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Chapter Three
Averages and
Variation
Understanding Basic Statistics
Fourth Edition
By Brase and Brase
Prepared by: Lynn Smith
Gloucester County College
Measures of Central Tendency
• Mode
• Median
• Mean
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The Mode
• the value that occurs most frequently in
a data set
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Find the mode:
6, 7, 2, 3, 4, 6, 2, 6
The mode is 6.
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Find the mode:
6, 7, 2, 3, 4, 5, 9, 8
There is no mode for this data.
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The Median
• the central value of an ordered
distribution
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To find the median of raw data:
• Order the data from smallest to largest.
• For an odd number of values pick the
middle value.
or
• For an even number of values compute
the average of the middle two values
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Find the median:
Data:
5, 2, 7, 1, 4, 3, 2
Rearrange: 1, 2, 2, 3, 4, 5, 7
The median is 3.
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Find the Median:
Data:
31, 57, 12, 22, 43, 50
Rearrange: 12, 22, 31, 43, 50, 57
The median is the average of the middle two
values = 31  43
2
 37
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Finding the median for a large data set
For an ordered data set of n values:
Position of the middle value =
n1
2
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The Mean
• An average that uses the exact value of
each entry
• Sometimes called the arithmetic mean
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The Mean
The mean of a collection of data is found by:
• summing all the entries
• dividing by the number of entries
mean
sum of all entries

number
of entries
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Find the Mean:
6, 7, 2, 3, 4, 5, 2, 8
6  7  2  3  4  5  2  8 37
mean 
  4.625  4.6
8
8
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Sigma Notation
• The symbol S means “sum the
following.”
•
S is the Greek letter (capital) sigma.
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Notations for mean
Sample mean
x
Population
mean

Greek letter
(mu)
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Number of entries in a set of data
• If the data represents a sample, the
number of entries = n.
• If the data represents an entire
population, the number of entries = N.
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Sample mean
x
x
n
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Population mean
x

N
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Resistant Measure
• a measure that is not influenced by
extremely high or low data values
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Which is less resistant?
• Mean
• Median
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• The mean is less
resistant. It can be
made arbitrarily
large by increasing
the size of one
value.
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Trimmed Mean
• a measure of center that is more
resistant than the mean but is still
sensitive to specific data values
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To calculate a (5 or 10%) trimmed mean
• Order the data from smallest to largest.
• Delete the bottom 5 or 10% of the data.
• Delete the same percent from the top
of the data.
• Compute the mean of the remaining 80
or 90% of the data.
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Compute a 10% trimmed mean:
15, 17, 18, 20, 20, 25, 30, 32, 36, 60
• Delete the top and bottom 10%
• New data list:
17, 18, 20, 20, 25, 30, 32, 36
10% trimmed mean =
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 x 198

 24 .8
n
8
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Weighted Average
• An average where more importance or
weight is assigned to some of the
numbers
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Weighted Average
If x is a data value and w is the weight
assigned to that value
Weighted average =
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Sxw
Sw
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Calculating a Weighted Average
In a pageant, the interview is worth 30% and
appearance is worth 70%. Find the
weighted average for a contestant with an
interview score of 90 and an appearance
score of 80.
0.30(90)  0.70(80)
Weighted average 
0.30  0.70
27  56

 83
1.00
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Measures of Variation
• Range
• Standard Deviation
• Variance
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The Range
• the difference between the largest and
smallest values of a distribution
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Find the range:
10, 13, 17, 17, 18
The range = largest minus smallest
= 18 minus 10 = 8
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The Standard Deviation
• a measure of the average variation of
the data entries from the mean
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Standard deviation of a sample
s
 (x  x)
n 1
2
mean of the
sample
n = sample size
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To calculate standard deviation of a sample
•
•
•
•
•
•
Calculate the mean of the sample.
Find the difference between each entry (x) and the
mean. These differences will add up to zero.
Square the deviations from the mean.
Sum the squares of the deviations from the mean.
Divide the sum by (n  1) to get the variance.
Take the square root of the variance to get
the standard deviation.
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The Variance
• the square of the standard deviation
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Variance of a Sample
(
x

x
)

2
s 
n 1
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2
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Find the standard deviation and variance
x
30
26
22
78
xx
4
0
-4
Mean = 26
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(x - x)
Sum = 0
2
16
0
16
___
32
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The variance
s 
2

