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Statistics: A Gentle Introduction By Frederick L. Coolidge, Ph.D. Sage Publications. Chapter 2 Descriptive Statistics: Understanding Distributions of Numbers. 0730 Q1 Results N=20. 1|5 2|1124456679 3|001124779. 0900 Q1 Results N=32. 1|249 2|0335567799 3|2224444445566889 4|001.

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## Statistics: A Gentle Introduction By Frederick L. Coolidge, Ph.D. Sage Publications

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**Statistics: A Gentle IntroductionBy Frederick L. Coolidge,**Ph.D.Sage Publications Chapter 2 Descriptive Statistics: Understanding Distributions of Numbers**0730 Q1 Results N=20**• 1|5 • 2|1124456679 • 3|001124779**0900 Q1 Results N=32**• 1|249 • 2|0335567799 • 3|2224444445566889 • 4|001**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**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?**Graphs and TablesWhat’s the point?**• Make appropriate comparisons. • Compare similar data. • Make the point more clearly. • Make data more understandable. • Eliminate doubt.**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.**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**Histogram Math Anxiety Scores**.30 .25 .20 .15 .10 .5 .5 2.5 4.5 6.5 8.5 10.5**“Blacks More Pessimistic than whites economic**opportunities”**Laws Covering Sales of Firearms: Increase Restrictions(**2000)?**Women and Firearm Restrictions: Frequency**Distribution(N=538)**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.**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)?**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?**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**Graphs and TablesThe nasty tricks of the trade**Is Brand One really any better than the others?**Stem-and-leaf plot**• Presents the frequency of data points without losing important information. Data set: 25, 27, 29 Stem 2 579 Leaves**Stem-and-leaf plot**• The first digit is the stem • The second digit is each leaf 25 27 29 Stem 2 579 Leaves**Stem-and-leaf plot**• The first digit is the stem • The second digit is each leaf 25 27 29 Stem 2 579 Leaves**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**Types of DistributionsFrequency Distribution**• Frequency distribution • Showing what you have • A way to illustrate how many of each thing.**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**Types of DistributionsPositively Skewed Distribution**• Not all data are created equal • Positive skew • Many data points near the origin of the graph**Types of DistributionsNegatively Skewed Distribution**• Negative skew • Many data points away from the origin of the graph**Types of DistributionsBimodal Distribution**• Bimodal • Two areas under the curve with many data points**Types of DistributionsNon-normal Distributions**• Nonnormal distributions • But not abnormal • Platykurtic: flat like a plate**Types of DistributionsNon-normal Distributions**• Leptokurtic: up & down (like leaping) • Bimodal: lumpy**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**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**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**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**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**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**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**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?**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).**Cumulative Frequency Distribution**• How many data points have accounted for as each group is displayed.**Cumulative Frequency Distribution**• Cumulative frequencies can also be illustrated using percentages.

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