9.2 oz?
WHAT ARE PERCENTILES?
The percentile of a distribution is the percent of
observations less than it.
What is the percentile of a bag weighing 9.17 oz?
What is the percentile of a bag weighing 9.15 oz?
From Percentiles to Scores: z in Reverse


Sometimes we start with areas and need to find
the corresponding z-score or even the original
data value.
Example: What z-score represents the first
quartile in a Normal model?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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From Percentiles to Scores: z in
Reverse (cont.)

Calculator method
nd DISTR
 2
 invNorm(percent as a decimal)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 22
In order to be in the top 40%, what
score on the SAT did you have to have,
given N(445, 55)?
40%
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 6- 23
What were the scores for the middle
50%?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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