introduction to statistics - university of texas at el pasoutminers.utep.edu/crboehmer/introduction...
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Introduction to Statistics
Measures of Central Tendency
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
• 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.
• 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.
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
What do you What do you ““MeanMean””??
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)/nMean of a population: µ=(∑X)/N
Problem of being Problem of being ““meanmean””• 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.
DonutDonut--Eating ProfessorsEating ProfessorsWeightWeightProfessor
248.3194.6
227227Calzone
199199Googles-Boop
132132Queenie
151151Boehmer
308308Zingers
251251Honkey-Doorey
148148Levin
165165Schnickerson
610187Homer
410189Pallitto
213213Bopsey
165165Schmuggles
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)
What is the Median?
194.6227Calzone199Googles-Boop132Queenie151Boehmer308Zingers251Honkey-Doorey148Levin165Schnickerson187Homer189Pallitto213Bopsey165Schmuggles
WeightProfessor
308251227213199189187165165151148132
Weight
Rank order and choose middle value.
If even then average between two in the middle
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.
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.
Figuring the Mode
227Calzone199Googles-Boop132Queenie151Boehmer308Zingers251Honkey-Doorey148Levin165Schnickerson187Homer189Pallitto213Bopsey165Schmuggles
WeightProfessor
What is the mode?
Answer: 165
Important descriptive information that may help inform your research and diagnose problems like lack of variability.
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
• Remember that we said in order to glean information from data, i.e. to make an inference, we need to see variability in our variables.
• Measures of dispersion give us information about how much our variables vary from the mean, because if they don’t it makes it difficult infer anything from the data. Dispersion is also known as the spread or range of variability.
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.
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.
Formula for Standard DeviationFormula for Standard Deviation
=square root∑=sum (sigma)X=score for each point in data_X=mean of scores for the variablen=sample size (number of observations or cases
1)-(n
2)( XX −∑S =
X X- mean x-mean squaredSmuggle 165 -29.6 875.2Bopsey 213 18.4 339.2 Pallitto 189 -5.6 31.2Homer 187 -7.6 57.5Schnickerson 165 -29.6 875.2Levin 148 -46.6 2170.0Honkey-Doorey 251 56.4 3182.8Zingers 308 113.4 12863.3Boehmer 151 -43.6 1899.5Queeny 132 -62.6 3916.7Googles-boop 199 4.4 19.5Calzone 227 32.4 1050.8Mean 194.6 2480.1 49.8We can see that the Standard Deviation equals 165.2 pounds. The weight of Zinger is still likely skewing this calculation (indirectly through the mean).
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.
Table 2: Development and Relative Risk of Territorial Claim
Probability* % Change
Baseline 0.0401development 0.0024 -94.3
pop density 0.0332 -17.3pop growth 0.0469 16.8Capability 0.0813 102.5Openness 0.0393 -2Capability and pop growth 0.0942 134.8
% Change in prob after 1 sd change in given x variable, holding others at their means
Let’s go to computers!
• Type in data in the Excel sheet.
VarianceVariance
1)-(n
2)( XX −∑S2 =
• 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
Organizing and Graphing Data
Goal of Graphing?
1. Presentation of Descriptive Statistics2. 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)
What is the Distribution?
• Gives us a picture of the variability and central tendency.
• Can also show the amount of skewness and Kurtosis.
Graphing Data: Types
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.
Ranking of Donut-eating Profs. (most to least)
132Queeny
148Levin
151Boehmer
165Smuggle
165Schnickerson
187Homer
189Pallitto
199Googles-boop
213Bopsey
227Calzone
251Honkey-Doorey
308Zingers
Here we have placed the Professors into weight classes and depict with a histogram in columns.
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.
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
Pie Charts:
Proportions of Donut-Eating Professors by Weight Class
130-150151-185186-210211-240241-270271-310311+
Actually, why not use a donut graph. Duh!
Proportions of Donut-Eating Professors by Weight Class
130-150151-185186-210211-240241-270271-310311+
See Excel for other options!!!!
Line Graphs: A Time Series
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Month
App
rova
l
Approval
Economic approval
Scatter Plot (Two variable)
Presidential Approval and Unemployment
0
20
40
60
80
100
0 2 4 6 8 10 12
Unemployment
App
rova
l
Approve