Introduction to statistics
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This document provides an introduction to descriptive statistics and measures of central tendency, including the mean, median, and mode. It discusses how the mean can be impacted by outliers, while the median is not. The standard deviation and variance are introduced as measures of dispersion that quantify how much values vary from the mean or from each other. Finally, the document discusses different ways of organizing and graphing data, including histograms, pie charts, line graphs, and scatter plots.
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Introduction to statistics
- 1. Introduction to Statistics Measures of Central Tendency Written by jwala
- 3. Two Types of StatisticsTwo Types of Statistics • Descriptive statistics of a POPULATION • Relevant notation (Greek): µ mean – N population size ∑ sum • Inferential statistics of SAMPLES from a population. – Assumptions are made that the sample reflects the population in an unbiased form. Roman Notation: – X mean – n sample size ∑ sum
- 4. • Be careful though because you may want to use inferential statistics even when you are dealing with a whole population. • Measurement error or missing data may mean that if we treated a population as complete that we may have inefficient estimates. – It depends on the type of data and project. – Example of Democratic Peace.
- 5. • Also, be careful about the phrase “descriptive statistics”. It is used generically in place of measures of central tendency and dispersion for inferential statistics. • Another name is “summary statistics”, which are univariate: – Mean, Median, Mode, Range, Standard Deviation, Variance, Min, Max, etc.
- 6. Measures of Central TendencyMeasures of Central Tendency • These measures tap into the average distribution of a set of scores or values in the data. – Mean – Median – Mode
- 7. What do you “Mean”?What do you “Mean”? The “mean” of some data is the average score or value, such as the average age of an MPA student or average weight of professors that like to eat donuts. Inferential mean of a sample: X=(∑X)/n Mean of a population: µ=(∑X)/N
- 8. Problem of being “mean”Problem of being “mean” • The main problem associated with the mean value of some data is that it is sensitive to outliers. • Example, the average weight of political science professors might be affected if there was one in the department that weighed 600 pounds.
- 9. Donut-Eating ProfessorsDonut-Eating Professors Professor Weight Weight Schmuggles 165 165 Bopsey 213 213 Pallitto 189 410 Homer 187 610 Schnickerson 165 165 Levin 148 148 Honkey-Doorey 251 251 Zingers 308 308 Boehmer 151 151 Queenie 132 132 Googles-Boop 199 199 Calzone 227 227 194.6 248.3
- 10. The Median (not the cement in the middle of the road) • Because the mean average can be sensitive to extreme values, the median is sometimes useful and more accurate. • The median is simply the middle value among some scores of a variable. (no standard formula for its computation)
- 11. What is the Median? Professor Weight Schmuggles 165 Bopsey 213 Pallitto 189 Homer 187 Schnickerson 165 Levin 148 Honkey-Doorey 251 Zingers 308 Boehmer 151 Queenie 132 Googles-Boop 199 Calzone 227 194.6 Weight 132 148 151 165 165 187 189 199 213 227 251 308 Rank order and choose middle value. If even then average between two in the middle
- 12. PercentilesPercentiles • If we know the median, then we can go up or down and rank the data as being above or below certain thresholds. • You may be familiar with standardized tests. 90th percentile, your score was higher than 90% of the rest of the sample.
- 13. The Mode (hold the pie and the ala) (What does ‘ala’ taste like anyway??) • The most frequent response or value for a variable. • Multiple modes are possible: bimodal or multimodal.
- 14. Figuring the Mode Professor Weight Schmuggles 165 Bopsey 213 Pallitto 189 Homer 187 Schnickerson 165 Levin 148 Honkey-Doorey 251 Zingers 308 Boehmer 151 Queenie 132 Googles-Boop 199 Calzone 227 What is the mode? Answer: 165 Important descriptive information that may help inform your research and diagnose problems like lack of variability.
- 15. Measures of DispersionMeasures of Dispersion (not something you cast…) • Measures of dispersion tell us about variability in the data. Also univariate. • Basic question: how much do values differ for a variable from the min to max, and distance among scores in between. We use: – Range – Standard Deviation – Variance
- 17. The RangeThe Range (no Buffalo roaming!!) • r = h – l – Where h is high and l is low • In other words, the range gives us the value between the minimum and maximum values of a variable. • Understanding this statistic is important in understanding your data, especially for management and diagnostic purposes.
