descriptive stats saturday

8/14/2019 Descriptive Stats Saturday

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Educational Research

Descriptive Statistics


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Preparing data for analysis Types of descriptive statistics

Central tendency Variation Relative position Relationships

Calculating descriptive statistics


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Preparing Data for Analysis

Issues Scoring procedures Tabulation and coding Use of computers


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Scoring Procedures

Instructions Standardized tests detail scoring instructions Teacher made tests require the delineation of scoring

criteria and specific procedures

Types of items Selected response items - easily and objectively

scored

Open-ended items difficult to score objectively witha single number as the result


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Tabulation and Coding

Tabulation is organizing data Identifying all relevant information to the

analysis Separating groups and individuals within

groups Listing data in columns


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Tabulation and Coding

Coding Assigning identification numbers to subjects Assigning codes to the values of non-

numerical or categorical variables Gender: 1=Female and 2=Male Subjects: 1=English, 2=Math, 3=Science, etc.


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Computerized Analysis

Need to learn how to calculate descriptivestatistics by hand Creates a conceptual base for understanding the

nature of each statistic Exemplifies the relationships among statistical

elements of various procedures

Use of computerized software

SPSS Windows Other software packages


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Purpose to describe or summarize datain a parsimonious manner

Four types Central tendency Variability Relative position

Relationships


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Graphing data afrequency polygon Vertical axis

represents thefrequency with which ascore occurs

Horizontal axisrepresents the scores

themselves


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Central Tendency

Purpose to represent the typical scoreattained by subjects

Three common measures Mode Median Mean


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Central Tendency

Mode The most frequently occurring score Appropriate for nominal data

Median The score above and below which 50% of all scoreslie (i.e., the mid-point)

Characteristics Appropriate for ordinal scales

Doesnt take into account the value of each and every scorein the data


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Central Tendency

Mean The arithmetic average of all scores Characteristics

Advantageous statistical properties Affected by outlying scores Most frequently used measure of central tendency

Formula


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Variability

Purpose to measure the extent to whichscores are spread apart

Four measures Range Quartile deviation Variance

Standard deviation


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Variability

Range The difference between the highest and

lowest score in a data set

Characteristics Unstable measure of variability Rough, quick estimate


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Variability

Quartile deviation One-half the difference between the upperand lower quartiles in a distribution

Characteristic - appropriate when the

median is being used


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Variability

Variance

The average squared deviation of allscores around the mean Characteristics

Many important statistical properties

Difficult to interpret due to squared metric Formula


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Variability

Standard deviation The square root of the variance Characteristics

Many important statistical properties Relationship to properties of the normal curve Easily interpreted

Formula


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The Normal Curve

A bell shaped curve reflecting the

distribution of many variables of interestto educators

See Figure 14.2

See the attached slide


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The Normal Curve

Characteristics Fifty-percent of the scores fall above the mean and

fifty-percent fall below the mean The mean, median, and mode are the same values Most participants score near the mean; the further a

score is from the mean the fewer the number ofparticipants who attained that score

Specific numbers or percentages of scores fall

between 1 SD, 2 SD, etc.


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The Normal Curve

Properties Proportions under the curve

1 SD 68%

1.96 SD 95%2.58 SD 99%

Cumulative proportions and percentiles


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Skewed Distributions

Positive many low scores and few high scores Negative few low scores and many high

scores

Relationships between the mean, median, andmode Positively skewed mode is lowest, median is in the

middle, and mean is highest Negatively skewed mean is lowest, median is in the

middle, and mode is highest


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Measures of Relative Position

Purpose indicates where a score is inrelation to all other scores in thedistribution

Characteristics Clear estimates of relative positions Possible to compare students performances

across two or more different tests provided

the scores are based on the same group


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Types Percentile ranks the percentage of scores

that fall at or above a given score

Standard scores a derived score based onhow far a raw score is from a reference pointin terms of standard deviation units Z-score T-score Stanine


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Z-score The deviation of a score from the mean in standard

deviation units The basic standard score from which all other

standard scores are calculated Characteristics

Mean = 0 Standard deviation = 1 Positive if the score is above the mean and negative if it is

below the mean Relationship with the area under the normal curve


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Z-score (continued)

Possible to calculate relative standings likethe percent better than a score, the percentfalling between two scores, the percentfalling between the mean and a score, etc.

