Post on 21-Jan-2017
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Created by Tom Wegleitner, Centreville, VirginiaEdited by Olga Pilipets, San Diego, California
Overview
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Descriptive Statistics summarize or describe the important
characteristics of a known set of data
Inferential Statistics use sample data to make inferences
(or generalizations) about a population
Copyright © 2007 Pearson Education, Inc Publishing as
Pearson Addison-Wesley.
Overview
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Education, Inc Publishing as Pearson Addison-Wesley.
Key Concept
In order to be able to describe, analyze, and compare data sets we need further insight into the nature of data. In this chapter we are going to look at such extremely important characteristics of data: center and variation. We will also learn how to compare values from different data sets.
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Definition
Measure of Centerthe value at the center or middle of a data set
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Arithmetic Mean(Mean)
the measure of center obtained by adding the values and dividing the total by the number of values
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Definition
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denotes the sum of a set of values. x is the variable usually used to represent the
individual data values. n represents the number of values in a
sample. N represents the number of values in a
population.
Copyright © 2007 Pearson Education, Inc Publishing as
Pearson Addison-Wesley.
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µ is pronounced ‘mu’ and denotes the mean of all values in a population
Copyright © 2007 Pearson Education, Inc Publishing as
Pearson Addison-Wesley.
Notation
x =n
xis pronounced ‘x-bar’ and denotes the mean of a set of sample values
x
Nµ =
x
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Medianthe middle value when the original data values are arranged in order of increasing (or decreasing) magnitude
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Definitions
often denoted by x (pronounced ‘x-tilde’)~
is not affected by an extreme value
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Modethe value that occurs most frequently
Mode is not always unique A data set may be:
BimodalMultimodalNo Mode
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Definitions
Mode is the only measure of central tendency that can be used with nominal data
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Midrange
the value midway between the maximum and minimum values in the original data set
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Definition
Midrange =maximum value + minimum value
2
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Because this section introduces the concept of variation, which is something so important in statistics, this is one of the most important sections in the entire book.
Place a high priority on how to interpret values of standard deviation.
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The range of a set of data is the difference between the maximum value and the minimum value.
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Range = (maximum value) – (minimum value)
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The standard deviation of a set of sample values is a measure of variation of values about the mean.
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(x - x)2
n - 1s =
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2 (x - µ)
N =
This formula is similar to the previous formula, but instead, the population mean and population size are used.
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Population variance: Square of the population standard deviation
The variance of a set of values is a measure of variation equal to the square of the standard
deviation.
Sample variance: Square of the sample standard deviation s
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standard deviation squared
s
2
2
}Notation
Sample variance
Population variance
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Carry one more decimal place than is present in the original set of data.
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Round only the final answer, not values in the middle of a calculation.
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Estimation of Standard DeviationRange Rule of Thumb
For estimating a value of the standard deviation s,Use
Where range = (maximum value) – (minimum value)
Range4
s
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Estimation of Standard DeviationRange Rule of Thumb
For interpreting a known value of the standard deviation s, find rough estimates of the minimum and maximum “usual” sample values by using:
Minimum “usual” value (mean) – 2 X (standard deviation) =Maximum “usual” value (mean) + 2 X (standard deviation) =
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Definition
Empirical (68-95-99.7) RuleFor 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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The Empirical Rule
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The Empirical Rule
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The Empirical Rule
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z Score (or standardized value)the number of standard deviations
that a given value x is above or below the mean
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Definition
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Sample
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Population
x - µz =
Round z to 2 decimal places
Measures of Position z score
z = x - xs
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Interpreting Z Scores
Whenever a value is less than the mean, its corresponding z score is negative
Ordinary values: z score between –2 and 2 Unusual Values: z score < -2 or z score > 2