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INFERENTIAL STATISTICS

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Descriptive statistics is used simply to describe what's going on in the data. Inferential statistics helps us reach conclusions that extend beyond the immediate data alone; i.e., to make inferences from our data to more general conditions E.g. to infer from the sample data what the population might think to make judgments of the probability of an observed difference between groups: is it a dependable one or one that might have happened by chance in this study?

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There are many ways to make more general inferences: Considering the sampling error Distribution of sample means Standard error of the mean (SEM) Confidence interval, etc Our focus will be on the most common inferential tests used in Social Sciences!

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P RACTICAL VERSUS S TATISTICAL S IGNIFICANCE Statistical significance: A result is statistically significant if it is unlikely to occur by chance! It is known as the significance level or p-value. In educational research, having.05 level of significance is accepted statistically significant ( p =.05). However, while interpreting the results, you should also consider its practical importance as well!

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P VALUE Statistical significance If the findings hold true 95% of the time then p = 0.05 If the findings hold true 99% of the time then p = 0.01 If the findings hold true 99.9% of the time then p = 0.001

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I NFERENCE T ECHNIQUES One of the most common inferential test is t -test : to compare the average performance of two groups on a single measure to see if there is a difference. E.g. You might want to know whether attitudes of eighth-grade boys and girls towards learning English differ. You need to compare the average performance between these two groups using the t -test.

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T HE T - TEST FOR M EANS ( PARAMETRIC TEST ) The t -test assesses whether the means of two groups are statistically different from each other..05 level of significance in the t -test means there is a real difference between the groups. There are two forms of t -test: t -test for independent means t -test for correlated means

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T HE T - TEST FOR INDEPENDENT MEANS It is used to compare the mean scores of two different, independent groups. E.g. Mean scores on the achievement test of class A is 80 and class B is 85. If the t -test score is at or below.05 significance level, it is possible to say that the difference between these score is statistically significant.

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T HE T - TEST FOR CORRELATED MEANS ( PAIRED T - TEST ) It is used to compare the mean scores of the same group before and after a treatment: it shows whether the change in the mean scores is significantly significant. PS. To calculate the t -test for correlated means, you need to pair the scores for each individual!

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S AMPLE T - TEST T ABLE NMeansdtp Group A309.391.951.2.625 Group B339.631.72 There is no statistical significance in the means of Groups A and Group B.

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There is a statistically significant difference in the attitudes of the two groups towards reading books in English for fun (t=3.26, p<.05). Groups B has a more positive attitude (X=73.42) towards reading books for fun than Group A (X=69.36). NMeansdtP Group A8269.368.053.26.002 Group B8073.427.78

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A NALYSIS OF V ARIANCE (ANOVA) It is used when we want to see the differences between the means of more than two groups: a more general form of the t -test. (e.g. Effect of reading for fun on the attitudes of Group A, B, & C) Variation both within and between each of the groups is analyzed statistically and is represented as the F value.

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S AMPLE SPSS O UTCOME

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S AMPLE ANOVA T ABLE (APA) E.g. Following is a table that shows means on test anxiety scales of 6 th, 7 th, and 8 th graders NMeansd Scheffe 6 th graders5270.288.42 7 th graders6080.389.77 8 th graders5082.809.33

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Sum of squares dfMean squares FP Between groups4583.1422291.5726.94.000 Within groups13520.0015985.037 Total18104.00161 Analysis of data shows that there is a statistically significant difference in the test anxiety levels of the groups (F=26.94). In other words, students’ test anxiety level shows difference at different grade levels. (The higher the grade level, the higher the anxiety level) According to the Scheffe results, 7 th (X=80.38) and 8 th (X=82.80) graders’ anxiety level is higher than that of the 6 th (X=20.28) graders.

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A NALYZING C ATEGORICAL D ATA ( X 2 ) ( NON - PARAMETRIC ) The Chi-Square Test ( X 2 ) Used to analyze the categorical data. E.g. How many male and female teachers favor the new curriculum? If they do not differ significantly, it means the same proportion of males and females would be in favor of (or opposed to) the new curriculum. If there is a significant difference, it means males (or females) favor the new curriculum.

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To see the significance here, we need chi-square

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Questions (Section B)YesNoX2X2 1Grading portfolios together with the teacher was very useful 44141.08 ** 2It is a good idea that an outsider, besides the teacher, grades the portfolios 23220.02 n.s. 3It was good that all my essays were graded32138.02 * 4It was good that I had my grades from my final drafts 45-- *p<.01 **p<.001 n.s.= statistically not significant

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R ELIABILITY C OEFFICIENCY Reliability in quantitative analysis: generally in two forms (but both calculate a coefficient of reliability) Split-half ( r ): consistency between the two halves of an instrument Cronbach alpha ( alpha ): correlation of each item with the sum of the other items.

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CORRELATION Correlation is a measure of association between two variables. The variables are not designated as dependent or independent. e.g. there is no cause and effect relationship in shoe size and hat size Shoe size Hat size12345 Perfect positive correlation: r= + 1

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CORRELATION AGESTRENGTH 3060 4050 5040 6030 Perfect negative correlation: r= – 1

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CORRELATIONS Correlations Spearman correlation for nominal and ordinal data Pearson correlation for interval and ratio data Statistical significance is a function of the co- efficient and the sample size: the smaller the sample, the larger the co- efficient has to be; the larger the sample, the smaller the co- efficient can be.

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Summary of Statistical Analysis Procedures

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AnalysisTypes of DataFeaturesExample Test of Causal Effects? t test Independent samples Indep variable= nominal Dep.= one interval-ratio measure Tests the differences between 2 treatment groups Does the problem- based treatment group surpass the traditional instruction treatment group? Yes t test Dependent samples Indep variable= nominal (repeated measure) Dep.= one interval-ratio measure Tests the differences between 2 treatment means for a given group Will participants change their attitude toward drugs, from pretest to posttest, following a videotape on drug effects? Yes

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Analysis Types of Data FeaturesExample Test of Causal Effects? Analysis of variance (ANOVA) Indep variable= nominal Dep.= one interval-ratio measure Tests the differences between 3 or more treatment means. If ANOVA is significant, follow- up comparisons of means are performed. Will there be differences in learning among three groups that paraphrase, summarize, or neither? Yes Multivariate analysis of variance (MANOVA) Indep variable= nominal Dep.= two or more interval- ratio measure Tests the difference between 2 or more treatment group means on 2 or more learning measures. If MANOVA is significant, an ANOVA on each individual measure is performed. Will there be differences among 3 feedback strategies on problem solving and knowledge learning? Yes

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AnalysisTypes of DataFeaturesExample Test of Causal Effects? Pearson r Two ordinal or interval-ratio measures Tests relationship between two variables Is anxiety related to test performance? No Multiple linear regression Indep variable= two or more ordinal or interval-ratio measures Dep.= one ordinal or interval-ratio measure Tests relationship between independent variables (predictor) and outcome variable. Shows the relative contribution of each predictor in accounting for variability in the outcome variable. How well do experience, age, gender, and grade point average predict time spent on completing a task? No Chi-Square test of independence Two nominal variables Tests relationship between two nominal variables Is there a relationship between gender (males vs females) and attitudes toward the instruction (liked, no opinion, disliked)? No

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