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Understanding Cumulative Frequencies and Measures of Central Tendency in Statistical Analysis

This chapter provides insight into using graphical representations, such as graphs and tables, to illustrate relationships between variables. It covers cumulative frequencies, cross-tabulations, and rates of change, including practical examples for calculating these metrics. Additionally, it explains the three main measures of central tendency: mode, median, and mean. It discusses their applications based on measurement level and the distribution's shape, helping readers choose the appropriate measure for their statistical analyses.

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Understanding Cumulative Frequencies and Measures of Central Tendency in Statistical Analysis

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  1. Chapter 2 Review • Using graphs/tables/diagrams to show variable relationships • Understand cumulative frequency/percentage and cross-tabulations • Perform rates of change

  2. Cumulative Frequencies

  3. Cumulative Frequencies N = 30

  4. Cross-Tabulations Attitude towards Lowering the Drinking Age to 19

  5. Cross-Tabulations Attitude towards Lowering the Drinking Age to 19

  6. Rate of Change • Rate of Change = (100) * (time 2f – time 1f) (time1f) • Allows us to compare the same population at two points in time. • Always be aware of the sign. • A negative percent signifies a reduction • A positive percent signifies an increase

  7. In 2011, there were 60 thefts from a building. In 2010, there were 40. What is the rate of change? • In 2010, there were 51 cases of forcible rape whereas in 2011, there was 35. What is the rate of change?

  8. Chapter 3Measures of Central Tendency

  9. Measures of Central Tendency • Three main types • Mode • Median • Mean • Choice depends upon level of measurement

  10. The Mode • The mode is the most frequently occurring value in a distribution. • Abbreviated as Mo • Sometimes there is more than one mode • EX: 96, 91, 96, 90, 93, 90, 96, 90 • Bimodal • Mode is the only measure of central tendency appropriate for nominal-level variables

  11. Mode - Example • What is the mode for the following set of numbers? • 20, 21, 30, 20, 22, 20 • Explains nothing about • Ordering of variables • Variation within variables • Distributions can be bimodal and/or multimodal • Several categories with same frequencies

  12. Position of the Mdn The Median • The median is the middle case of a distribution • Abbreviated as Mdn • Appropriate for ordinal data because it only shows direction and not distance • Used if distribution is skewed • How to find the median? • If even, there will be two middle cases – interpolate • If odd, choose the middle-most case • Cases must be ordered

  13. What is the median? odd or even? (7+1)/2=4th case Where is the 4th case? Sort distribution from lowest to highest 1 5 2 9 13 11 4 Example of median: Years in Prison

  14. (8+1)/2=4.5 Half way between the 4th and 5th case (2 + 3) / 2 = 2.5 Median = 2.5 1 1 2 2 3 4 4 6 Example of median with even # of cases Position of the Mdn

  15. The Mean • Most popular measure of central tendency • Assumes equality of intervals • Basis of many higher order formulas for statistical procedures • Use either μ or X depending on whether population or sample estimate

  16. The Mean • The mean is appropriate for interval and ratio level variables X = raw scores in a set of scores N = total number of scores in a set

  17. What is the mean? 4.6 7.9 11.4 2.2 Example: Prison Sentences

  18. An Illustration: Measures of Central Tendency in a Skewed Distribution • Mean = $50,000 • Median = $40,000 • Mode = $30,000

  19. Illustration • Suppose we’re interested in finding out the number of evenings freshmen are drinking each month. Using the first ten responses from a survey, we find the following responses: • 2, 5, 0, 3, 11, 1, 3, 1, 1, 2 • What is the mode, median, and mean?

  20. Comparing the Mode, Median, and Mean • Three factors in choosing a measure of central tendency • Level of measurement • Shape or form of the distribution of data • Skewness • Kurtosis • Research Objective

  21. Level of Measurement

  22. Shape of the Distribution • In symmetrical distribution – mode, median, and mean have identical values • In skewed data, the measures of central tendency are different • Mean heavily influenced by extreme outliers

  23. Research Objective • Choice of reported central tendency depends on the level of precision required. • Most published research requires median and/or mean calculations. • In skewed data, median more balanced view • For advanced statistical analyses, mean usually preferred • In large data sets, mean most stable measure

  24. Summary • Three best known measure of central tendency – mode, median, mean • Three factors determine appropriateness • Level of measurement • Shape of the distribution • Research objective

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