Formula


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T-score a transformation of a z-score

where t = 10(Z) + 50 Characteristics

Mean = 50 Standard deviation = 10 No negative scores


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Stanine a transformation of a z-scorewhere the stanine = 2(Z) + 5 rounded tothe nearest whole number Characteristics

Nine groups with 1 the lowest and 9 the highest Categorical interpretation Frequently used in norming tables


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Measures of Relationship

Purpose to provide an indication of therelationship between two variables

Characteristics of correlation coefficients Strength or magnitude 0 to 1 Direction positive (+) or negative (-)

Types of correlations coefficients

dependent on the scales of measurement ofthe variables Spearman Rho ranked data Pearson r interval or ratio data


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Measures of Relationship

Interpretation correlation does notmean causation

Formula for Pearson r


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Calculating Descriptive Statistics

Symbols used in statistical analysis General rules form calculating by hand

Make the columns required by the formula Label the sum of each column Write the formula Write the arithmetic equivalent of the problem Solve the arithmetic problem


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Calculating Descriptive Statistics

Using SPSS Windows Means, standard deviations, and standard

scores

The DESCRIPTIVESprocedures Interpreting output

Correlations The CORRELATIONprocedure Interpreting output


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Formula for the Mean


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Formula for Variance


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Formula for Standard Deviation


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Educational Research

Inferential Statistics


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Concepts underlying inferential statistics Types of inferential statistics

Parametric T-tests ANOVA

One-way Factorial Post-hoc comparisons

Multiple regression ANCOVA

Non-parametric Chi-Square


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Important Perspectives

Inferential statistics Allow researchers to generalize to a population of

individuals based on information obtained from asample of those individuals

Assesses whether the results obtained from asample are the same as those that would have

been calculated for the entire population Probabilistic nature of inferential analyses


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Underlying Concepts

Sampling distributions Standard error Null and alternative hypotheses

Tests of significance Type I and Type II errors One-tailed and two-tailed tests

Degrees of freedom Tests of significance


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Sampling Distributions

A distribution of sample statistics A distribution of mean scores A distribution of the differences between two mean

scores A distribution of the ratio of two variances

Known statistical properties of samplingdistributions The mean of the sampling distribution of means isan excellent estimate of the population mean The standard error of the mean is an excellent

estimate of the standard deviation of the


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Standard Error

Sampling error the expected random orchance variation of means in samplingdistributions

The calculation of standard errors toestimate sampling error Standard error of the mean

Standard error of the differences between twomeans


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Null and Alternative Hypotheses

The null hypothesis represents astatistical tool important to inferential

tests of significance The alternative hypothesis usually

represents the research hypothesis

related to the study


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Comparisons between groups Null: no difference between the means scores of the

groups Alternative: differences between the mean scores of

the groups Relationships between variables

Null: no relationship exists between the variablesbeing studied

Alternative: a relationship exists between thevariables being studied


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Acceptance of the nullhypothesis The difference between

groups is too small toattribute it to anything butchance

The relationship betweenvariables is too small toattribute it to anything butchance

Rejection of the nullhypothesis The difference between

groups is so large it can beattributed to something

other than chance (e.g.,experimental treatment) The relationship between

variables is so large it canbe attributed to somethingother than chance (e.g., a

real relationship)


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Tests of Significance

Statistical analyses to help decide whether toaccept or reject the null hypothesis

Alpha level An established probability level which serves as the

criterion to determine whether to accept or reject thenull hypothesis

Common levels in education .01 .05 .10


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Specific tests are used in specificsituations based on the number ofsamples and the statistics of interest One sample tests of the mean, variance,proportions, correlations, etc. Two sample tests of means, variances,

proportions, correlations, etc.


