1. Descriptive Statistics: Maarten Buis Lecture 1: Central tendency, scales of measurement, and shapes of distributions

2. Outline Practicalities Central tendency Scales of measurement Shapes of distributions

3. Two statistics courses Descriptive Statistics (McCall, part 1) Inferential Statistics (McCall, part 2)

4. Course Material McCall: Fundamental Statistics for Behavioral Sciences. SPSS (available from Surfspot.nl) and chapter 2 of Field Lectures: 2 x a week computer labs: 1 x a week. course mailing list: descriptive_statistics@yahoogroups.com course website

5. setup of lectures Recap of material assumed to be known New Material Student Recap

6. How to pass this course Read assigned portions of McCall before each lecture Do the exercises Do the computer lab assignments, and hand them in before Tuesday 17:00! come to the computer lab come to the lectures ask questions: during class or to the course mailing list answer questions

7. Recap: mean, median, mode Mean of 1, 1, 2, 4 is (1 + 1 + 2 + 4)/4 = 2 Median of 1, 1, 2, 4 is the middle observation, here two middle observations: 1 and 2. Use mean of middle observations, which is 1.5. Mode of 1, 1, 2, 4 is the most common value: 1.

8. Recap: quadratic 1 2 = 1 x 1 =1 2 2 = 2 x 2 = 4 3 2 = 3 x 3 = 9 squaring makes large numbers much larger than small numbers

9. Recap: absolute value |3| = 3 |-3| = 3 just loose the minus sign if it is there

10. Data: rents of rooms rent room 1175room 11240 room 2180room 12250 room 3185room 13250 room 4190room 14280 room 5200room 15300 room 6210room 16300 room 7210room 17310 room 8210room 18325 room 9230room 19620 room 10240

11. What is a reasonable summary One always makes errors What if you choose that number that minimizes the sum of the absolute errors? If you want to put more weight on preventing large errors you could minimize the sum of the squared errors

12. mean and median mean is that summary that minimizes the sum of the squared errors median is that summary that minimizes the sum of the absolute errors

13. Measurement assigning numbers to observations: for example rents to rooms scale of measurement: –nominal –ordinal –interval/ratio

14. nominal == assigning numbers to classify observations in categories The categories are exclusive, but have no further relationship with one another. typical example: religion

15. ordinal == assigning numbers with the purpose to order observation It is meaningful to speak of more or less, or lower or higher typical example: education

16. Interval == assigning numbers to compare differences It is meaningful to say that the “distance between A and C is larger than between B and C” Typical example: temperature, intelligence Hard to find really good examples, often combined with ratio

17. ratio == assigning numbers to compare ratios of observations requires an absolute zero point It is meaningful to say “A is twice B” typical examples: age, income, percentage immigrant children in a classroom

18. What is the scale of measurement of: Choice of Party during an election Gender exam grades highest achieved level of education: primary, secondary, or tertiary

19. what is the scale of measurement of : income percentile of income (top 5% or bottom 20%) highest level of education in years

20. Why bother? Determines which statistical techniques are meaningful: mean religion or most common religion Use common sense

21. Central Tendency 2 Nominal  Mode Ordinal  Mode or Median Interval/ratio  Mode, Median, or Mean

22. Dichotomous variable only two answers possible: yes/no, male/female, 1/0 Every variable can be dichotomized Dichotomous variables can be treated as interval variables: mean is meaningful: percentage “yes”.

23. Frequency distribution A frequency distribution shows how many times a value occurs within a variable Can be visualized in a histogram, frequency polygon, pie chart

24. rentFreq.PercentCum. 1751 5,26% 1801 5,26%10,53% 1851 5,26%15,79% 1901 5,26%21,05% 2001 5,26%26,32% 2103 15,79%42,11% 2301 5,26%47,37% 2402 10,53%57,89% 2502 10,53%68,42% 2801 5,26%73,68% 3002 10,53%84,21% 3101 5,26%89,47% 3251 5,26%94,74% 6201 5,26%100,00% Total19100.00%

25. Shapes of distribution The mean, median and mode are equal in unimodal symmetric distributions. The mean and median are equal in multimodal symmetric distributions Skewness, in a right skewed distribution the mean is right of the median, the income distribution is an example of a right skewed distribution.

26. Shapes of distributions Kurtosis: leptokurtotic (flat) or platycurtotic (peaked) Uniform distribution, each value is equally likely For the fans: Mean, variance, skewness, and kurtosis are the first four moments of a distribution (p. 49 McCall)

27. Do before Wednesday Read: –McCall Ch 1: 6-14 –McCall Ch 2: entirely –McCall Ch 3: 54-63 Exercises: –1.1-1.3 –2.1-2.7, 2.11, 2.13

28. Student recap