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THE STATS UNIT

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BACKGROUND Statistics can be used as a tool to help demystify research data. Everyday examples of the use of statistics include: election polls market research, exercise regimens drug abuse surveys Statistics is a means of organizing and analyzing data (numbers) systematically so that they have meaning. Descriptive statistics organize data so that we can communicate about that data. Inferential statistics answer the question, "What can we infer about the population from data gathered from the sample?"

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MEASUREMENT SCALES Nominal scale - numbers are used to name or categorize: driver's license gender (#1-female, #2-male) car color (assign numbers to denote color: red #1, blue #2, white #3, black #4) Ordinal scale - numbers represent serial position: finish in a race 1=small; 2=medium; 3=large Interval scale - consistent units of measurement, and equal spacing between measurement units, but has no true zero point: Fahrenheit or Centigrade temperature

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Ratio scale - same consistent units of measurement as in the interval scale but with the added property of a true zero point. This measurement scale has the advantage of allowing ratio comparisons. For example, 10 pounds is twice as heavy as 5 pounds, or twelve ducks is four times as many as three ducks. Other examples are: weight height time length score on most exams

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LOOKING AT DATA IN A MEANINGFUL WAY Frequency distributions: An organized list enables us to see clusters or patterns in data that would be less obvious in an unorganized list. Scores are listed in ascending or descending order so that groupings of same scores can be recognized. For example, scores on an exam might include: 91, 92, 87, 99, 83, 84, 82, 93, 89, 91, 85, 94, 91, 98, 90

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Frequency distribution of ungrouped scores: 99 1 98 1 97 0 96 0 95 0 94 1 93 1 92 1 91 3 90 1 89 1 88 0 87 1 86 0 85 1 84 1 (Frequency counts of 0 are always included in the table, as shown.) 83 1 82 1 81 0 80 0 N= 15 (N represents the total number of observations or scores.)

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Grouped Frequency Distribution of same scores: 95-99 2 90-94 7 The width of the intervals in grouped frequency tables 85-89 3 must be equal. There should be no overlap between the intervals. 80-84 3 N=15 The grouped distribution shows that the scores tend to cluster in the 90-94 range. The class can then see how their own score compares in relation to the groupings - or the teacher can evaluate the testing/teaching process by observing this clustering of scores. This clustering is not apparent from the original random listing of the test scores.

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GRAPHS Pie Chart : A pie chart is a circle within which all of the data points or numbers are contained in the form of percentages. Suppose that you are a car dealer and are interested in knowing what percentage of cars you should order in each color. A pie chart divided by the percentages of each car color would be a quick visual representation of the data from the previous year's car sales. Bar Graph: A common method used for representing nominal data is with a bar graph. The height of the bars indicate the percentage or frequency of each category.

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LINE GRAPH Line Graph: Line graphs indicate changes that occur during experiments. It shows the change in relationship between the independent and dependent variables. Perhaps you are investigating the amount of change in SAT scores between groups retaking the test: those who take an SAT course and those who do not. The independent variable (the two groups in this case) is always on the vertical axis and the dependent variable (i.e., SAT scores) is always on the horizontal axis.

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ORGANIZING YOUR DATA Graphical representation of data is typically the first organizational step. Frequency distributions, histograms, and/or frequency polygons are usually prepared in this process. A frequency distribution, often the first organizational step, is an ordered arrangement of all variables, which shows the number of occurrences in each category. Table 1 shows a frequency distribution concerning how much time students spent studying for an exam. Note that the total number tallied (counted) in each category by the researcher equals the number listed in the frequency column.

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GRAPHS Histogram: Frequency histograms are bar graphs. Figure 1 shows a frequency histogram derived from the data in the frequency distribution in Table 1. The frequency (number of students) determined from the tally is the ordinate (vertical, or Y, axis), and the number of hours studied is the abscissa (horizontal, or X, axis). Each one-hour interval is presented sequentially, and the height of each bar represents the number of students who studied that number of hours.

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GRAPHS Frequency Polygon: Frequency polygons are graphs in which the frequency of occurrence of the variable measured is shown by using connected points rather than bars. Figure 2 shows, in a frequency polygon, the same data displayed in Figure 1. (Note that if the midpoints of each of the bars in Figure 1 were connected, the result would be this frequency polygon.)

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SKEWING THE DATA

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ASSIGNMENT Draw a graph of the grouped frequency distribution of the sample grades. Select between a histogram and a polygon. Be sure that you list the grouped scores on the horizontal (or X) axis and the frequencies on the vertical (or Y axis). Identify any skew in the distribution. Graph the frequencies again using different groupings. This will give you experience as to how the same data can be used to convey different impressions. Statistics and graphs don't lie, but they can be deceptive or used inappropriately.