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Type I and Type II Errors

Correct decisions The null hypothesis is true and it is accepted

The null hypothesis is false and it is rejected Incorrect decisions

Type I error - the null hypothesis is true and it isrejected

Type II error the null hypothesis is false and it isaccepted


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Type I and Type II Errors

Reciprocal relationship between Type I andType II errors

Control of Type I errors using alpha level As alpha becomes smaller (.10, .05, .01, .001,

etc.) there is less chance of a Type I error

Value and contextual based nature ofconcerns related to Type I and Type II errors


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One-Tailed and Two-Tailed Tests

One-tailed an anticipated outcome in a specificdirection Treatment group is significantly higher than the

control group

Treatment group is significantly lower than the controlgroup Two-tailed anticipated outcome not directional

Treatment and control groups are equal

Ample justification needed for using one-tailedtests


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Degrees of Freedom

Statistical artifacts that affect thecomputational formulas used in tests ofsignificance

Used when entering statistical tables toestablish the critical values of the test

statistics


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Parametric and non-parametric Four assumptions of parametric tests

Normal distribution of the dependent variable Interval or ratio data Independence of subjects Homogeneity of variance

Advantages of parametric tests More statistically powerful More versatile


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Types of Inferential Statistics

Two issues discussed Steps involved in testing for significance Types of tests


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Steps in Statistical Testing

State the null and alternative hypotheses Set alpha level Identify the appropriate test of significance Identify the sampling distribution Identify the test statistic Compute the test statistic


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Identify the criteria for significance If computing by hand, identify the critical value of the

test statistic If using SPSS Windows, identify the probability level

of the observed test statistic Compare the computed test statistic to the

criteria for significance If computing by hand, compare the observed test

statistic to the critical value If using SPSS Windows, compare the probability level

of the observed test statistic to the alpha level


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Accept or reject the null hypothesis Accept

The observed test statistic is smaller than the criticalvalue

The observed probability level of the observed statistic issmaller than alpha

Reject The observed test statistic is larger than the critical value The observed probability level of the observed statistic is

smaller than alpha


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Specific Statistical Tests

T-test for independent samples Comparison of two means from independent samples

Samples in which the subjects in one group are not related tothe subjects in the other group

Example - examining the difference between themean pretest scores for an experimental and controlgroup

Computation of the test statistic

SPSS Windows syntax


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T-test for dependent samples Comparison of two means from dependent

samples

One group is selected and mean scores are comparedfor two variables Two groups are compared but the subjects in each group

are matched

Example examining the difference betweenpretest and posttest mean scores for a singleclass of students

Computation of the test statistic SPSS Windows syntax


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Simple analysis of variance (ANOVA) Comparison of two or more means Example examining the difference between

posttest scores for two treatment groups anda control group Computation of the test statistic SPSS Windows syntax


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Multiple comparisons Omnibus ANOVA results

Significant difference indicates whether a difference existsacross all pairs of scores

Need to know which specific pairs are different

Types of tests A-priori contrasts Post-hoc comparisons

Scheffe Tukey HSD Duncans Multiple Range

Conservative or liberal control of alpha


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Multiple comparisons (continued) Example examining the difference

between mean scores for Groups 1 & 2,Groups 1 & 3, and Groups 2 & 3


SPSS Windows syntax


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Two factor ANOVA Comparison of means when two independent

variables are being examined

Effects Two main effects one for each independent

variable One interaction effect for the simultaneous

interaction of the two independent variables


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Two factor ANOVA (continued)

Example examining the mean scoredifferences for male and female students inan experimental or control group


SPSS Windows syntax


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Analysis of covariance (ANCOVA) Comparison of two or more means with statistical

control of an extraneous variable Use of a covariate Advantages

Statistically controlling for initial group differences (i.e.,equating the groups)

Increased statistical power Pretest is typically the covariate



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Multiple regression Correlational technique which uses

multiple predictor variables to predict asingle criterion variable Characteristics

Increased predictability with additional variables

Regression coefficients Regression equations


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Multiple regression (continued)

Example predicting college freshmensGPA on the basis of their ACT scores, highschool GPA, and high school rank in class


descriptive stats saturday

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