Review of Statistics
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Statistics is the science of data gathering and analysis. It involves collecting data, analyzing it, and interpreting results to infer information about a population based on a sample. There are different scales of measurement for data, such as nominal, ordinal, interval, and ratio scales. Frequency distributions organize data by categories and show how often each occurs. Measures of central tendency like the mean, median, and mode describe the center of a distribution. Measures of variability like range and standard deviation describe how spread out the scores are. Normal distributions and standard scores are commonly used in statistics. Correlation measures the relationship between variables while regression predicts relationships between variables.
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Review of Statistics
- 2. WHAT IS STATISTICS? • The science of data gathering and treatment. • Collection of data • Data Analysis • Interpretation of Data • Inferring the Sample Data to a Population
- 3. SCALES OF MEASUREMENT • Measurement • Act of assigning numbers or symbols to characteristics of things • Can be categorized as discrete or continuous • Error • Always involved • Collective influence of factors on a test score or measurement beyond what is specifically measured
- 4. SCALES OF MEASUREMENT • Nominal • Catregorization/distinction • Ordinal • Classification • Rank-ordering • Interval • Equal intervals • Exactly equal to any other unit • Ratio • Has an absolute zero point
- 5. FREQUENCY DISTRIBUTION • Distribution • Set of test scores arrayed for recording/study • Raw Score • Straightforward, unmodified number • Frequency Distribution • List of scores along with the number of times a score has occurred • Can be listed on intervals (grouped)
- 6. PERCENTILES • Replaces simple ranks when trying to adjust for the score in a group • “What percent of scores fall below a particular score?” • Divides the total frequency for a set of observations into hundredths
- 7. MEASURES OF CENTRAL TENDENCY • “The middlemost score/s in a distribution” • Mean • average • Median • Middle score in a distribution • 50th Percentile • Mode • Frequently occurring score
- 8. MEASURES OF VARIABILITY • “the average distance between scores” • Range • Difference between highest and lowest score • Interquartile Range • Difference between the third and first quartile • Semi-Quartile Range • Interquartile range divided by 2 • Variance • Square of deviation scores • Standard Deviation • Square root of variance
- 9. SKEWNESS • “Absence of symmetry” • Positive Skew: relatively few scores falls on the high end; • Q3 – Q2 > Q2 – Q1 • Negative Skew: relatively few scores falls on the low end • Q3 – Q2 < Q2 – Q1
- 10. KURTOSIS • Steepness of a distribution to the center
- 11. NORMAL CURVE • Began in the mid-18th Century in the work of Abraham DeMoivre and Marquis de LaPlace • Karl Pearson termed it as the normal curve • Bell-shaped • Smooth • Asymptotic • Ranges from negative infinity to positive infinity
- 12. STANDARD SCORES • Raw scores converted from one scale to another, where the other score has arbitrarily set mean and standard deviation • Z-scores: difference between the raw score and a mean divided by the standard deviation • T-scores: mean is set at 50 and std. deviation is set at 10 • Sten: standard ten; mean is set at 5.5 and std. deviation is set at 2; used in SAT and GRE • Stanine: used by US Airforce; mean is set at 5 and std. deviation is set at 2; no decimals • A Scores: mean is set at 500 and std. deviation is set at 100 • IQ Scores: mean is set a5 100 and std. deviation is set at 15; used for IQ score interpretation
- 13. STANDARD SCORES
- 14. CORRELATION • Expression of the degree and direction of correspondence between variables • Runs from -1 to +1 • Graph is demonstrated in a scatterplot
- 15. Correlation Coefficients When to use? Pearson’s r 2 set of scores from the same respondents in interval-ratio level of measurement Spearman’s rho 2 sets of scores from the same respondents with a sample size less than 30 and in an ordinal level of measurement Kendall’s tau 2 sets of ordinal data from participants either less than or more than 30 in an ordinal level of measurement Kendall’s W More than two sets of ranking in ordinal level and the rankings come from several raters Phi-Coefficient 2 or more sets of frequencies; nominal in nature Point Biserial Only 2 dichotomous variables Multiple correlation 3 or more sets of Pearson
- 16. REGRESSION • A method used to make predictions about scores on one variable from knowledge of scores on another variable • Y = bX + a • b = slope • a = y-intercept
- 17. REGRESSION • For any variable, the mean is the point of least squares • Residual: the difference between Y and the predicted value of Y; the best-fitting line keeps this to a minimum • Standard Error of Estimate: measures the accuracy of prediction; standard deviation of the residuals • Coefficient of Determination: total variation in scores; squared value of r • Coefficient of Alienation: measure of non-association between variables.
- 18. MULTIPLE REGRESSION • Find linear combination of three variables • Transform all variables into single units • Two predictor variables that are highly correlated with the criterion will not both have large regression coefficients if they are highly correlated with each other as well.
- 19. DISCRIMINANT ANALYSIS • Used to determine the linear combination of variables that provide maximum discrimination between categories • Determine whether a group of variables predict success or failure
- 20. FACTOR ANALYSIS • Study interrelationships among set of variables without reference to criterion