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Chapter 12: Descriptive Statistics

Objectives



Describe the process of tabulating
and coding data.
Define frequency and central
tendency, and differentiate among
mean, median, and mode.
Define variability, and differentiate
among the range, quartile
deviation, variance, and standard
deviation.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
1
© 2009 Pearson Education, Inc.
All rights reserved.
Chapter 12: Descriptive Statistics



Describe the major characteristics of
normal and skewed distributions.
Define and differentiate among
measures of relative position,
including percentile ranks and
standard scores.
Define and differentiate among two
measures of relationship, the Pearson
r and the Spearman rho.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
2
© 2009 Pearson Education, Inc.
All rights reserved.
Descriptive Statistics

Statistics is a set of procedures for
describing, synthesizing, analyzing, and
interpreting quantitative data.


The mean is an example of a statistic.
One can calculate statistics by hand or
can use the assistance of statistical
programs.

Excel, SPSS, and many other programs exist.
Some programs are also available on the Web
to analyze datasets.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
3
© 2009 Pearson Education, Inc.
All rights reserved.
Preparing Data for Analysis



After data are collected, the first step
toward analysis involves converting
behavioral responses into a numerical
system or categorical organization.
It is critical that all data are scored
accurately and consistently.
Data scoring should be doublechecked for consistency and accuracy
(i.e., at least 25% of all cases should
be checked).
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
4
© 2009 Pearson Education, Inc.
All rights reserved.
Preparing Data for Analysis


Open-ended items should be scored
by two scorers to check reliability.
All data scoring and coding
procedures should be documented
and reported in the written report.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
5
© 2009 Pearson Education, Inc.
All rights reserved.
Preparing Data for Analysis

After instruments are scored, the resulting data
are tabulated and entered into a spreadsheet.

Tabulation involves organizing the data systematically
(e.g., by participant).
ID
Cond.
Gender
Grade
Ach
Mot
001
1
2
10
78
52
002
2
2
11
82
62
003
1
1
11
86
74
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
6
© 2009 Pearson Education, Inc.
All rights reserved.
Preparing Data for Analysis





In this potential dataset,
ID represents participant number
Cond. is the experimental condition (1 or 2)
Gender is represented by female=1; male=2
Achievement (Ach) and motivation (Mot) were also
variables assessed
ID
Cond.
Gender
Grade
Ach
Mot
001
1
2
10
78
52
002
2
2
11
82
62
003
1
1
11
86
74
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
7
© 2009 Pearson Education, Inc.
All rights reserved.
Types of Descriptive Statistics


After data are tabulated and entered,
the next step is to conduct descriptive
statistics to summarize data.
In some studies, only descriptive
statistics will be conducted.


If the indices are calculated for a sample,
they are referred to as statistics.
If indices are calculated for the entire
population, they are referred to as
parameters.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
8
© 2009 Pearson Education, Inc.
All rights reserved.
Types of Descriptive Statistics

Frequencies


The frequency refers to the number of
times something occurs.
Frequencies are often used to describe
categorical data.


We might want to have frequency counts of
how many males and females were in a study
or how many participants were in each
condition.
Frequency counts are not as helpful in
describing interval and ratio data.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
9
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Central Tendency


Measures of central tendency are
indices that represent a typical score
among a group of scores.
Measures of central tendency provide
a way to describe a dataset with a
single number.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
10
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Central Tendency

The three most common measures of
central tendency are the mean,
median, and mode.
 Mean: Appropriate for describing
interval or ratio data
 Median: Appropriate for describing
ordinal data
 Mode: Appropriate for describing
nominal data
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
11
© 2009 Pearson Education, Inc.
All rights reserved.
(XX )
Measures of Central Tendency

The mean is the most commonly used
measure of central tendency.
 The formula for the mean is:
X= ∑Xi/n

To calculate the mean, all the scores
are summed and then divided by the
number of scores.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
12
© 2009 Pearson Education, Inc.
All rights reserved.
(XX )
The Mean or Average Value
Example
What is the mean of
4 3 6 8 4
Mean = (4+3+6+8+4)/5 = 5
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
13
© 2009 Pearson Education, Inc.
All rights reserved.
(XX )
The Median Value



The median is the midpoint in a distribution:
50% of the scores are above the median
and 50% are below the median.
To determine the median, all scores are
listed in order of value.
If the total number of scores is odd, the
median is the middle score.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
14
© 2009 Pearson Education, Inc.
All rights reserved.
(XX )
The Median Value


If the total number of scores is even, the
median is halfway between the two middle
scores.
Median values are useful when there is large
variance in a distribution.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
15
© 2009 Pearson Education, Inc.
All rights reserved.
(XX )
The Median Value
Example 1: Use the previous data put in order
3 4 4 6 8
The Median is (the middle value) = 4
Example 2:
3 4 4 6 8 9
The Median is (4+6/2) = 5
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
16
© 2009 Pearson Education, Inc.
All rights reserved.
(XX )
The Mode



The mode is the most frequently occurring
score in a distribution.
The mode is established by looking at a set of
scores or at a graph of scores and determining
which score occurs most frequently.
The mode is of limited value.

Some distributions have more than one mode (e.g.,
bi-modal, or multi-modal distributions)
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
17
© 2009 Pearson Education, Inc.
All rights reserved.
(XX )
Measures of Central Tendency

Deciding among measures of central
tendency




Generally the mean is most preferred.
The mean takes all scores into account.
The mean, however, is greatly influenced by
extreme scores- outlying data.
When there are extreme scores present in a
distribution, the median is a better measure of
central tendency.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
18
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Variability


Measures of variability provide an index of
the degree of spread in a distribution of
scores.
Measures of variability are critical to
examine and report because some
distributions may be very different but yet
still have the same mean or median.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
19
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Variability

Three common measures of variability are
the range, quartile deviation, and
standard deviation.
 Range: The difference between the
highest and lowest score.


The range is not a stable measure.
The range is quickly determined.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
20
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Variability

Quartile Deviation: One half the difference
between the upper quartile and the lower
quartile in a distribution.


By subtracting the cutoff point for the lower
quartile from the cutoff point for the upper
quartile and then dividing by two we obtain a
measure of variability.
A small number indicates little variability and
illustrates that the scores are close together.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
21
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Variability

Variance: The amount of spread among
scores. If the variance is small the scores
are close together. If the variance is large
the scores are spread out.


Calculation of the variance shows how far each
score is from the mean.
The formula for the variance is:
∑(X–X)2/n
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
22
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Variability

Standard deviation: The square root of
the variance.


The standard deviation is used with interval
and ratio data.
The standard deviation is the most commonly
used measure of variability.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
23
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Variability


If the mean and the standard deviation are
known, the distribution can be described fairly
well.
SD represents the standard deviation of a
sample and the symbol  (i.e., the Greek lower
case sigma) represents the standard deviation
of the population.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
24
© 2009 Pearson Education, Inc.
All rights reserved.
Standard Deviation – 3 cases with the same Mean
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
25
© 2009 Pearson Education, Inc.
All rights reserved.