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**QUANTITATIVE DATA ANALYSIS**

Chapter 11

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LEVELS OF MEASUREMENT Variable attributes: the characteristics or qualities that describe a variable Variable attributes can be defined at four different levels of measurement Nominal Ordinal Interval Ratio

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**Nominal Measurement The lowest level of measurement**

Attributes or response categories of a variable are mutually exclusive

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**Ordinal Measurement Second highest level of measurement**

Attributes or responses categories or a variable are Mutually exclusive Rank ordered

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**Interval Measurement Third highest level of measurement**

Attributes or responses categories or a variable are Mutually exclusive Rank ordered Equal distance from each other

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**Ratio Measurement Highest level of measurement**

Attributes or responses categories or a variable are Mutually exclusive Rank ordered Equal distance from each other Based on a true 0 point

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**COMPUTER APPLICATIONS**

Variables must be coded (assigned a distinct value) for data to be processed by computer software The researcher must know the level of measurement for each variable to determine which statistical tests to use

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**DESCRIPTIVE STATISTICS**

Summarize a variable of interest and portray how that particular variable is distributed in the sample or population Frequency distributions Measures of Central Tendency Measures of Variability

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**Frequency Distributions**

A counting of the occurrences of each response value of a variable, which can be presented in Table form Graphic form (Frequency Polygon)

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**Measures of Central Tendency**

The value that represents the typical or average score in a sample or population Three types: Mode, Median, and Mean Normal Curve: a bell-shaped frequency polygon in which the mean, median, and mode represent the average equally (See Figure 17.4)

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Mode The score or response value that occurs most often (i.e., has the highest frequency) in a sample or population Minimum level of measurement is nominal

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Median The score or response value that divides the a distribution into two equal halves Minimum level of measurement is ordinal

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Mean Calculated by summing individual scores and dividing by the total number of scores The most sophisticated measure of central tendency Minimum level of measurement is interval

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**Measures of Variability**

A value or values that indicated how widely scores are distributed in a sample or population; a measure of dispersion Two common types Range Standard Deviation

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Range The distance between the minimum and maximum score in a distribution The larger the range, the greater the amount of variation of scores in a distribution Minimum level of measurement is ordinal

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Standard Deviation A mathematically calculated value that indicates the degree to which scores in a distribution are scattered or dispersed about the mean The mean and standard deviation define the basic properties of the normal curve Minimum level of measurement is interval

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**INFERENTIAL STATISTICS**

Make it possible to study a sample and “infer” the findings of that study to the population from which the sample was randomly drawn Based on chance or probability of error Commonly accepted levels of chance are p < .01 (1 in 100) and p < .05 (5 in 100)

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**Statistics that Determine Associations**

Statistics that determine whether or not a relationship exists between two variables The values of one variable co-vary with the values of another variable Chi-square (2) Correlation (r)

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**Chi-Square (2) Used with nominal or ordinal levels of measurement**

Provides a measure of association based on observed (actual scores) and expected (statistically estimated) frequencies The direction or strength of the relationship between the two variables is not specified

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Correlation (r) Typically used with interval and ratio levels of measurement A measure of association between two variables that also indicates direction and strength of the relationship r=0 (no relationship), r=1.00 (perfect relationship) A +r value (a direct relationship), -r value (an inverse relationship)

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**Statistics that Determine Differences**

Statistics used to determine whether group differences exist on a specified variable Differences between Two related groups: Dependent t-test Two unrelated groups: Independent t-test More than two groups: ANOVA

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Dependent t-test Used to compare two sets of scores provided by one group of individuals Example: pretest and posttest scores

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Independent t-test Used to compare two sets of scores, each provided by a different group of individuals Example: Fathers and Mothers

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**One-Way Analysis of Variance**

Used to compare three or more sets of scores, each provided by a different group of individuals Example: Fathers, Mothers, and Children

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**SUMMARY Statistics are used to analyze quantitative data**

The level of measurement must be specified for each variable Descriptive and Inferential statistics are used to build knowledge about a sample or population

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