practice 4 for cj stat
ismails95
4MeasuresofVariability-FORPRACTICE.ppt© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved
Fox/Levin/Forde, Elementary Statistics in Criminal Justice Research, 4e
Chapter 4: Measures of Variability
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© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved
Calculate the range
Calculate the variance and standard deviation
Obtain the variance and standard deviation from a simple frequency distribution
Understand the meaning of the standard deviation
Calculate the coefficient of variation
CHAPTER OBJECTIVES
4.1
4.2
4.3
4.4
4.5
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Calculate the range
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
4.1
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Introduction
4.1
Measures of Central Tendency
Measures of Variability
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- Summarizes what is average or typical of a distribution
- Summarizes how scores are scattered around the center of the distribution
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4.1
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The difference between the highest and lowest scores in a distribution
- Provides a crude measure of variation
- Can be strongly affected by one case
- As such may not give a precise indication of variability
- should be considered a preliminary or rough index
The Range
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Calculate the variance and standard deviation
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
4.2
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4.2
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We need a measure of variability that takes into account every score.
- Deviation: the distance of any given raw score from the mean
- The sum of actual deviations will always be zero
- Squaring deviations eliminates the minus signs
The Variance
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4.2
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Variance
- Summing the squared deviations and dividing by N gives us the average of the squared deviations
The Variance
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4.2
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With the variance, the unit of measurement is squared.
- It is difficult to interpret squared units
- We can remove the squared units by taking the square root of both sides of the equation
- This will give us the standard deviation
The Standard Deviation
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Step by Step – Finding the Standard Deviation
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mean = 5
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mean = 5
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Summary of Steps for Standard Deviation
Step 1 Find the mean for the distribution
Step 2 Subtract the mean from each raw score to get the deviation
Step 3 Square each deviations before adding together the squared deviations
Step 4 Divide by N and take the square root of the result
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4.2
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There is an easier way to calculate the variance and standard deviation.
- Raw score method
The Raw-Score Formulas
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Step by Step
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Summary of Steps
Step 1 Square each raw score and then add them together
Step 2 Obtain the mean and square it
Step 3 Insert results from Step 1 and 2 into the formulas
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Obtain the variance and standard deviation from a simple frequency distribution
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
4.3
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Summary of Steps
Step 1 Multiple each score value (X) by its frequency (f) to obtain the fX products and then sum the fX products
Step 2 Square each score value (X2) and multiply by its frequency (f) to obtain the fX2 products and then sum the fX2 column
Step 3 Obtain the mean and square it
Step 4 Calculate the variance using the results from previous steps
Step 5 Calculate the standard deviation (the square root of the variance)
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Example
4.3
Obtaining the variance and standard deviation from a simple frequency distribution
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| X | f | fX | fX2 |
| 31 | 1 | 31 | 961 |
| 30 | 1 | 30 | 900 |
| 29 | 1 | 29 | 841 |
| 28 | 0 | 0 | 0 |
| 27 | 2 | 54 | 1,458 |
| 26 | 3 | 78 | 2,028 |
| 25 | 1 | 25 | 625 |
| 24 | 1 | 24 | 576 |
| 23 | 2 | 46 | 1,058 |
| 22 | 2 | 44 | 968 |
| 21 | 2 | 42 | 882 |
| 20 | 3 | 60 | 1,200 |
| 19 | 4 | 76 | 1,444 |
| 18 | 2 | 36 | 648 |
| 575 | 13,589 |
Calculate the coefficient of variation
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
4.5
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Coefficient of Variation
- sometimes researchers want to compare the variability for two or more characteristics that have been measured in different units
- study variability of hours worked per week as well as hourly wages hours among COs in a state penitentiary
which has greater spread wages per hour or hours per week
- might think to calculate the SD
however, the value of this is meaningless as it depends on the unit of measurement
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4.5
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- coefficient of variation is based on the size of the SD but its value is independent of the unit of measurement
- expresses the SD as a percentage of the mean (see p 72)
The Coefficient of Variation
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Understand the meaning of the standard deviation
Learning Objectives
After this lecture, you should be able to complete the following Learning Outcomes
4.4
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4.4
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The standard deviation converts the variance to units we can understand.
But how do we interpret this new score?
- The standard deviation represents the average variability in a distribution.
- It is the average of deviations from the mean.
- The greater the variability, the larger the standard deviation.
The Meaning of the Standard Deviation
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SD allows us to
- measure the degree of variability in a distribution
- OR to compare the variability in different distributions
- used to calibrate the relative standing of individual scores within a distribution
- deviations above the mean are plus
- deviations below the mean are minus
- about 2/3 of scores (in a normal distribution) fall within 1 SD above and below the mean
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Comparing Measures of Variability
range – rough index of the variability of a distribution
simple and quick, but not very reliable
size of SD is an approximately 1/6 of the range
smaller number of cases will result in fewer SDs to cover the range
variance and the SD take into account every score in a distribution
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© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved
Researchers can calculate the range for a crude measure of variation.
The variance and standard deviation provide the researcher with a measure of variation that takes into account every score.
The variance and standard deviation can also be calculated for data presented in a simple frequency distribution.
The standard deviation can be understood as the average of deviations from the mean.
The coefficient of variation is used to compare the variability for two or more characteristics that have been measured in different units.
CHAPTER SUMMARY
4.1
4.2
4.3
4.4
4.5
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RHL
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range
highest score in a distribution
lowest score in a distribution
R
H
L
=
=
=
(
)
2
2
XX
s
N
-
=
å
(
)
2
2
variance
sum of the squared deviations from the
mean
total number of scores
s
XX
N
=
-=
=
å
(
)
2
XX
s
N
-
=
å
2
22
X
sX
N
=-
å
2
2
X
sX
N
=-
å
2
2
variance
standard deviation
total number of scores
mean squared
s
s
N
X
=
=
=
=
2
2
2
2
2
2
2
575
23
25
(23)529
13,589
529543.5652914.56
25
14.563.82
fX
X
N
X
fX
sX
N
fX
sX
N
===
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=-=-=-=
=-==
å
å
å
100
s
CV
X
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=
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coefficient of variation
standard deviation
mean
CV
s
X
=
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