( x  x)
n1
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2
= 32  2 =16
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The standard deviation
s=
16  4
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Find the mean, the
standard deviation and variance
mean = 5
x
xx
(x - x)
4
1
1
5
0
0
5
0
0
7
2
4
4
1
1
25
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2
6
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The mean, the
standard deviation and variance
Mean = 5
S tan dard deviation 
Variance
1 .5  1 .22
6

 1 .5
4
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Computation Formulas for Sample Variance
and Standard Deviation:

Sx 
x


2
2
Sample variance  s 
2
n 1
n

x
2

x


2
Sample standard devaition  s 
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n1
n
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To find S x2
• Square the x values, then add.
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To find ( S x ) 2
Sum the x values, then square.
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Use the computing formulas to find s and s2
x
x2
4
16
5
25
5
25
7
49
4
25
16
131
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s2 
131  625
51
5  1.5
s  1.5  1.22
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Population Mean
population
x

mean   
N
where N  number of data values in the population
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Population Standard Deviation
 x  x 
2

N
where N  number of data values in the population
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Coefficient Of Variation:
• A measurement of the relative
variability (or consistency) of data.
s

CV   100 or
 100
x

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CV is used to compare
variability or consistency
A sample of newborn infants had a mean weight
of 6.2 pounds with a standard deviation of 1
pound.
A sample of three-month-old children had a
mean weight of 10.5 pounds with a standard
deviation of 1.5 pound.
Which (newborns or 3-month-olds) are more
variable in weight?
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To compare variability,
compare Coefficient of Variation
• For newborns:
CV = 16%
Higher CV: more
variable
• For 3-month-olds:
CV = 14%
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Lower CV: more
consistent
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Use Coefficient of Variation
• To compare two groups of data, to
answer:
• Which is more consistent?
• Which is more variable?
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CHEBYSHEV'S THEOREM
For any set of data and for any number k,
greater than one, the proportion of the
data that lies within k standard
deviations of the mean is at least:
1
1 
k
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2
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Results of Chebyshev’s Theorem
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Using Chebyshev’s Theorem
• A mathematics class completes an
examination and it is found that the
class mean is 77 and the standard
deviation is 6.
• According to Chebyshev's Theorem,
between what two values would at least
75% of the grades be?
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Mean = 77
Standard deviation = 6
At least 75% of the grades would be in the
interval:
x  2 s to x  2 s
77 – 2(6) to 77 + 2(6)
77 – 12 to 77 + 12
65 to 89
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Percentiles
• For any whole number P (between 1
and 99), the Pth percentile of a
distribution is a value such that P% of
the data fall at or below it.
• The percent falling at or above the Pth
percentile will be (100 – P)%.
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A histogram showing the 60th Percentile
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Percentiles
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Quartiles
• Percentiles that divide the data into
fourths
• Q1 = 25th percentile
• Q2 = the median
• Q3 = 75th percentile
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Quartiles
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Computing Quartiles
• Order the data from smallest to largest.
• Find the median, the second quartile.
• Find the median of the data falling
below Q2. This is the first quartile.
• Find the median of the data falling
above Q2. This is the third quartile.
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Inter-quartile range
IQR  Q3  Q1
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Find the quartiles:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
22
34
The data has been ordered.
The median is 24.
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Find the quartiles:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
22
34
The data has been ordered.
The median is 24.
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3 | 62
Find the quartiles:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
22
34
For the data below the median, the median is 17.
17 is the first quartile.
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Find the quartiles:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
22
34
For the data above the median, the median is 33.
33 is the third quartile.
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Find the interquartile range:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
22
34
IQR = Q3 – Q1 = 33 – 17 = 16
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Five-Number Summary of Data
•
•
•
•
•
Lowest value
First quartile
Median
Third quartile
Highest value
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Box-and-Whisker Plot
• a graphical presentation of the fivenumber summary of data
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Box-and-Whisker Plot
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Making a Box-and-Whisker Plot
• Draw a vertical scale including the lowest
and highest values.
• To the right of the scale, draw a box from Q1
to Q3.
• Draw a solid line through the box at the
median.
• Draw lines (whiskers) from Q1 to the lowest
and from Q3 to the highest values.
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Construct a Box-and-Whisker Plot:
12
23
41
15
24
45
16
25
51
16
30
17
32
18
33
22
33
Lowest = 12
Q1 = 17
Median = 24
Q3 = 33
22
34
Highest = 51
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Box-and-Whisker Plot
Highest = 51
Q3 = 33
Median = 24
Q1 = 17
Lowest = 12
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