Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e © 2010 Pearson Education, Inc. All rights reserved. Basic Statistical Concepts Sang

Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e

© 2010 Pearson Education, Inc.All rights reserved.

Basic Statistical ConceptsSang Min Lee

Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1eSheperis/Daniels/Young


15-2

Learning Goals1. The basic principles of statistical

analysis.2. The differences between various

scales of measurement.3. Concepts of reliability and validity.4. Basic Descriptive statistics.5. Variability.6. Measures of central tendency.



15-3

Measurement

In statistics, the level of measurement of a variable is a classification intended to describe the nature of information contained within numbers assigned to objects, and therefore within the variable.



15-4

Four Levels of Measurement

1. Nominal2. Ordinal3. Interval4. Ratio



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Nominal

Examples

Key Characteristics

Mathematical Operations

Appropriate Statistics

Nominal

GenderEthnicityMaritalStatusReligion

Identity None

Chi Square (2)Percent ofTotal

Mode



15-6

Ordinal

Examples

Key Characteristics



Ordinal

Top 10 rating for musicThe rankorder ofanything

Identity

Magnitude

Ranking Median

Interquartile

range



15-7

Interval

Examples Key Characteristics



Interval

TemperatureIQCurved Grades

Identity

Magnitude

Equal interval

Addition

Subtraction

MeanStandard DeviationCorrelational&RegressionANOVAFactor Analysis



15-8

Ratio

Examples

Key Characteristics



Ratio

AgeWeightAnnual Income

IdentityMagnitudeEqual IntervalTrue Zero

Addition SubtractionMultiplicationDivision

All possiblestatistics



15-9

Reliability and Validity

Two concepts important for defining and measuring bias and distortion.

Reliability refers to the extent to which assessments are consistent.

Validity is the extent to which a test measures the construct it is intended to measure.



15-10

Four Reliability Indices

1. Test-Retest Reliability: Reliability coefficient is obtained by administering the same test twice and correlating the scores.

2. Parallel Form Reliability: reliability coefficient is obtained by administering similar, but not identical tests and correlating the scores.



15-11

Four Reliability Indices

3. Split-Half Reliability: reliability score is obtained by diving the test into halves, correlating the scores on each half.

4. Internal Consistency: Reliability score is obtained by correlating the individual items of a test to each other.



15-12

Reliability Limitations Memory Effect: When using a test-retest

approach, responses may be overly consistent because participants remember some of the questions asked previously.

Practice Effect: With a parallel forms approach, respondents have taken one version of a test then take a similar form at a later date which may impact results.



15-13

Reliability Concepts Internal Consistency: focuses on the

degree to which the individual items are correlated with each other.

Homogeneous items measure a single construct and correlate highly with each other.

Kuder-Richardson Formula 20 (KR-20) and Cronbach’s Alpha are measures of homogeneity.



15-14

Three Major Approaches to Validity

Content: the extent to which the measurement adequately samples the content domain .

Construct: The extent to which the test is an accurate measure of a particular construct or variable.

Criterion: the extent to which a test is related to some external criterion of the construct being measured.



15-15

Construct Validity One way of assessing construct validity is to

demonstrate the evidence of convergent and Discriminant validity.

Convergent validity refers to the degree to which scores correlated highly with other tests designed to assess the same construct.

Discriminant validity refers to the degree scores do not correlate highly with other tests that are not designed to assess the same trait.



15-16

Criterion Validity

Two types of Criterion Validity:1. Concurrent validity: yields scores at

the time of administration, and2. Predictive validity: yields scores at

a later time from administration.



15-17

Descriptive Statistics Descriptive Statistics are used to

describe the basic features of the data in a study including when the data are or what they show.

The Frequency Distribution is a summary of the frequency of individual values or ranges of values for a variable.



15-18

Frequency DistributionYears of

Experience (X)

Frequency (f)

Percentage

Cumulative

Percentage

13-15 1 10.0% 100.00%

10-12 0 0.0% 90.0%

7-9 1 10.0% 90.0%

4-6 5 50.0% 80.0%

1-3 3 30.0% 30.0%

N=10 100.00%



15-19

Histogram

Histogram: also known as a bar graph, uses bars to indicate the frequencies for each value of X (which are placed along the horizontal x-axis). The height of the bars indicates the frequencies (which are on the vertical Y-axis).

0%

10%

20%

30%

40%

50%

60%

1~3 4~6 7~9 10~12 13~15



15-20

Frequency PolygonPlots data on the x-axis and frequency on the y-axis. More effective when comparing two or more distributions within the same figure.

0%

10%

20%

30%

40%

50%

60%

1~3 4~6 7~9 10~12 13~15



15-21

Types of Distribution For quantitative variables, the pattern

of scores in the distribution can be determined by visually examining the frequency polygon.

In large data sets if a pattern exists it typically fits one of three patterns: Symmetrical Skewed Multimodal



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Symmetrical Distribution

Positively SkewedNormal Distribution

Negatively SkewedNormal Distribution



15-23

Measures of Central Tendency

There are several common methods to estimate measure of central tendency.

Intended to describe the most average scores in the distribution.

Most common measures are the mean, the median, and the mode.



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Mean Most commonly used

method of describing central tendency.

Arithmetic average of all scores.

Add up all of the values and divide by the number of values.

Represented by the following equation:

Add equation 19.4



15-25

Median The midpoint of the

ordered list of values.

List the values in rank order, then find the point below which one-half of the scores lie.

Represented by this mathematical equation:

Add equation 19.3



15-26

Mode The easiest measure to understand

since it is determined by inspection rather than computation.

Reports the most frequent score in the variable.

Useful when studying nominal variables.

Not often a useful indicator of central tendency in a distribution.



15-27

Variability Refers to the extent to which the scores

in a distribution differ from each other. A distribution lacking variability is

referred to as homogenous. Distribution with much variability is

referred to as heterogeneous. Three frequently used measures of

variability are the range, variance, and standard deviation.



15-28

Range

The simplest measure of variability. Take the highest score and subtract the

lowest score. A quick measure of variability but

excludes a information from the other scores.



15-29

Variance Represents how

close the scores in the distribution are to the mean.

Variance is the average of the squared deviations from the mean.

Represented by the following equation:

Add equation 19.6



15-30

Standard Deviation

The standard deviation is the square root of the variance.

Indicates the average difference between individual scores and the group mean.

Add equation 19.9



15-31

Standard Normal Distribution & Standard Scores Z Scores: if the mean and standard deviation

are known, individual scores can be represented relative to the entire set of scores in the distribution through standardization.

When you standardize a raw score to a Z score, it provides information about how far a person is from the mean.

One standard deviation above the mean has a Z-score of 1, one below = -1, and at the mean = 0.



15-32

Standard Normal Distribution and Z Scores



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T Scores Standardized scores

which are widely used to report performance on standardized tests and inventories.

A score of 50 represents the mean.

A difference of 10 from the mean indicates a difference of one standard deviation.

T= 50 + 10z



15-34

Standard Normal Distribution and T-Scores



15-35

Stanines Relating stanines to the normal curve,

each stanine score represents a wide band of raw scores and percentile ranks.

A normal distribution is divided into 9 intervals, each of which has a width of one half of a standard deviation excluding the first and last intervals.

Useful in comparing performance across content areas.



15-36

Standard Normal Distribution and Stanines

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Counseling Research: Quantitative, Qualitative, and Mixed Methods, 1e © 2010 Pearson Education, Inc. All rights reserved. Basic Statistical Concepts Sang