Measures of Central Tendency and Variation

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Measures of Central Tendency and Variation
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
Holt McDougal Algebra 2
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Warm Up 1/13/15 Simplify each expression. 1.
2. 3. 4. Find the mean and median. 5.
1, 2, 87 6. 3, 2, 1, 10
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30 2
4 2.5
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Objectives
Find measures of central tendency and measures of
variation for statistical data. Examine the
effects of outliers on statistical data.
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Vocabulary
expected value probability distribution variance s
tandard deviation outlier
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Recall that the mean, median, and mode are
measures of central tendencyvalues that describe
the center of a data set.
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Check It Out! Example 1a
Find the mean, median, and mode of the data set.
6, 9, 3, 8
Mean
Median
3 6 8 9
Mode
None
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Check It Out! Example 1b
Find the mean, median, and mode of the data set.
2, 5, 6, 2, 6
Mean
Median
2 2 5 6 6
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Mode
2 and 6
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A weighted average is a mean calculated by using
frequencies of data values. Suppose that 30
movies are rated as follows
weighted average of stars
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For numerical data, the weighted average of all
of those outcomes is called the expected value
for that experiment.
The probability distribution for an experiment is
the function that pairs each outcome with its
probability.
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Example 2 Finding Expected Value
The probability distribution of successful free
throws for a practice set is given below. Find
the expected number of successes for one set.
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Example 2 Continued
Use the weighted average.
Simplify.
The expected number of successful free throws is
2.05.
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Check It Out! Example 2
The probability distribution of the number of
accidents in a week at an intersection, based on
past data, is given below. Find the expected
number of accidents for one week.
Use the weighted average.
expected value 0(0.75) 1(0.15) 2(0.08)
3(0.02)
0.37
Simplify.
The expected number of accidents is 0.37.
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A box-and-whisker plot shows the spread of a data
set. It displays 5 key points the minimum and
maximum values, the median, and the first and
third quartiles.
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The quartiles are the medians of the lower and
upper halves of the data set. If there are an odd
number of data values, do not include the median
in either half.
The interquartile range, or IQR, is the
difference between the 1st and 3rd quartiles, or
Q3 Q1. It represents the middle 50 of the data.
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Example 3 Making a Box-and-Whisker Plot and
Finding the Interquartile Range
Make a box-and-whisker plot of the data. Find the
interquartile range. 6, 8, 7, 5, 10, 6, 9, 8, 4
Step 1 Order the data from least to greatest. 4,
5, 6, 6, 7, 8, 8, 9, 10
Step 2 Find the minimum, maximum, median, and
quartiles.
4, 5, 6, 6, 7, 8, 8, 9, 10
Mimimum
Median
Maximum
Third quartile 8.5
First quartile 5.5
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Example 3 Continued
Step 3 Draw a box-and-whisker plot.
Draw a number line, and plot a point above each
of the five values. Then draw a box from the
first quartile to the third quartile with a line
segment through the median. Draw whiskers from
the box to the minimum and maximum.
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Example 3 Continued
IRQ 8.5 5.5 3
The interquartile range is 3, the length of the
box in the diagram.
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Check It Out! Example 3
Make a box-and-whisker plot of the data. Find the
interquartile range. 13, 14, 18, 13, 12, 17, 15,
12, 13, 19, 11, 14, 14, 18, 22, 23
Step 1 Order the data from least to greatest. 11,
12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18, 18,
19, 22, 23
Step 2 Find the minimum, maximum, median, and
quartiles.
11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18,
18, 19, 22, 23
Mimimum
Median
Maximum
First quartile 13
Third quartile 18
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Check It Out! Example 3 Continued
Step 3 Draw a box-and-whisker plot.
IQR 18 13 5
The interquartile range is 5, the length of the
box in the diagram.
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The data sets 19, 20, 21 and 0, 20, 40 have
the same mean and median, but the sets are very
different. The way that data are spread out from
the mean or median is important in the study of
statistics.
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A measure of variation is a value that describes
the spread of a data set. The most commonly used
measures of variation are the range, the
interquartile range, the variance, and the
standard deviation. START HERE TUES 1/14/13
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The variance, denoted by s2, is the average of
the squared differences from the mean. Standard
deviation, denoted by s, is the square root of
the variance and is one of the most common and
useful measures of variation.
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Low standard deviations indicate data that are
clustered near the measures of central tendency,
whereas high standard deviations indicate data
that are spread out from the center.
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Example 4 Finding the Mean and Standard Deviation
Find the mean and standard deviation for the data
set of the number of people getting on and off a
bus for several stops. 6, 8, 7, 5, 10, 6, 9, 8,
4
Step 1 Find the mean.
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Example 4 Continued
Step 2 Find the difference between the mean and
each data value, and square it.
