Download Standard Deviation

Measure of Center
The value at the center or middle of a
data set
Measures of
Center
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1. Mean
2. Median
3. Mode
4. Midrange (rarely used)
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Mean
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Notation
 denotes the sum of a set of values.
The measure of center obtained by
adding the values and dividing the total
by the number of values
x is the variable used to represent the
individual data values.
n represents the number of data values in a
sample.
What most people call an average.
N represents the number of data values in a
population.
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Mean
Advantages
Is relatively reliable.
Takes every data value into account
This is the sample mean
Disadvantage
Is sensitive to every data value, one
extreme value can affect it dramatically; is
not a resistant measure of center
µ is pronounced ‘mu’ and denotes the
mean of all values in a population
This is the population mean
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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
0.42
Example 2
0.73
0.48
1.10
0.66
5.40
Order from smallest to largest:
0.42
0.48
0.66
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0.73
1.10
1.10
1.10
0.42
0.73
0.48
1.10
Order from smallest to largest:
5.40
0.42
0.48
0.73
1.10
5.40
Middle values
Middle value
MEDIAN is 0.73
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1.10
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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Midrange
Midrange
Sensitive to extremes
because it uses only the maximum and
minimum values.
The value midway between the maximum
and minimum values in the original data set
Midrange is rarely used in practice
Midrange =
maximum value + minimum value
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Round-off Rule for
Measure of Center
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Common
Distributions
Carry one more decimal place than is
present in the original set of values
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Symmetry and skewness
Skewed and Symmetric
Symmetric
Distribution of data is symmetric if
the left half of its histogram is roughly
a mirror image of its right half.
Skewed
Distribution of data is skewed if it is
not symmetric and extends more to
one side than the other.
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Measure of Variation
The spread, variability, of data width
of a distribution
Measures of
Variation
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1. Standard Deviation
2. Variance
3. Range (rarely used)
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Standard Deviation
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Sample Standard Deviation
Formula
The standard deviation of a set of
sample values, denoted by s, is a
measure of variation of values about
the mean.
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Sample Standard Deviation
Formula (Shortcut Formula)
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Population Standard Deviation
Formula
is pronounced ‘sigma’
This formula only has a theoretical
significance, it cannot be used in practice.
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Variance
Example
The measure of variation equal to the
square of the standard deviation.
Sample variance: s2 - Square of the
sample standard deviation s
Population variance: 2 - Square of
the population standard deviation 
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Notation
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Example
s = sample standard deviation
s = 7.0
s2 = sample variance
s2 = 49.0
 = population standard deviation
 = 5.7
 2 = population variance
 2 = 32.5
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Values: 1, 3, 14
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Range
(Rarely Used)
The difference between the maximum data
value and the minimum data value.
Using StatCrunch
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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(1) Enter values into first column
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(2) Select Stat, Summary Stats, Columns
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(3) Select var1 (the first column)
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(4) Click Next
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(5) The highlighted stats will be displayed
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Optional: Select Unadf. Variance Unadj. Std. Dev.
(the population variance and standard deviation)
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(6) Click Calculate and see the results
Usual and
Unusual Events
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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
Measures of
Relative Standing
Also called “standardized value”
The number of standard deviations
that a given value x is above or below
the mean
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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:
Unusual values:
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–2 ≤ z score ≤ 2
z score < –2 or z score > 2
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Percentiles
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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Percentile of value x =
L=
k
100
•n
k
L
Pk
• 100
total number of values
Round it off to the nearest whole number
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Converting from the
kth Percentile to the
Corresponding Data Value
Finding the Data Value of the
k-th Percentile
n
number of values less than x
total number of values in the
data set
percentile being used
locator that gives the
position of a value
kth percentile
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Finding Percentiles Using StatCrunch
Finding Percentiles Using StatCrunch
(1) From The menu shown before, enter the
percentiles you wish to calculate.
Example: “10,20,80,90”
for the 10th, 20th, 80th, and 90th percentiles
(2) The Percentiles will be listed with the
other statistics.
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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%.
(minimum)
Other Statistics
Q3 – Q1
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For a set of data, the 5-number
summary consists of
1.The minimum value
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 Midquartile:
Q1 Q2 Q3(maximum)
5-Number Summary
 Interquartile Range (or IQR): Q3 – Q1
 Semi-interquartile Range:
25% 25%
(median)
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25%
2.First quartile (Q1)
Q3 + Q1
3.Median (Q2)
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 10 - 90 Percentile Range: P90 – P10
4.Third quartile (Q3)
5.The maximum value.
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