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# Descriptivestatistics

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### Descriptivestatistics

1. 1. Descriptive Statistics Notes from McMillan & Schumacher
2. 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. 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. 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. 5. Descriptive to Inferential Descriptive Statistics Of Sample Population Sample Estimate Population Based on Descriptive Statistics
6. 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. 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. 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. 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. 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. 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. 12. Histograms and Bar Graphs • Graphic way of representing frequency distribution • Histogram – frequencies rank- ordered • Bar Graph – order arbitrary
13. 13. Frequency Polygon • Illustrates frequency distribution • Single points rather than bars are graphed • Normal curve – curves the straight lines
14. 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. 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. 16. Normal Distribution
17. 17. Bell Curve – Normal Distribution Mean = Median = Mode0 100 Normal curve – theoretical distribution used to transform data and calculate many statistics
18. 18. Positively Skewed Mode Mean Median0 100 Most of the scores are at the low end of the distribution
19. 19. Negatively Skewed Mean Median Mode 0 100 Most of the scores are at the high end of the distribution
20. 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. 21. Standard Deviation • Indicates the amount on average that the set of scores deviates from the mean
22. 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. 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. 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. 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. 26. Scatterplot • Graphic representation of relationship of variables • Relationships – Positive – Negative – None – Curvilinear
27. 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. 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. 29. Example of Correlation - SPSS