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Example 4 Continued
Step 3 Find the variance.
Find the average of the last row of the table.
Step 4 Find the standard deviation.
The standard deviation is the square root of the
variance.
The mean is 7 people, and the standard deviation
is about 1.83 people.
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Check It Out! Example 4
Find the mean and standard deviation for the data
set of the number of elevator stops for several
rides.
0, 3, 1, 1, 0, 5, 1, 0, 3, 0
Step 1 Find the mean.
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Check It Out! Example 4 Continued
Step 2 Find the difference between the mean and
each data value, and square it.
Data Value x 0 3 1 1 0 5 1 0 3 0
x x -1.4 1.6 -0.4 -0.4 -1.4 3.6 -0.4 -1.4 1.6 -1.4
(x x)2 1.96 2.56 0.16 0.16 1.96 12.96 0.16 1.96 2.56 1.96
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Check It Out! Example 4 Continued
Step 3 Find the variance.
Find the average of the last row of the table
Step 4 Find the standard deviation.
The standard deviation is the square root of the
variance.
The mean is 1.4 stops and the standard deviation
is about 1.6 stops.
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An outlier is an extreme value that is much less
than or much greater than the other data values.
Outliers have a strong effect on the mean and
standard deviation. If an outlier is the result
of measurement error or represents data from the
wrong population, it is usually removed. There
are different ways to determine whether a value
is an outlier. One is to look for data values
that are more than 3 standard deviations from the
mean.
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Example 5 Examining Outliers
Find the mean and the standard deviation for the
heights of 15 cans. Identify any outliers, and
describe how they affect the mean and the
standard deviation.
Can Heights (mm) Can Heights (mm) Can Heights (mm)
92.8 92.8 92.9
92.9 92.9 92.8
92.7 92.9 92.1
92.7 92.8 92.9
92.9 92.7 92.8
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Example 5 Continued
Step 1 Enter the data values into list L1 on a
graphing calculator.
Step 2 Find the mean and standard deviation.
On the graphing calculator, press ,
scroll to the CALC menu, and select 11-Var
Stats.
The mean is about 92.77, and the standard
deviation is about 0.195.
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Example 5 Continued
Step 3 Identify the outliers. Look for the data
values that are more than 3 standard deviations
away from the mean in either direction. Three
standard deviations is about 3(0.195) 0.585.
Values less than 92.185 and greater than 93.355
are outliers, so 92.1 is an outlier.
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Example 5 Continued
Check
92.1 is 3.4 standard deviations from the mean, so
it is an outlier.
Step 4 Remove the outlier to see the effect that
it has on the mean and standard deviation.
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Example 5 Continued
All Data Without outlier
The outlier in the data set causes the mean to
decrease from 92.82 to 92.77 and the standard
deviation to increase from ? 0.077 to ? 0.195.
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Check It Out! Example 5
In the 2003-2004 American League Championship
Series, the New York Yankees scored the following
numbers of runs against the Boston Red Sox 2, 6,
4, 2, 4, 6, 6, 10, 3, 19, 4, 4, 2, 3. Identify
the outlier, and describe how it affects the mean
and standard deviation.
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Check It Out! Example 5 Continued
Step 1 Enter the data values into list L1 on a
graphing calculator.
Step 2 Find the mean and standard deviation.
On the graphing calculator, press ,
scroll to the CALC menu, and select 11-Var Stats.
The mean is about 5.4, and the standard deviation
is about 4.3.
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Check It Out! Example 5 Continued
Step 3 Identify the outliers. Look for the data
values that are more than 3 standard deviations
away from the mean in either direction. Three
standard deviations is about 3(4.3) 12.9.
Values less than 7.5 and greater than 18.3 are
outliers, so 19 is an outlier.
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Check It Out! Example 5 Continued
Step 4 Remove the outlier to see the effect that
it has on the mean and standard deviation.
Without outlier
All data
The outlier in the data set causes the mean to
increase from ? 4.3 to ? 5.4, and the standard
deviation increases from ? 2.2 to ? 4.3.
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Lesson Quiz Part I
Use the data set for 1 and 369, 4, 7, 8, 5, 8,
24, 5 1. Find the mean, median, and mode. 2.
The probability distribution of the number of
people entering a store each day based on past
data is given below. Find the expected number of
people for one day.
mean 8.75, median 7.5, modes 5 and 8
85
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Lesson Quiz Part II
Use the data set for 1 and 369, 4, 7, 8, 5, 8,
24, 5 3. Make a box-and-whisker plot of the data
in 1. Find the interquartile range. 4. Find the
variance and the standard deviation of the data
set. 5. Use the standard deviation to identify
any outliers in the data set.
IQR 3.5
var ? 35.94 std. dev. ? 5.99
none by this method