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CRP phase 4-

Analyzing andInterpreting Quantitative

Data

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Describing Quantitative Data

Quantitative data -Data that can be quantified and verified, and is amenable

to statistical manipulation.

Quantitative data defines whereas qualitative datadescribes.

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I.MAKING SENSE OF NUMBERS:

STATISTICAL DATA ANALYSIS

Data analysis is the process of examining whatdata mean to researchers. Statistical data

analysis refers to the process of examining what

quantitative data mean to researchers. Statisticsrefers to any numerical indicator of a set of data.

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Descriptive Statistical Data Analysis:

- method used to describe the population we

are studying.Frequency distributions, measures of central tendency

(mean, median, and mode), and graphs like pie charts and bar charts

that describe the data are all examples of descriptive statistics.

Inferential Data Analysis:

- making predictions or inferences about a

population from observations and analyses of a

sample.

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2 PURPOSES

Estimates the characteristics of a population

from data gathered on a sample.

Tests for significant DIFFERENCES between

groups and significant RELATIONSHIPS

between variables.

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II. DESCRIBING DATA

THROUGH SUMMARYSTATISTICS

Summary statistics provide an efficient way to

describe an entire set of quantitative data.

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A. Measures of CENTRAL

TENDENCY

describe the central point of distribution(representative values)

The measure of central tendency describes theone score that best represents the entiredistribution, the most CHARACTERISTIC

SCORE. The most characteristic score is thatwhich describes the center point of adistribution of scores.

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B. Measures of DISPERSION

Describe how scores differ; how scores are

SPREAD (measures of VARIABILITY)

Do not exist for nominal data because categories

are used rather than meaningful numbers.

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RANGE (or span) - number that reports the distancebetween the highest and lowest scores in a distribution.

VARIANCE - number that represents the mathematical

index of the average distance of the scores on an intervalor ratio scale from the mean in squared units.

STANDARD DEVIATION- number that represents

a summary statistic of how much scores vary from themean - expressed in the original units of measurement.

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C. DESCRIBING DATA IN

STANDARD SCORES Standard scores provide a common unit of measurementindicating how far away any particular score is from themean. Used to compare numbers(scores) from differentdistributions.

Z score- most popular standard score used by researchers. The z score is calculated by dividing the deviation score by the

standard deviation (X - X bar) divided by the standarddeviation.

Each z score indicates how many standard deviations that score

is from the mean of the distribution.

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B. FREQUENCY Distributions

Counting and reportingHOW OFTEN

different categories or points on a measurement

scale occur.

A list of the frequency of responses for each

category or measurement point is called a

frequency table.

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Show changes over time Frequency Tables - a total number of times particular values on

a measurement scale occur in a data set.

Pie Charts - illustrate the frequency counts of categories. Bar Charts - visually illustrate frequency counts for a nominal

and ordinal variables.

Line Graphs - use a single point to represent the frequency

count on a dependent variable for each of the groups. Interval and Ratio level variable frequencies

are illustrated using... Frequency Histograms - distribution of frequencies where

blocks touch Frequency Polygons - similar to line graphs except a line

connects the points representing the frequency count for eachpoint on the measurement scale rather than each category.

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CRP Phase 4- Inferring

from Data: Estimationand Significance Testing

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INFERENTIAL STATISTICS :

The set of statistical procedures that allows aresearcher to go beyond the group that has

been measured, and make statements about the

characteristics of a much larger group.

Inferential statistics:accomplish ESTIMATION and

SIGNIFICANCE TESTING.

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I I. ESTIMATION (Estimation Theory)

A. ESTIMATION Estimates of Population PARAMETERS from Sample

STATISTICS are possible as long as 2 ASSUMPTIONSmet: Normal Distribution and Random Sample.

1. Assumes that the charactetristic of interest is measured on anINTERVAL or RATIO scale and that the characteristics aredistributed NORMALLY

2. Assumes that the SAMPLE has been selected RANDOMLY so

it reflects the population.

