The document discusses key concepts in descriptive statistics including measures of central tendency (mean, median, mode) and measures of variation (range, standard deviation, variance). It defines these concepts and provides formulas for calculating them. Examples are given of how to interpret standard deviation values and apply the empirical rule regarding how data is distributed within a certain number of standard deviations from the mean.

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Created by Tom Wegleitner, Centreville, Virginia
Edited 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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Copyright © 2007 Pearson
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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Copyright © 2007 Pearson
Education, Inc Publishing as
Pearson Addison-Wesley.
Definition
 Measure of Center
the 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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Education, Inc Publishing as
Pearson Addison-Wesley.
Definition

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Σ 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
Σ x
is pronounced ‘x-bar’ and denotes the mean of a set
of sample values
x
N
µ =
Σ x

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 Median
the 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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 Mode
the value that occurs most frequently
 Mode is not always unique
 A data set may be:
Bimodal
Multimodal
No 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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Copyright © 2007 Pearson
Education, Inc Publishing as
Pearson Addison-Wesley.
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.
Copyright © 2007 Pearson
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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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Σ (x - x)2
n - 1
s =

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2
Σ (x - µ)
N
σ =
This formula is similar to the previous formula, but
instead, the population mean and population size
are used.

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Education, Inc Publishing as
Pearson Addison-Wesley.
 Population variance: Square of the population
standard deviation σ
 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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Education, Inc Publishing as
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standard deviation squared
s
σ
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.
Copyright © 2007 Pearson
Education, Inc Publishing as
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Round only the final answer, not values in
the middle of a calculation.

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Education, Inc Publishing as
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Estimation of Standard Deviation
Range Rule of Thumb
For estimating a value of the standard deviation s,
Use
Where range = (maximum value) – (minimum value)
Range
4
s ≈

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Copyright © 2007 Pearson
Education, Inc Publishing as
Pearson Addison-Wesley.
Estimation of Standard Deviation
Range 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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Copyright © 2007 Pearson
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Definition
Empirical (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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The Empirical Rule

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The Empirical Rule

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Copyright © 2007 Pearson
Education, Inc Publishing as
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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 =
σ
Round z to 2 decimal places
Measures of Position z score
z = x - x
s

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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