Transcript sec 14.6

Probability and Statistics
Copyright © Cengage Learning. All rights reserved.
Descriptive Statistics
14.6
(Graphical)
Copyright © Cengage Learning. All rights reserved.
Objectives
► Histograms and the Distribution of Data
► The Normal Distribution
Histogram and the Distribution
of Data
Example 1 –
Describe what this histogram tells you.
Histogram of heart rate data
Figure 2
cont’d
Histograms and the Distribution of Data
A histogram gives a visual representation of how
the data are distributed in the different bins.
The histogram allows us to determine whether
the data are symmetric about the mean, as in
Figure 3.
Symmetric and skew distributions
Figure 3
Histograms and the Distribution of Data
The heart rate data in Figure 2 are approximately
symmetric. If the histogram has a long “tail” on the
right, we say that the data are skewed to the right.
Similarly, if there is a long tail on the left, the data are
skewed to the left. Since the area of each bar in the
histogram is proportional to the number of data points
in that category, it follows that the median of the data
is located at the x-value that divides the area of the
histogram in half.
Histograms and the Distribution of Data
The extreme values have a large effect on the
mean but not on the median.
So if the data are skewed to the right, the mean
is to the right of the median; if the data are
skewed to the left, the mean is to the left of the
median.
The Normal Distribution
Suppose we measured the right foot length of 30 teachers and graphed the
results.
Assume the first person had a 10 inch foot. We could create a bar graph and plot
that person on the graph.
Number of People with
that Shoe Size
If our second subject had a 9 inch foot, we would add her to the graph.
As we continued to plot foot lengths, a pattern would
begin to emerge.
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Length of Right Foot
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Number of People with
that Shoe Size
Notice how there are more people (n=6) with a 10 inch right foot than any other length.
Notice also how as the length becomes larger or smaller, there are fewer and fewer
people with that measurement. This is a characteristics of many variables that we
measure. There is a tendency to have most measurements in the middle, and fewer
as we approach the high and low extremes.
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If we were to connect the top of each bar, we would create a
frequency polygon.
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Length of Right Foot
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Number of People with
that Shoe Size
You will notice that if we smooth the lines, our data almost creates a bell
shaped curve.
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Length of Right Foot
You will notice that if we smooth the lines, our data almost creates a bell
shaped curve.
Number of People with
that Shoe Size
This bell shaped curve is known as the “Bell Curve” or the “Normal Curve.”
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Length of Right Foot
Number of Students
Whenever you see a normal curve, you should imagine the HISTOGRAM within it.
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Points on a Quiz
The Normal Distribution
Most real-world data are distributed in a special way called a normal
distribution. For example, the heart rate data in Example 1 are
approximately normally distributed. That is, the shape of the
distribution is relatively symmetric, centered at the mean of the
distribution with extreme values at either tail.
Can you think of other measures that might have a Normal
Distribution?
Properties of Normal Distributions
Normal distribution
• The most important continuous probability
distribution in statistics.
• The graph of a normal distribution is called
the normal curve.
x
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The Normal Distribution
All normal distributions have the same general
shape, called a bell curve. Graphs of several
normal distributions are shown in Figure 5.
Normal curves
Figure 5
Means and Standard Deviations
• A normal distribution can have any mean and
any positive standard deviation.
• The mean gives the location of the line of
symmetry.
• The standard deviation describes the spread
of the data.
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μ = 3.5
σ = 1.5
μ = 3.5
σ = 0.7
μ = 1.5
σ = 0.7
The 68-95-99.7 Rule
• Normal models give us an idea of how
extreme a value is by telling us how likely it is
to find one that far from the mean.
• We can find these numbers precisely, but until
then we will use a simple rule that tells us a
lot about the Normal model…
Slide 6- 19
The 68-95-99.7 Rule (cont.)
• It turns out that in a Normal model:
– about 68% of the values fall within one standard
deviation of the mean;
– about 95% of the values fall within two standard
deviations of the mean; and,
– about 99.7% (almost all!) of the values fall within
three standard deviations of the mean.
Slide 6- 20
The 68-95-99.7 Rule (cont.)
• The following shows what the 68-95-99.7 Rule
tells us:
Slide 6- 21
The Normal Distribution
Example 3 – Using the Normal Distribution (Empirical Rule)
IQ scores are normally distributed with mean 100 and
standard deviation 15. Find the proportion of the population
with IQ scores in the given interval. Also find the probability
that a randomly selected individual has an IQ score in the
given interval.
(a) Between 85 and 115
(b) At least 130
(c) At most 130
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Example 3(a) – Solution
IQ scores between 85 and 115 are within one standard
deviation of the mean:
100 – 15 = 85 and 100 + 15 = 115
By the Empirical Rule, about 68% of the population have IQ
scores between 85 and 115. So the probability that a
randomly selected individual has an IQ between 85 and
115 is 0.68.
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Example 3(b) – Solution
cont’d
IQ scores between 70 and 130 are within two standard
deviations of the mean:
100 – 2(15) = 70 and 100 + 2(15) = 130
By the Empirical Rule, about 95% of the population have IQ
scores between 70 and 130. The remaining 5% of the
population have IQ scores above 130 or below 70.
Since normally distributed data are symmetric about the
mean, it follows that 2.5% have IQ scores above 130 (and
2.5% below 70).
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Example 3(b) – Solution
cont’d
So the probability that a randomly selected individual has
an IQ of at least 130 is 0.025.
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Example 3(c) – Solution
cont’d
By part (b), 2.5% of the population have IQ scores above
130. It follows that the rest of the population have IQ scores
below 130.
Thus 97.5% of the population have IQ scores of 130 or
below. So the probability that a randomly selected
individual has an IQ of 130 at most is 0.975.
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