Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Psychometrics wikipedia , lookup
Bootstrapping (statistics) wikipedia , lookup
Taylor's law wikipedia , lookup
Foundations of statistics wikipedia , lookup
Statistical inference wikipedia , lookup
History of statistics wikipedia , lookup
Resampling (statistics) wikipedia , lookup
Statistical Analysis of Data Graziano and Raulin Research Methods: Chapter 5 Graziano & Raulin (2000) Individual Differences A fact of life – – People differ from one another People differ from one occasion to another Most psychological variables have effects that are small compared to individual differences Statistics give us a way to detect such subtle effects in a sea of individual differences Graziano & Raulin (2000) Descriptive Statistics Are used to describe the data Many types of descriptive statistics – – – Frequency distributions Summary measures such as measures of central tendency, variability, and relationship Graphical representations of the data A way to visualize the data The first step in any statistical analysis Graziano & Raulin (2000) Frequency Distributions First step in organization of data – Can see how the scores are distributed Used with all types of data Illustrate relationships between variables in a cross-tabulation Simplify distributions with a large range by using a grouped frequency distribution Graziano & Raulin (2000) Histograms A bar graph, as shown at the right Can be used to graph either – – Data representing discrete categories Data representing scores from a continuous variable Sample Histogram 60 50 40 Freq 30 20 10 0 Graziano & Raulin (2000) 1 2 3 4 5 6 Scores Histograms (2 distributions) Possible to graph two or more distributions on the same histogram to see how they compare Note that one of the two groups in this histogram was the same group graphed previously Sample Histogram 80 70 60 50 Freq 40 30 20 10 0 Graziano & Raulin (2000) 1 2 3 4 5 6 Scores Frequency Polygon (1 group) Like a histogram except that, instead of a bar representing the mean score or frequency, a dot is used, with the dots connected as shown Data could be either discrete or continuous Sample Frequency Polygon 60 50 40 30 20 10 0 Graziano & Raulin (2000) 1 2 3 4 Scores 5 6 Frequency Polygon (2 groups) Can compare two of more frequency polygons on the same scale as shown Easier to compare frequency polygons because the graph appears less cluttered than multiple histograms Sample Frequency Polygon 80 60 40 20 0 Graziano & Raulin (2000) 1 2 3 4 Scores 5 6 Shapes of Distributions Many variables in psychology are distributed normally The distribution is skewed if scores bunch up at one end Illustrate on the left are symmetric and skewed distributions Graziano & Raulin (2000) Measures of Central Tendency Mode: the most frequently occurring score – Median: the middle score in a distribution – Easy to compute from frequency distribution Less affected than the mean by a few deviant scores Mean: the arithmetic average – – Most commonly used central tendency measure Used in later inferential statistics Graziano & Raulin (2000) Computing the Mean Compute the mean of 3, 4, 2, 5, 7, & 5 Sum the numbers – Count the numbers – 26 6 Plug these values into the equation at the right Graziano & Raulin (2000) X X N 26 X 4.33 6 Measuring Variability Range: lowest to highest score Average Deviation: average distance from the mean Variance: average squared distance from the mean – Used in later inferential statistics Standard Deviation: square root of variance – expressed on the same scale as the mean Graziano & Raulin (2000) Measures of Relationship Pearson product-moment correlation – Spearman rank-order correlation – Used with interval or ratio data Used when one variable is ordinal and the second is at least ordinal Scatter plots – – Visual representation of a correlation Helps to identify nonlinear relationships Graziano & Raulin (2000) Regression Using a correlation (relationship between variables) to predict one variable from knowing the score on the other variable Usually a linear regression (finding the best fitting straight line for the data) Best illustrated in a scatter plot with the regression line also plotted (see Figure 5.6) Graziano & Raulin (2000) Reliability Indices Test-retest reliability and interrater reliability are indexed with a Pearson product-moment correlation Internal consistency reliability is indexed with coefficient alpha Details on these computations are included on the CD supplement Graziano & Raulin (2000) Standard Scores (Z-scores) A way to put scores on a common scale Computed by subtracting the mean from the score and dividing by the standard deviation Interpreting the Z-score – – Positive Z-scores are above the mean; negative Z-scores are below the mean The larger the absolute value of the Z-score, the further the score is from the mean Graziano & Raulin (2000) Inferential Statistics Used to draw inferences about populations on the basis of samples from the populations The “statistical tests” that we perform on our data are inferential statistics Provide an objective way of quantifying the strength of the evidence for our hypothesis Graziano & Raulin (2000) Populations and Samples Population: the larger groups of all participants of interest to the researcher Sample: a subset of the population Samples almost never represent populations perfectly (termed “sampling error”) – Not really an error; just the natural variability that you can expect from one sample to another Graziano & Raulin (2000) The Null Hypothesis States that there is NO difference between the population means Compare sample means to test the null hypothesis Population parameters & sample statistics – – Population parameter is a descriptive statistic computed from everyone in the population Sample statistics is a descriptive statistic computed from everyone in your sample Graziano & Raulin (2000) Statistical Decisions We can either Reject or Fail to Reject the null hypothesis – – – Rejecting the null hypothesis suggests that there is a difference in the populations sampled Failing to reject suggests that no difference exists Decision is based on probability (reject if it is unlikely that the null hypothesis is true) Alpha: the statistical decision criteria used Traditionally alpha is set to small values (.05 or .01) Always a chance for error in our decision Graziano & Raulin (2000) Statistical Decision Process Reject Null Hypothesis Retain Null Hypothesis Null Hypothesis is True Type I Error Correct Decision Null Hypothesis is False Correct Decision Type II Error Graziano & Raulin (2000) Testing for Mean Differences t-test for independent groups: tests mean difference of two independent groups Correlated t-test: tests mean difference of two correlated groups Analysis of Variance: tests mean differences in two or more groups – – Groups may or may not be independent Also capable of evaluating factorial designs Graziano & Raulin (2000) Power of a Statistical Test Sensitivity of the procedure to detect real differences between the populations Not just a function of the statistical test, but also a function of the precision of the research design and execution Increasing the sample size increases the power because larger samples estimate the population parameters more precisely Graziano & Raulin (2000) Statistical versus Practical Significance Statistical significance means that the observed mean differences are not likely due to sampling error – Can get statistical significance, even with very small population differences, if the sample size is large enough Practical significance looks at whether the difference is large enough to be of value in a practical sense Graziano & Raulin (2000) Effect Size Gives an indication of the size of the difference between groups Unlike the statistical test, the effect size is NOT affected by the size of the sample More details on effect size – – In Chapter 15 On the CD supplement Graziano & Raulin (2000) Meta-Analysis A relatively new statistical technique Allows researchers to statistically combine the results of several studies of the same phenomenon to get a sense of how powerful the effect is Discussed in more detail in Chapter 15 Graziano & Raulin (2000) Summary Statistics allow us to detect and evaluate group differences that are small compared to individual differences Descriptive versus inferential statistics – – Descriptive statistics describe the data Inferential statistics are used to draw inferences about population parameters on the basis of sample statistics Statistics objectify evaluations, but do not guarantee correct decisions every time Graziano & Raulin (2000)