Pols 7000x Statistics in Political Science Class 3 Brooklyn college – CUNY Shang E. Ha

1. Pols 7000xStatistics in Political ScienceClass 3Brooklyn college – CUNYShang E. Ha Leon-Guerrero and Frankfort-Nachmias, Essentials of Statistics for a Diverse Society

2. Chapter 3: Measures of Central Tendency • The Mode • The Median • The Mean • Finding the Mean in a Frequency Distribution • The Shape of the Distribution • Considerations for Choosing a Measure of Central Tendency • Statistics in Practice: Representing Income

3. What is a measure of Central Tendency? • Numbers that describe what is average or typical of the distribution • You can think of this value as where the middle of a distribution lies.

4. The Mode • The category or score with the largest frequency (or percentage) in the distribution. • The mode can be calculated for variables with levels of measurement that are: nominal, ordinal, or interval-ratio.

5. The Mode: An Example • Example: Number of Votes for Candidates for Mayor of Camarillo, California. The mode, in this case, gives you the “central” response of the voters: the most popular candidate. Sheriff Tupper – 11,769 votes The Mode: Jessica Fletcher – 39,443 votes “Dr. Seth Hazlett” Dr. Seth Hazlett – 78,331 votes

6. The Median • The score that divides the distribution into two equal parts, so that half the cases are above it and half below it. • The median is the middle score, or average of middle scores in a distribution.

7. The Mean • The arithmetic average obtained by adding up all the scores and dividing by the total number of scores.

8. Formula for the Mean Where ΣY = sum of all scores N = the number of scores.

9. An Example • Annual per capita carbon dioxide emissions (metric tons) for n = 10 largest nations in population size • Bangladesh 0.3, Brazil 1.8, China 2.3, India 1.2, Indonesia 1.4, Pakistan 0.7, Russia 9.9, U.S. 20.1, Japan 1.4, Nigeria 0.6 • Ordered sample: 0.3, 0.6, 0.7, 1.2, 1.4, 1.4, 1.8, 2.3, 9.9, 20.1 • Mode = 1.4 • Median = (1.4 + 1.4)/2 = 1.4 • Mean = (0.3 + 0.6 + 0.7 + …+ 20.1)/10 = 3.97

10. Calculating the mean with grouped scores where: f Y= a score multiplied by its frequency

11. Example: Mean of Grouped Scores

12. Number of People Age 18 or older living in a U.S. Household in 1996 (GSS 1996) Number of People Frequency 1 190 2 316 3 54 4 17 5 2 6 2 TOTAL 581 Grouped Data: the Mean & Median Calculate the median and mean for the grouped frequency below.

13. Shape of the Distribution • Symmetrical (mean is about equal to median) • Skewed • Negatively (example: years of education) mean < median • Positively (example: income) mean > median • Bimodal (two distinct modes) • Multi-modal (more than 2 distinct modes)

14. Distribution Shape

15. Considerations for Choosing a Measure of Central Tendency • For a nominal variable, the mode is the only measure that can be used. • For ordinal variables: -Use the mode to show what is the most common value in the distribution. -Use the median to show which value is located exactly in the middle of the distribution. • For interval-ratio variables, the mode, median, and mean may all be calculated. The mean provides the most information about the distribution, but the median is preferred if the distribution is skewed.

16. Central Tendency

17. Chapter 4: Measures of Variability • The Importance of Measuring Variability • The Range • The Inter-Quartile Range • The Variance and the Standard Deviation • Considerations for Choosing a Measure of Variation • Reading the Research Literature: Differences in College Aspirations and Expectations Among Latino Adolescents

18. The Importance of Measuring Variability • Central tendency - Numbers that describe what is typical or average (central) in a distribution • Measures of Variability - Numbers that describe diversity or variability in the distribution. These two types of measures together help us to sum up a distribution of scores without looking at each and every score. Measures of central tendency tell you about typical (or central) scores. Measures of variation reveal how far from the typical or central score that the distribution tends to vary.

19. Notice that both distributions have the same mean, yet they are shapeddifferently

20. The Range Range = highest score - lowest score • Range – A measure of variation in interval-ratio variables. It is the difference between the highest (maximum) and the lowest (minimum) scores in the distribution.

21. Percentiles • A score below which a specific percentage of the distribution falls. • For example, the 75th percentile is a score that divides the distribution so that 75% of the cases are below it. • For example, the 25th percentile is a score that divides the distribution so that 25% of the cases are below it.

22. Inter-Quartile Range • Inter-Quartile Range (IQR) – A measure of variation for interval-ratio data. It indicates the width of the middle 50 percent of the distribution and is defined as the difference between the lower and upper quartiles (Q1 and Q3.) • IQR = Q3 – Q1 • Q3 = 75th percentile • Q1 = 25th percentile

23. The Difference Between the Range and IQR These values fall together closely Shows greater variability Importance of the IQR Yet the ranges are equal!

24. Variance • Variance– A measure of variation for interval-ratio variables; it is the average of the squared deviations from the mean

25. Standard Deviation • Standard Deviation – A measure of variation for interval-ratio variables; it is equal to the square root of the variance.

26. Finding the Mean

27. Finding the Standard Deviation

28. Considerations for Choosing a Measure of Variability • For nominal variables, you can only use IQV (Index of Qualitative Variation) • Not discussed in this class! • For ordinal variables, you can calculate the IQV or the IQR (Inter-Quartile Range). Though, the IQR provides more information about the variable. • For interval-ratio variables, you can use IQV, IQR, or variance/standard deviation. The standard deviation (also variance) provides the most information, since it uses all of the values in the distribution in its calculation.