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Parametric statistics

Parametric statistics. 922. Outline. Measuring the accuracy of the mean Practical notes for practice Inferential statistics T-test ANOVA. Measuring the accuracy of the mean. The mean is the simplest statistical model that we use This statistic predicts the likely score of a person

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Parametric statistics

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  1. Parametric statistics 922

  2. Outline • Measuring the accuracy of the mean • Practical notes for practice • Inferential statistics • T-test • ANOVA

  3. Measuring the accuracy of the mean The mean is the simplest statistical model that we use This statistic predicts the likely score of a person The mean is a summry statistic

  4. Measuring the accuracy of the mean • The model we choose (mean/ median / mode) should represent the state in the real world • Does the model represent the world precisely? • The mean is a prefect representation only if all the scores we collect are the same as the mean.

  5. Mean • When the mean is a perfect fit: there is no difference between the mean and each data point

  6. Mean • Usually, there are differences between the mean and the raw scores • If the mean is representative of the data these differences are small.

  7. Deviation • The differences between the model prediction (=mean) and each raw score is the deviation

  8. Deviation • Compute the deviation of each score from the mean • Measure the overall deviation (sum)

  9. Deviation

  10. Deviation • Sum of squared deviations (also called the sum of squared errors) is a good measure of the accuracy of the mean • Except that it gets bigger with more scores

  11. Variance • Divide sum of squared deviations by the number of scores minus 1 • We can compare variance across samples • Square root of variance is standard deviation

  12. Accuracy of the mean Sum of squared deviations (sum of squared errors), variance and standard deviation all measure the same thing: variability of the data Standard deviation (SD) measures how well the mean represents the data: small SD indicate data points close to the mean

  13. Standard Deviation (SD) SD close to the mean

  14. Standard Deviation (SD) SD far from the mean

  15. Why use number of scores minus 1? • We are using a sample to estimate the variance in the population • Population?

  16. Sample-population The intended population of psycholinguistic research can be all people / all children aged 3 / etc. Actually, we collect data only from a sample of the population we are interested in. We use the sample to make a guess about the linguistic behavior of the relevant population.

  17. Sample - population Size of the sample The mean as a model is resistant to sampling variation: different samples from the same populations usually have a similar mean

  18. Why use number of scores minus 1? • We are using a sample to estimate the variance in the population • Population?

  19. Why use number of scores minus 1? We are using a sample to estimate the variance in the population Variance in the sample: observations can vary (5, 6, 2, 9, 3) mean=5 But: if we assume that the sample mean is the same as the population (mean=5)

  20. Why use number of scores minus 1? For the next sample, not all observations are free to vary. For a sample of (5, 7, 1, 8, ?) we already need to assume that the mean is 5. (?=4)

  21. Why use number of scores minus 1? This does not mean we fix the value of the observation, but simply that for various statistics we have to calculate the number of observations that are free to vary. This number is called: Degrees of freedom and it must be one less than the sample size (N-1).

  22. To summarize Mean  represents the sample Sample  represents population

  23. Many samples - population Theoretically, if we take several samples from the same population Each sample will have its own Mean and SD If the samples are taken from the same population, they are expected to be reasonably similar.

  24. Many samples

  25. Sampling distribution Average of all sample means will give the value for the population mean

  26. Sampling distribution • How accurate is a sample likely to be? • Calculate the SD of the sampling distribution • This is called the standard error of the mean (SE)

  27. Standard Error (SE) • We do not collect many samples, but compute SE • SE= SD/N

  28. Standard Error (SE)

  29. Why are the samples different? • Source of the variance: • Different population or • Sampling error (random effect, can be calculated) • Can we take results from the sample to make generalizations about the population?

  30. Accuracy of sample means Calculate the boundaries within which most sample means will fall. Looking at the means of 100 samples, the lowest mean is 2, and the highest mean is 7. The mean of any additional sample will fall within these limits.

  31. Confidence Interval The limits within which a certain percent (typically we look at 95%) of sample means will fall. If we collect 100 samples, 95 of them will have a mean within the confidence interval

  32. Practice

  33. Experiment: • Compare children with specific language impairment (SLI) and children who are typically developing (TD). • Hypothesis: • effect of word order SVO vs. VSO

  34. Task : repeat a sentence 30 Children, each was presented with 10 sentences (5 SVO, 5 VSO).

  35. Compute: Mean? Frequency?

  36. Basic analysis with Excel • Descriptive statistics: Sum, Average, Percentage • Drawing graphs • Parametric statistics: Mean, Standard Deviation, t-test • Smart sheets: COUNTIF

  37. Inferential statistics

  38. Statistical hypothesis • Hypothesis for the effect of a linguistic phenomenon • Findings from a sample • Do the findings support the hypothesis? • Do they show a linguistic effect? • To answer this, we consider a null hypothesis

  39. The null hypothesis (H0) H0= the experiment has no effect The purpose of statistical inference is to reject this hypothesis H1 = the mean of the population affected by the experiment is different from the general population

  40. Rejecting the null hypothesis Compare the mean of the sample to two populations (under H1 or H0). We cannot show the sample belongs to the population under H1. All we can do is compare the sample to population under H0 and consider the likelihood that it belongs to it.

  41. Rejecting the null hypothesis Check if our sample belongs to the population under H1 or H0 Consider confidence interval, SE Compare means Compare varience

  42. Level of significance (alpha) Is the difference between the sample and the population big enough to reject H0? Determine a critical value (alpha) as criterion for including the sample in the population  < 0.05

  43. Parametric statistics • Variables are on interval scale (at least) • Compute means of raw grade (several items to one condition) • t-tests • ANOVA • ANACOVA

  44. t-Tests • t-tests are used in order to compare two samples and decide whether they are significantly different or not. • The t-tests represent the difference between the means of the two samples which takes into consideration the degree to which these means could differ by chance.

  45. t-Tests • The degree to which the means could differ by chance is the Standard Error (SE) • We do not calculate the t-value ourselves, but we use it to determine the effect of the experiment on the sample. • How do we know if the t-value is significant (p<0.05)?

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