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Chapter 9
Statistics
Copyright © 2012 Pearson Education, Inc. All rights reserved
9.1
Frequency Distributions;
Measures of Central
Tendency
Copyright © 2012 Pearson Education, Inc. All rights reserved
Example
A survey asked a random sample of 30 business
executives for their recommendations as to the
number of college credits in management that a
business major should have. The result are shown
as below. (1)Group the data into intervals and
find the frequency of each interval. (2) Draw a
histogram and a frequency polygon this
distribution.
 3, 25, 22, 16, 0, 9, 14, 8, 34, 21,
 15,12, 9, 3, 8,15,20,12,28,19
 17,16,23,19,12,14,29,13,24,18

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Figure 1
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Your Turn 1
Find the mean of the following data: 12, 17, 21, 25, 27, 38, 49.
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Your Turn 2
Find the mean of the following grouped frequency.
Interval Midpoint, Frequency Product x
x
f
f
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0-6
3
2
6
7-13
10
4
40
14-20
17
7
119
21-27
24
10
240
28-34
31
3
93
35-41
38
1
38
Total = 27
Total = 536
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Your Turn: Find the median of the data 12, 17, 21, 25, 27, 38, 49.
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9.2
Measures of Variation
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Variation
Variation
is
the heart of statistics
no variation, no need to do statistical
analysis
the mean would describe the
distribution
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Quest for Variation

Measures of dispersion
 consider

the spread between scores
Calculations include
 range
 variance
 standard
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deviation
The Range

Range
 distance
between the highest and lowest score
in a distribution.

Calculation
 Range
H
= H minus L
= the highest score in the data set
 L = the lowest score in the data set
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Example: Find the range of each data set

Data Set A: 5,3,8,2,7,4,14,3,12,10

Data Set B: 1,1,1,1,1,1,1,1,1,100
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Variance and Standard
Deviation

Most commonly used measures of
dispersion
 based
upon the distance of scores in a
distribution from the mean
 the mean is used as the central point
The Variance is defined as:
The average of the squared differences from
the Mean.
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Variance
 The
variance is
a
measure of the spread of scores in a
distribution around its mean
 The
larger the variance
 the
greater the spread of scores around the
mean
 The
smaller the variance
 the
more closely the scores are distributed
around the mean
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Standard Deviation ( = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒) )
 Standard
deviation
 is
a measure of dispersion of the scores around
the mean
 The
higher the standard deviation
 the
 The
greater the spread in the scores
lower the standard deviation
 the
closer the scores are on average from the
mean of the distribution
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Example: Variance and Standard
Deviation of a population.

http://www.mathsisfun.com/data/standarddeviation.html
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Example 1
Find the range, variance, and standard deviation for the list of
numbers: 7, 11, 16, 17, 19, 35.
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Example: Variance and Standard
Deviation of a sample
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Example 2
Find the standard deviation for the grouped frequency
distribution.
Interval
x
x2
f
fx2
Total
Total =
12,528
0-6
7-13
14-20
21-27
28-34
35-41
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Group Class Work
Suppose we were interested in how many siblings are in
statistics students' families.
Number of
Children (x)
Frequency
(f)
1
2
5
12
3
4
5
8
3
0
6
0
7
1
Totals
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(1) Find the mean of
the data.
(2) Find the standard
deviation.
Summary
 Explaining
variation is the basis for
statistical analysis
 We begin with basic measures of
dispersion
 range
 variance
 standard
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deviation
9.3
The Normal
Distribution
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Normal Distribution Curve
A normal distribution curve is
symmetrical, bell-shaped curve
defined by the mean and standard
deviation of a data set.
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Right Skewed, Left Skewed, and the Bell
-Shaped
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One standard deviation away from the mean (  ) in either
direction on the horizontal axis accounts for around 68 percent
of the data. Two standard deviations away from the mean
accounts for roughly 95 percent of the data with three standard
deviations representing about 99.7 percent of the data.
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Standard Normal Distribution
z
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x

z-scores
When a set of data values are normally
distributed, we can standardize each score
by converting it into a z-score.
z-scores make it easier to
compare data values measured
on different scales.
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Standard Normal Distribution
The mean of the data in a standard
normal distribution is 0 and the
standard deviation is 1.
A standard normal distribution is the
set of all z-scores.
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A z-score reflects how many standard
deviations above or below the mean a raw
score is.
The z-score is positive if the data value
lies ___________ the mean and
negative if the data value lies
___________ the mean.
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z-score formula
z
x

Where x represents an element
of the data set, the mean is
represented by  and standard
deviation by .

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Analyzing the data
Suppose SAT scores among college students
are normally distributed with a mean of 500
and a standard deviation of 100. If a student
scores a 700, what would be her z-score?
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Your Turn 2
Find a value of z satisfying the following conditions.
(a) 2.5% of the area is to the left of z.
(b) 20.9% of the area is to the right of z.
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Your Turn 3
Find the z-score for x = 20 if a normal distribution has a mean
35 and standard deviation 20.
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Your Turn 4
Dixie Office Supplies finds that its sales force drives an average
of 1200 miles per month per person, with a standard deviation
of 150 miles. Assume that the number of miles driven
by a salesperson is closely approximated by a normal distribution.
Find the probability that a sales person drives between 1275 and
1425 miles.
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