Estimation procedures are often referred to as PARAMETRICSTATISTICS

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B.The Normal Distribution Normal Curve

(symmetrical bell shaped curve)

Distributions NOT in the shape of a normal curve can bedescribed using two characteristics:

1. Skewness- the majority of the scores are toward one end of adistribution, which makes the curve tail off in that direction. a. Positive - tail runs to right side of the curve;

b. Negative - tail runs to the left side of the curve.

2. Kurtosis- refers to how pointed or flat the shape of a distribution is

when it is not a normal curve. a. Peaked - scores are clusterd around the middle; LEPTOKURTIC

distribution

b. Flat - scores are dispersed evenly across the distribution; PLATYKURTICdistribution

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C. Use of Random Sampling - each

population member has an equal chance of

being selected for the sample. 1. Central Limits Theorem - for samples of a

sufficiently large size, the real distribution

of means is almost always approximately normal. 2. Large samples give more accurate results than

do smaller samples.

3. Random samples allow researchers to operatesafely under the normal curve.

4. Sample must be representative of population.

KISS

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D.Inferring from a Random Sample

to a Population1. Sampling Distribution

a. Sampling distribution meanb. Standard ERROR (SE) of the mean tells researchershow much these random sample means are likely to

differ from the mean of the sampling distribution.

2. Confidence Level - the degree of assurance that asample statistic can be used to infer a population

parameter.

3. Confidence Interval - the range of scores associatedwith this confidence level. Be cautious!

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II. SIGNIFICANCE TESTING

The process of analyzing

quantitative data for the puprose oftesting whether a nulll hypothesis is

probably either correct or

false. Significance testing =hypothesis testing.

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A. The Logic of Significance Testing

1. Testing a Null Hypothesis -- every research hypothesis has a correspondingnull hypothesis.

2. Rejecting a Null Hypothesis -- We do not "PROVE" a research hypothesis.

3. Deciding on the Probability Level (p value)a. The 95% confidence level is referred to as the .05 significance level.

b. A significant difference or relationship found at the .05 level ofprobability means that the researchers are 95% confident that the differenceor relationship is NOT DUE TO CHANCE OR ERROR.

c. The result is only likely to happen BY CHANCE 5 out of 100 times.

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B. The Practice of Significance Testing

1. Step One: Posing a Research Question

or Hypothesis and a Null Hypothesis

2. Step Two: Conducting the Study

3. Step Three: Testing the NullHypothesis

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C. Type I Error and Type II Error

1. Type I error

a. ALPHA errorb. False Positivec. Researcher rejects the null and

accepts a research hypothesisd. Much more serious

2. Type II error

a. BETA error

b. False Negativec. Researcher accepts the null andrejects a research hypothesis

d. Not as serious

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D. Statistical Power 1. POWER Analysis

a. The ability to detect effects of a specific size, given theparticular variance and sample size of the studyb. Equal to 1 minus the probability of committing a Type II

error (failing to reject a false null hypothesis)c. Minimum statistical power is 0 and maximum is 1.0d. Statistical Power should be .80 at a minimum

2. Increasing Statistical Power

a. Employ the most sensitive (and appropriate) statisticaltest

b. Test one-tailed hypotheses and research questionsc. Increase the size of the random sample studied

3. Effect Size

a. An estimate of the degree to which a phenomenon ispresent in a population and/or

b. The extent to which the null hypothesis is false

c. Three Types of Effect Sizes1) Small n= 3932) Medium n= 643) Large n= 26

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III. IMPORTANT CONCEPTUAL

DEFINITIONS

Parameters - a characteristic of a population or a universe.

Statistic - the measurement of a sample with respect to avariable.

Nonparametric Statistics -statistics used only to describe thecharacteristics of a sample, without being able to generalize

back to its population.

Parametric Statistics - statistics used to estimate thecharacteristics (parameters) of a population based on the

characteristics (statistics) of a sample.

Data Analysis - the methods researchers use to infermeaning from data, to determine what conclusions arejustifed.