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Chapter 3
Averages and
Variation
Understanding Basic Statistics
Fifth Edition
By Brase and Brase
Prepared by Jon Booze
Measures of Central Tendency
• Average – a measure of the center value or
central tendency of a distribution of values.
• Three types of average:
– Mode
– Median
– Mean
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Mode
The mode is the most frequently occurring value in
a data set.
Example: Sixteen students
are asked how many
college math classes they
have completed.
{0, 3, 2, 2, 1, 1, 0, 5, 1,
1, 0, 2, 2, 7, 1, 3}
The mode is 1.
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Median
Finding the median:
1). Order the data from smallest to largest.
2). For an odd number of data values:
Median = Middle data value
3). For an even number of data values:
Sum of middle two values
Median 
2
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Median
Find the median of the following data set.
{ 4, 6, 6, 7, 9, 12, 18, 19}
a). 6
b). 7
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c). 8
d). 9
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Median
Find the median of the following data set.
{4, 6, 6, 7, 9, 12, 18, 19}
a). 6
b). 7
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c). 8
d). 9
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Mean
Sample mean
Population mean
x

x
x


n
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N
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Mean
Sample mean
Population mean
x

x
x


N
n
Find the mean of the following data set.
{3, 8, 5, 4, 8, 4, 10}
a). 8
b). 6.5
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c). 6
d). 7
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Mean
Sample mean
Population mean
x

x
x


N
n
Find the mean of the following data set.
{3, 8, 5, 4, 8, 4, 10}
a). 8
b). 6.5
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c). 6
d). 7
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Trimmed Mean
• Order the data and remove k% of the data
values from the bottom and top.
• 5% and 10% trimmed means are common.
• Then compute the mean with the remaining
data values.
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Resistant Measures of Central Tendency
• A resistant measure will not be affected by
extreme values in the data set.
• The mean is not resistant to extreme values.
• The median is resistant to extreme values.
• A trimmed mean is also resistant.
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Critical Thinking
• Four levels of data – nominal, ordinal, interval,
ratio (Chapter 1)
• Mode – can be used with all four levels.
• Median – may be used with ordinal, interval, of
ratio level.
• Mean – may be used with interval or ratio level.
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Critical Thinking
• Mound-shaped
data – values of
mean, median and
mode are nearly
equal.
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Critical Thinking
• Skewed-left data – mean < median < mode.
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Critical Thinking
• Skewed-right data – mean > median > mode.
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Weighted Average
• At times, we may need to assign more
importance (weight) to some of the data values.
xw

Weighted Average 
w
• x is a data value.
• w is the weight assigned to that value.
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Measures of Variation
Three measures of variation:
range
variance
standard deviation
• Range = Largest value – smallest value
Only two data values are used in the
computation, so much of the information in
the data is lost.
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Sample Variance and Standard Deviation
Sample Variance
Sample Standard Deviation
n
2
(
x

x
)
 i
i1
2
s 
n 1
s s
2
Find the standard deviation of the data set.
{2,4,6}
a). 2
b). 3
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c). 4
d). 3.67
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Sample Variance and Standard Deviation
Sample Variance
Sample Standard Deviation
n
2
(
x

x
)
 i
i1
2
s 
n 1
s s
2
Find the standard deviation of the data set.
{2,4,6}
a). 2
b). 3
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c). 4
d). 3.67
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Population Variance
and Standard Deviation
Population Variance
N
 (x  )
i
 
2
Population Standard
Deviation
2
i 1
N
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 
2
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The Coefficient of Variation
For Samples
s
CV  100
x
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For Populations

CV  100

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Chebyshev’s Theorem
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Chebyshev’s Theorem
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Critical Thinking
• Standard deviation or variance, along with the
mean, gives a better picture of the data
distribution than the mean alone.
• Chebyshev’s theorem works for all kinds of data
distribution.
• Data values beyond 2.5 standard deviations
from the mean may be considered as outliers.
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Percentiles and Quartiles
• For whole numbers P, 1 ≤ P ≤ 99, the Pth
percentile of a distribution is a value such that
P% of the data fall below it, and (100-P)% of the
data fall at or above it.
• Q1 = 25th Percentile
• Q2 = 50th Percentile = The Median
• Q3 = 75th Percentile
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Quartiles and Interquartile Range (IQR)
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Computing Quartiles
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Five Number Summary
• A listing of the following statistics:
– Minimum, Q1, Median, Q3, Maximum
• Box-and-Whisder plot – represents the fivenumber summary graphically.
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Box-and-Whisker Plot Construction
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Critical Thinking
• Box-and-whisker plots display the spread of data
about the median.
• If the median is centered and the whiskers are
about the same length, then the data distribution
is symmetric around the median.
• Fences – may be placed on either side of the
box. Values lie beyond the fences are outliers.
(See problem 10)
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Critical Thinking
Which of the following box-and-whiskers plots
suggests a symmetric data distribution?
(a)
(b)
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(c)
(d)
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Critical Thinking
Which of the following box-and-whiskers plots
suggests a symmetric data distribution?
(a)
(b)
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(c)
(d)
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