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Descriptivestatistics

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  • 1. Descriptive Statistics Notes from McMillan & Schumacher
  • 2. Statistics • Methods of organizing and analyzing quantitative data • Tools designed to help researcher organize and interpret numbers derived from measuring a trait or variable • International language that only manipulates numbers • Numbers do not interpret themselves • Meaning derived from the research design
  • 3. Categories of Statistics Descriptive and Inferential Descriptive Statistics • Summarize, organize, and reduce large numbers of observations to a few numbers • Describe or characterize the data • Assigning numbers to things in order to differentiate one thing from another
  • 4. Inferential Statistics • Use to make inferences or predictions from the sample to the population • Depend on descriptive statistics • Estimation of population characteristics from sample • Experimental Designs – True Experimental – random assignment – Quasi Experimental – Pre-test/post-test, time series – Single subject
  • 5. Descriptive to Inferential Descriptive Statistics Of Sample Population Sample Estimate Population Based on Descriptive Statistics
  • 6. Scales of Measurement Nominal – Categories (sex, ethnicity, party) Ordinal – Rank order (percentile norms) Interval – Equal difference between #s Ratio – Equal amounts from zero
  • 7. Nominal Scale of Measurement • Nominal - Name • Categories and Classifications • Naming of mutually exclusive categories • People, events, phenomena • Eye color, gender, political party, group • No order or value implied • Assign number for coding - arbitrary • Numbers do not represent quantity or degree
  • 8. Ordinal Scale of Measurement • Ordinal – by ranked order • Categories rank-ordered from highest to lowest – Equal = – Greater than > – Less than < • Ranking by grade point average, percentile, achievement test score, social class
  • 9. Interval Scale of Measurement • Shares characteristics with ordinal • Equal intervals between each category • Equal difference between variables or attributes being measures • Constant unit of measurement • Difference between 5&6 = 18&19 • Year, Centigrade, Fahrenheit
  • 10. Ratio Scale of Measurement • Most refined type of measurement • Ordinal and Interval • Numbers can be compared by ratios • Numbers represent equal amounts from absolute zero • Age, dollars, time, speed, class size • Most educational measurement – not ratio
  • 11. Graphic Portrayals of Data • Frequency Distribution – f • Number of times each score was attained • Rank order and then tally • Show most/least occurring scores • General Shape of Distribution • Outliers
  • 12. Histograms and Bar Graphs • Graphic way of representing frequency distribution • Histogram – frequencies rank- ordered • Bar Graph – order arbitrary
  • 13. Frequency Polygon • Illustrates frequency distribution • Single points rather than bars are graphed • Normal curve – curves the straight lines
  • 14. Measures of Central Tendency • Mean – – arithmetic average of all scores • Median – – point which divides a rank-ordered distribution into halves that have an equal number of scores – 50% above and 50% below • Mode – score that occurs most frequently
  • 15. Relationships among Measures of Central Tendency • Normal Distribution – Mean, median, and mode about the same – Bell shaped symmetrical curve – Large numbers – normal distribution • Skewed Distribution – Positively skewed – Most scores at low end – Negatively skewed – Most scores at high end – Lower numbers distributed unevenly
  • 16. Normal Distribution
  • 17. Bell Curve – Normal Distribution Mean = Median = Mode0 100 Normal curve – theoretical distribution used to transform data and calculate many statistics
  • 18. Positively Skewed Mode Mean Median0 100 Most of the scores are at the low end of the distribution
  • 19. Negatively Skewed Mean Median Mode 0 100 Most of the scores are at the high end of the distribution
  • 20. Measures of Variability • Shows how spread out the distribution of the scores is from the mean of the distribution – dispersion of scores • How much, on average, does each score differ from the mean? • Variability measures – Range – highest and lowest (no mean) – Standard Deviation – numerical index indicating average variability of scores
  • 21. Standard Deviation • Indicates the amount on average that the set of scores deviates from the mean
  • 22. SD in Normal Distribution -1 SD0 100 34% 34% +1 SD 68% +1 SD = 84th percentile -1 SD = 16th percentile 50% below the mean 50% above the mean
  • 23. Box and Whisker Plot • Use to give picture of variability • Size of rectangular box – 25th to 75th percentiles • Whiskers draw from ends of box to 10th and 90th percentiles
  • 24. Standard Scores • Makes it easier to analyze several distributions if means and standard deviations are different for each distribution • Raw scores converted to standard scores • Provide constant normative or relative meaning • Obtained from the mean and standard deviation of the raw score distribution
  • 25. The Z-Score • Most basic standard score • Mean of 0 • Standard deviation of 1 • Z-score of +1 = 84th percentile • Z-score of –1 = 16th percentile • Example – IQ tests 100 = mean 15/16 = standard deviation
  • 26. Scatterplot • Graphic representation of relationship of variables • Relationships – Positive – Negative – None – Curvilinear
  • 27. Correlation Coefficient • Calculated number representing the relationships between variables • Range from –1.00 to +1.00 • High Positive Relationship (.85 .90. 96) • Low Positive Relationship (.15 .20 . 08) - 1 +10 High negative High positive
  • 28. Types of Correlation Coefficients • See Table 7.5 – page 172 • Most common – Pearson product-moment • r • Both continuous – Spearman • rs • Both rank-ordered
  • 29. Example of Correlation - SPSS