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Statistics: A Gentle Introduction

This book chapter provides an overview of descriptive statistics and understanding distributions of numbers. It covers topics such as graphs and tables, the importance of statistical data sources and characteristics, appropriate comparisons, frequency distributions, cumulative frequency distributions, stem-and-leaf plot, and different types of distributions.

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Statistics: A Gentle Introduction

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  1. Statistics: A Gentle IntroductionBy Frederick L. Coolidge, Ph.D.Sage Publications Chapter 2 Descriptive Statistics: Understanding Distributions of Numbers

  2. 0730 Q1 Results N=20 • 1|5 • 2|1124456679 • 3|001124779

  3. 0900 Q1 Results N=32 • 1|249 • 2|0335567799 • 3|2224444445566889 • 4|001

  4. Overview • Graphs and tables • What’s the point? • The nasty tricks of the trade • Types of distributions • Grouping data • Cumulative frequency distributions • Stem-and-leaf plot

  5. Graphs and TablesWhat’s the point? • What’s the point? • Document the sources of statistical data and its characteristics. • Where did you get it? • What is it measuring?

  6. Graphs and TablesWhat’s the point? • Make appropriate comparisons. • Compare similar data. • Make the point more clearly. • Make data more understandable. • Eliminate doubt.

  7. Frequency Distributions • A table reporting the number of observations falling into each category of the variable; • Frequency count for data value is # of times value occurs in data set; • Ungrouped frequency distribution lists the data values w/frequency count with which each value occurs; • Relative frequency for any class is obtained by dividing frequency for that class by total # of observations.

  8. Cumulative Frequency(CF) and Cumulative Relative Freq(CRF) • CF- a specific value in a frequency table is sum of frequencies for all values at or below the given value; • CRF- the sum of the relative frequencies for all values at or below the given value expressed as a proportion; • Grouped Frequency distribution is obtained by constructing intervals for data and listing frequency count in each interval

  9. MathAnxietyScore7:30class(Grouped Freq Distribution

  10. Histogram Math Anxiety Scores .30 .25 .20 .15 .10 .5 .5 2.5 4.5 6.5 8.5 10.5

  11. “Blacks More Pessimistic than whites economic opportunities”

  12. Laws Covering Sales of Firearms: Increase Restrictions( 2000)?

  13. Men and Firearm Restrictions: Frequency Distribution(N=493)

  14. Women and Firearm Restrictions: Frequency Distribution(N=538)

  15. Graphs and TablesWhat’s the point? • Demonstrate the mechanisms of cause and effect and express the mechanisms quantitatively. • If you vary the cause and the results change in a predictable and uniform manner, then you make a stronger case for cause and effect.

  16. Graphs and TablesWhat’s the point? • Recognize the inherent multivariate (more than one cause) nature of the problem. • Is there anything with just one cause? • Temperature of boiling water: • Altitude of water • What is in the water (salt)?

  17. Graphs and TablesWhat’s the point? • Inspect and evaluate alternative hypotheses. • Cigarette smoking is related to a lower incidence of Alzheimer’s disease. • Is it the cigarettes? • Is it the dying at an earlier age, before Alzheimer’s is diagnosable?

  18. Graphs and TablesThe nasty tricks of the trade • The nasty tricks of the trade • Adjust the scale to make the point • Show only part of the scale • Omit the units of measure • Change the scale along the graph • Include too much junk • Not enough to bother graphing

  19. Graphs and TablesThe nasty tricks of the trade Is Brand One really any better than the others?

  20. Stem-and-leaf plot • Presents the frequency of data points without losing important information. Data set: 25, 27, 29 Stem  2 579  Leaves

  21. Stem-and-leaf plot • The first digit is the stem • The second digit is each leaf 25 27 29 Stem  2 579  Leaves

  22. Stem-and-leaf plot • The first digit is the stem • The second digit is each leaf 25 27 29 Stem  2 579  Leaves

  23. Stem-and-leaf plot • Let’s try it Data set: 30, 32, 32, 34, 37, 37, 39 Data set: 5, 9, 10, 11, 11, 23, 25, 27

  24. Types of DistributionsFrequency Distribution • Frequency distribution • Showing what you have • A way to illustrate how many of each thing.

  25. Types of DistributionsFrequency Distribution

  26. Types of DistributionsNormal Distribution • Normal distribution • Also known as the bell-shaped curve • An illustration of the expectation of what most types of data will look like • A few data points at each extreme • Most data points in the middle area

  27. Types of DistributionsNormal Distribution

  28. Types of DistributionsPositively Skewed Distribution • Not all data are created equal • Positive skew • Many data points near the origin of the graph

  29. Types of DistributionsNegatively Skewed Distribution • Negative skew • Many data points away from the origin of the graph

  30. Types of DistributionsBimodal Distribution • Bimodal • Two areas under the curve with many data points

  31. Types of DistributionsNon-normal Distributions • Nonnormal distributions • But not abnormal • Platykurtic: flat like a plate

  32. Bi-Modal Distribution: Spring 2010 Quiz Scores

  33. Types of DistributionsNon-normal Distributions • Leptokurtic: up & down (like leaping) • Bimodal: lumpy

  34. Grouping data • A way of organizing data so that they are manageable. Which is easier to understand? 3, 1, 7, 4, 1, 2, 3, 5, 4, 9 or 1, 1, 2, 3, 3, 4, 4, 5, 7, 9

  35. Grouping dataTips for grouping data • Tips for grouping lots of data • Choose interval widths that reduce your data to 5 to 10 intervals. 5 10 15 20 25 30 35

  36. Grouping dataTips for grouping data • Choose meaningful intervals. • Which is easier to understand at a glance? 5 10 15 20 25 30 35 or 4 7 10 13 16 19 22

  37. Grouping dataTips for grouping data • Interval widths must be the same. 5 10 15 20 25 30 35 NOT 5 10 20 22 30 33 35

  38. Grouping dataTips for grouping data • Intervals cannot overlap. 5-10 11-15 16-20 21-25 26-30 31-35 36-40 NOT 5-10 10-15 14-20 20-26 25-30 30-35 35

  39. Grouping dataAn example • The data are displayed using • A frequency table of individual data points • A frequency table by intervals • Graph of data by intervals

  40. Grouping dataAn example

  41. Grouping dataAn example

  42. Grouping dataAn example

  43. Freq Distribution Using Stated limits

  44. Problem w/ Stated Limits • Gap of one between adjacent intervals • Problem for scores with fractional values; where classify a woman 49.25 years old? Here age would actually fall between intervals 40-49 and 50-59!! • Real limits extend upper and lower limits by .5

  45. Freq Distribution Using Real Upper and Lower limits

  46. Upper/Lower limits &Fractional Values • Scores falling exactly at upper real limit or lower real limit are rounded to closest even number; EX=59.5 rounded to 60 and included in interval • 59.5-69.5 • Where would you classify respondent 49.25 years? How about 59.4?

  47. Cumulative Frequency Distribution • Cumulative frequency distribution • Shows how many cases (data points) have been accounted for out of the total number of cases (data points).

  48. Cumulative Frequency Distribution • How many data points have accounted for as each group is displayed.

  49. Cumulative Frequency Distribution • Cumulative frequencies can also be illustrated using percentages.

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