- 18. The Standard Deviation The Standard Deviation • A standardized measure of distance from the mean. • Very useful and something you do read about when making predictions or other statements about the data.
- 19. =square root ∑=sum (sigma) X=score for each point in data _ X=mean of scores for the variable n=sample size (number of observations or cases S = Formula for Standard DeviationFormula for Standard Deviation 1)-(n 2)( XX −∑
- 20. We can see that the Standard Deviation equals 165.2 pounds. The weight of Zinger is still likely skewing this calculation (indirectly through the mean). X X- mean x-mean squared Smuggle 165 -29.6 875.2 Bopsey 213 18.4 339.2 Pallitto 189 -5.6 31.2 Homer 187 -7.6 57.5 Schnickerson 165 -29.6 875.2 Levin 148 -46.6 2170.0 Honkey-Doorey 251 56.4 3182.8 Zingers 308 113.4 12863.3 Boehmer 151 -43.6 1899.5 Queeny 132 -62.6 3916.7 Googles-boop 199 4.4 19.5 Calzone 227 32.4 1050.8 Mean 194.6 2480.1 49.8
- 21. Example of S in useExample of S in use • Boehmer- Sobek paper. –One standard deviation increase in the value of X variable increases the Probability of Y occurring by some amount.
- 22. Table 2: Development and Relative Risk of Territorial Claim Probability* % Change Baseline 0.0401 development 0.0024 -94.3 pop density 0.0332 -17.3 pop growth 0.0469 16.8 Capability 0.0813 102.5 Openness 0.0393 -2 Capability and pop growth 0.0942 134.8 % Change in prob after 1 sd change in given x variable, holding others at their means
- 23. Let’s go to computers! • Type in data in the Excel sheet.
- 24. VarianceVariance 1)-(n 2)( XX −∑ • Note that this is the same equation except for no square root taken. • Its use is not often directly reported in research but instead is a building block for other statistical methods S2 =
- 26. Goal of Graphing? 1. Presentation of Descriptive Statistics 2. Presentation of Evidence 3. Some people understand subject matter better with visual aids 4. Provide a sense of the underlying data generating process (scatter- plots)
- 27. What is the Distribution? • Gives us a picture of the variability and central tendency. • Can also show the amount of skewness and Kurtosis.
- 29. Creating Frequencies • We create frequencies by sorting data by value or category and then summing the cases that fall into those values. • How often do certain scores occur? This is a basic descriptive data question.
- 30. Ranking of Donut-eating Profs. (most to least) Zingers 308 Honkey-Doorey 251 Calzone 227 Bopsey 213 Googles-boop 199 Pallitto 189 Homer 187 Schnickerson 165 Smuggle 165 Boehmer 151 Levin 148 Queeny 132
- 31. Weight Class Intervals of Donut-Munching Professors 0 0.5 1 1.5 2 2.5 3 3.5 130-150 151-185 186-210 211-240 241-270 271-310 311+ Number Here we have placed the Professors into weight classes and depict with a histogram in columns.
- 32. Weight Class Intervals of Donut-Munching Professors 0 0.5 1 1.5 2 2.5 3 3.5 130-150 151-185 186-210 211-240 241-270 271-310 311+ Number Here it is another histogram depicted as a bar graph.
- 33. Pie Charts: Proportions of Donut-Eating Professors by Weight Class 130-150 151-185 186-210 211-240 241-270 271-310 311+
- 34. Actually, why not use a donut graph. Duh! Proportions of Donut-Eating Professors by Weight Class 130-150 151-185 186-210 211-240 241-270 271-310 311+ See Excel for other options!!!!
- 35. Line Graphs: A Time Series 0 10 20 30 40 50 60 70 80 90 100 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Month Approval Approval Economic approval
- 36. Scatter Plot (Two variable) Presidential Approval and Unemployment 0 20 40 60 80 100 0 2 4 6 8 10 12 Unemployment Approval Approve