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Chapter 3
Statistics for Describing,
Exploring, and Comparing Data
3.2 Measures of Center
3.3 Measures of Variation
3.4 Measures of Relative Standing
and Boxplots
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Notation Guide
 denotes the sum of a set of values.
x is the variable used to represent the
individual data values.
n represents the number of data values
in a Sample
N represents the number of data values
in a Population
Example For a sample: 1 2 5 8 6 4
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Measures of Center
The value at the center or middle of a
data set
1. Mean
2. Median
3.Mode
4. Midrange (rarely used)
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Mean
The measure of center obtained by
adding the values and dividing the total
by the number of values
What most people call an average.
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Mean
Advantages
• Relatively reliable.
• Takes every data value into account
Disadvantage
• Sensitive to every data value.
• One extreme value can affect it dramatically
• Is not a resistant measure of center
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18 19 20 21 21 22 24 24 25 26
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Median
The middle value when the original data
values are arranged in order of increasing
(or decreasing) magnitude
is not affected by an extreme value,
resistant measure of the center
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Finding the Median
First sort the values (arrange them in
order), then follow one of these rules:
1. If the number of data values is odd,
the median is the value located in the
exact middle of the list.
2. If the number of data values is even,
the median is found by computing the
mean of the two middle numbers.
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Example 1
5.40
1.10
Find the median of the set:
0.42
0.73
0.48
1.10
First, Order from smallest to largest:
0.42 0.48 0.66 0.73 1.10 1.10
0.66
5.40
Middle value
MEDIAN is 0.73
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Example 2
5.40
Find the median of the set:
1.10
0.42
0.73
0.48
1.10
First, Order from smallest to largest:
0.42 0.48 0.73 1.10 1.10 5.40
Two middle values
Since there are two middle values,
the median is their average:
MEDIAN is 0.915
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Mode
The value that occurs with the greatest frequency.
Data set can have one, more than one, or no mode
Bimodal
Two data values occur with the
same greatest frequency
Multimodal More than two data values occur
with the same greatest frequency
No Mode
No data value is repeated
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Examples
a. 5.40 1.10 0.42 0.73 0.48 1.10
Mode is 1.10
b. 27 27 27 55 55 55 88 88 99
Bimodal -
c. 1 2 3 6 7 8 9 10
No Mode
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27 & 55
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Midrange
The value midway between the maximum
and minimum values in the original data set
Midrange =
maximum value + minimum value
2
Sensitive to extremes
because it uses only the max. and min. values.
Midrange is rarely used in practice
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Round-off Rule for
Measure of Center
Carry one more decimal
place than is present in
the original set of values
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Measures of Variation
The spread, variability, of data width
of a distribution
1.Variance
2. Standard Deviation
3. Range (rarely used)
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Deviation
The Deviation of a point is a measure of
how far the point is from the mean.
• Very large for points far from the mean.
• Very small for points near the mean.
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Variance
The Variance of a population (denoted σ2)
is a the average deviation for every point
in the population.
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We use ‘n-1’ instead of ‘n’
(for technical reasons) 17
Example Find the population variance (σ2) of
the following ages from a set of 10 values:
18 19 20 21 21 22 24 24 25 26
1. Find the mean:
3. Find the average of the deviations
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Standard Deviation
The Population Standard Deviation (σ) is
simply the square root of the population
Variance.
The Sample Standard Deviation (s) is
simply the square root of the sample
Variance.
Mathematicians like to use standard deviations so the units are
the same (e.g. inches, seconds, pounds, etc.)
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Notation
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Range
(Rarely Used)
The difference between the maximum data
value and the minimum data value.
Range = (maximum value) – (minimum value)
It is very sensitive to extreme values;
therefore not as useful as the other
measures of variation.
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Usual and
Unusual Events
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Usual Values
Values in a data set are those that are
typical and not too extreme.
Max. Usual Value = (Mean) – 2*(s.d.)
Min. Usual Value = (Mean) + 2*(s.d.)
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Rule of Thumb
Based on the principle that for many
data sets, the vast majority (such as
95%) of sample values lie within two
standard deviations of the mean.
A value is unusual if it differs
from the mean by more than two
standard deviations.
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Expirical Rule (68-95-99.7 Rule)
For data sets having a distribution that is
approximately bell shaped, the following
properties apply:
 About 68% of all values fall within
1 standard deviation of the mean.
 About 95% of all values fall within
2 standard deviations of the mean.
 About 99.7% of all values fall within
3 standard deviations of the mean.
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Measures of
Relative Standing
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Z-Score
Also called “standardized value”
The number of standard deviations away
a point x is from the mean.
If the value is less than the mean, then z is negative
If the value is greater than the mean, then z is positive.
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Sample
Population
Round z scores to 2 decimal places
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Interpreting Z-Scores
Whenever a value is less than the mean, its
corresponding z score is negative
Ordinary values:
Unusual values:
–2 ≤ z score ≤ 2
z score < –2 or z score > 2
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Percentiles
The measures of location. There are 99
percentiles denoted P1, P2, . . . P99,
which divide a set of data into 100
groups with about 1% of the values in
each group.
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Finding the Percentile of a Value
Percentile of value x =
number of values less than x
• 100
total number of values
Round it off to the nearest whole number
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Quartiles
The measures of location (denoted Q1, Q2, Q3)
dividing a set of data into four groups with
about 25% of the values in each group.
 Q1 (First Quartile)
separates the bottom 25% of sorted values
from the top 75%.
 Q2 (Second Quartile)
(Same as median)
Separates the bottom 50% of sorted values
from the top 50%.
 Q3 (Third Quartile)
separates the bottom 75% of sorted values
from the top 25%.
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Q1, Q2, Q3
Divide ranked scores into four equal parts
25%
(minimum)
25%
25% 25%
Q1 Q2 Q3(maximum)
(median)
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Other Statistics
 Interquartile Range (or IQR): Q3 – Q1
 Semi-interquartile Range:
Q3 – Q1
2
 Midquartile:
Q3 + Q1
2
 10 - 90 Percentile Range: P90 – P10
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5-Number Summary
For a set of data, the 5-number
summary consists of
1.The minimum value
2.First quartile (Q1)
3.Median (Q2)
4.Third quartile (Q3)
5.The maximum value.
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