Download AP-Stats-2.4-Variability

AP Stats
1)
9/17
Which set has the largest standard deviation? The smallest?
a.
b.
c.
2)
BW
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19
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40
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40
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40
Without calculating, which data set has the greatest standard deviation?
Which has the smallest?
How are the data sets the same? How do they differ?
Section2.4 – Normal Distributions
SWBAT:
Identify and analyze patterns of distributions using shape,
center and spread.
Source: www.stat119review.com
Normal Distributions
Many real-life data sets have distributions that are approximately symmetric
and bell-shaped. Sometimes, the overall pattern of a large data set can be
described by a smooth curve
Many measurements, including heights,
weights, and IQ’s will have a normal curve
when the data set is large.
The distribution whose shape is described by
a normal curve is called a normal
distribution.
Normal Distribution, cont’d
Characteristics of Normal Distributions:
• The mean and the median are the same value.
• Symmetric and bell-shaped.
• The area under the curve is exactly 1 or 100%.
• The spread is completely measured by a single
number, the standard deviation.
• Good approximation of chance outcomes
Normal Distribution, cont’d
The standard deviation of a normal curve can be found
directly from the curve.
The point where the normal curve changes “curvature” on
either side of the curve is the standard deviation.
Same mean - different standard deviations.
Emperical Rule (68 – 95 – 99.7 Rule)
For data with a bell-shaped distribution, the standard
deviation has the following characteristics:
•About 68% of the data lie within one standard
deviation of the mean.
•About 95% of the data lie within two standard
deviations of the mean.
•About 99.7% of the data lie within three standard
deviations of the mean.
Emperical Rule (68 – 95 – 99.7 Rule)
50% of the data will fall above the mean / 50% below
16% of the data falls more than 1 std deviation
above/below the mean
99.85% of the data falls above -3 std deviations.
Emperical Rule EXAMPLE 1
The distribution of scores on tests such as the SAT exam is close to normal. SAT
scores are adjusted so that the mean score is about μ = 500 and the standard
deviation is about σ = 100.
1. What percent of scores
fall between 200 and
800?
200
2. What percent of scores
fall above 700?
3. What percentile is a
student that scores a 600
on the SAT?
300
400
500
600
700
800
1) (2.35+13.5+34+34+13.5+2.35) = 99.7%
2) (2.35+ 0.15) = 2.5%
3) (0.15 + 2.35 + 13.5 + 34 +34) = 84TH percentile
Emperical Rule EXAMPLE 2
Battery A: 37, 38, 38, 39, 39, 39, 40, 40, 40, 40, 40, 41, 41, 41, 42, 42, 43
Using calculator, we find x-bar = 40 and s = 1.58
The Emperical Rule says appx 68% of the data should fall within 1 std dev of
the mean. So if we look at 39 to 41, we see there are 11 items out of the 17
So estimating 11/17 = 65%
Let’s graph the actual curve:
Mean = 40
± s: 38.42 – 41.58
± 2s: 36.84 – 43.16
± 3s: 35.26 – 44.74
35.26 36.84 38.42 40
68% of data lies between 38.42 and 41.58 which is
pretty close to the estimated 39 – 41.
41.58 43.16
44.74
Emperical Rule EXAMPLE 2, cont’d
We can also take a look at the frequency histogram for Battery A and see
that it has a normal distribution. Battery B is also shown. What can you
conclude about Battery B?
Both batteries have a mean (and median) of 40, however Battery B has a
greater standard deviation than Battery A.
You try….
IQ scores are normally distributed with a mean
of 100 and a standard deviation of 15.
1) What is the percentile for someone with an
IQ of 130?
HOMEWORK:
Normal Curve Worksheet and Catch UP