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Notation Guide
 denotes the sum of a set of values.
Chapter 3
x is the variable used to represent the
individual data values.
Statistics for Describing,
Exploring, and Comparing Data
n represents the number of data values
in a Sample
3.2 Measures of Center
N represents the number of data values
in a Population
3.3 Measures of Variation
3.4 Measures of Relative Standing
and Boxplots
Example For a sample: 1 2 5 8 6 4
𝑛=6
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∑𝑥 = 1 + 2 + 5 + 8 + 6 + 4 = 26
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Mean
Measures of Center
The measure of center obtained by
adding the values and dividing the total
by the number of values
The value at the center or middle of a
data set
What most people call an average.
1. Mean
2. Median
3.Mode
4. Midrange (rarely used)
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Mean
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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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Median
Finding the Median
First sort the values (arrange them in
order), then follow one of these rules:
The middle value when the original data
values are arranged in order of increasing
(or decreasing) magnitude
1. If the number of data values is odd,
the median is the value located in the
exact middle of the list.
is not affected by an extreme value,
2. If the number of data values is even,
the median is found by computing the
mean of the two middle numbers.
resistant measure of the center
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Example 1
5.40
1.10
Example 2
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
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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
5.40
Two middle values
Middle value
Since there are two middle values,
the median is their average:
MEDIAN is 0.73
MEDIAN is 0.915
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Mode
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Examples
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
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
27 & 55
Multimodal More than two data values occur
with the same greatest frequency
No Mode
No data value is repeated
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Round-off Rule for
Measure of Center
Midrange
The value midway between the maximum
and minimum values in the original data set
Midrange =
Carry one more decimal
place than is present in
the original set of values
maximum value + minimum value
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Sensitive to extremes
because it uses only the max. and min. values.
Midrange is rarely used in practice
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Deviation
Measures of Variation
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.
The spread, variability, of data width
of a distribution
1.Variance
2. Standard Deviation
3. Range (rarely used)
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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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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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We use ‘n-1’ instead of ‘n’
(for technical reasons) 17
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Standard Deviation
Notation
The Population Standard Deviation (σ) is
simply the square root of the population
Variance.
𝝈𝟐 = population variance
𝝈 = population standard deviation
The Sample Standard Deviation (s) is
simply the square root of the sample
Variance.
𝒔𝟐 = sample variance
𝒔 = sample standard deviation
Mathematicians like to use standard deviations so the units are
the same (e.g. inches, seconds, pounds, etc.)
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Range
(Rarely Used)
Usual and
Unusual Events
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 Values
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Rule of Thumb
Values in a data set are those that are
typical and not too extreme.
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.
Max. Usual Value = (Mean) – 2*(s.d.)
Min. Usual Value = (Mean) + 2*(s.d.)
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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Z-Score
Also called “standardized value”
Measures of
Relative Standing
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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Interpreting Z-Scores
Sample
Population
Whenever a value is less than the mean, its
corresponding z score is negative
Round z scores to 2 decimal places
Ordinary values:
z score < –2 or z score > 2
Unusual values:
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Percentiles
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Finding the Percentile of a Value
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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–2 ≤ z score ≤ 2
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
Q1, Q2, Q3
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.
Divide ranked scores into four equal parts
 Q1 (First Quartile)
separates the bottom 25% of sorted values
from the top 75%.
25%
 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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(minimum)
25%
25% 25%
Q1 Q2 Q3(maximum)
(median)
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Other Statistics
5-Number Summary
 Interquartile Range (or IQR): Q3 – Q1
 Semi-interquartile Range:
Q3 – Q1
1.The minimum value
2
 Midquartile:
For a set of data, the 5-number
summary consists of
2.First quartile (Q1)
Q3 + Q1
3.Median (Q2)
2
 10 - 90 Percentile Range: P90 – P10
4.Third quartile (Q3)
5.The maximum value.
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