Introduction to StatisticsIntroduction to Statistics

Measures of Central Tendency and Dispersion

• The phrase “descriptive statistics” is used generically in place of measures of central tendency and dispersion for inferential statistics.

• These statistics describe or summarize the qualities of data.

• 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 “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)/nMean of a population: =(X)/N

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.

Donut-Eating ProfessorsDonut-Eating ProfessorsProfessor 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

The Median The Median (not the cement in the middle (not the cement in the middle of the road)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?Professor Weight

Schmuggles 165Bopsey 213Pallitto 189Homer 187Schnickerson 165Levin 148Honkey-Doorey 251Zingers 308Boehmer 151Queenie 132Googles-Boop 199Calzone 227  194.6

Weight

132148151165165187189199213227251308

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 ModeThe Mode (hold the pie and the ala)(hold the pie and the ala)(What does ‘ala’ taste like anyway??) (What does ‘ala’ taste like anyway??)

• The most frequent response or value for a variable.

• Multiple modes are possible: bimodal or multimodal.

Figuring the ModeProfessor Weight

Schmuggles 165Bopsey 213Pallitto 189Homer 187Schnickerson 165Levin 148Honkey-Doorey 251Zingers 308Boehmer 151Queenie 132Googles-Boop 199Calzone 227

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 (standard deviation squared)

• 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 Normal CurveThe Normal Curve• Bell-shaped distribution or curve• Perfectly symmetrical about the mean.

Mean = median = mode• Tails are asymptotic: closer and closer to

horizontal axis but never reach it.

Sample Distribution • What does Andre do

to the sample distribution?

• What is the probability of finding someone like Andre in the population?

• Are you ready for more inferential statistics?

Normal curves and probability

Andre would be here Dr. Boehmer would be here

The Standard Deviation The Standard Deviation • A standardized measure of distance from

the mean.

• In other words, it allows you to know how far some cases are located from the mean. How extreme our your data?

• 68% of cases fall within one standard deviation from the mean, 97% for two deviations.

=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

S =

Formula for Standard DeviationFormula for Standard Deviation

1)-(n

2)( XX

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 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 1050.8

Mean 194.6 2480.1 49.8

Std. Deviation practiceStd. Deviation practice• What is the value of Democracy one std.

deviation above and below the mean?

Descriptive Statistics

319 -10.00 10.00 3.4859 6.71282319

DemocValid N (listwise)

N Minimum Maximum Mean Std. Deviation

The answer is 10.20872 and -3.22692What percentage of all the cases fall within 10.2 and -3.2?Roughly 68%

Std. Deviation practiceStd. Deviation practice

Descriptive Statistics

139 19.77 97.12 66.1166 17.74849139

UrbanpopValid N (listwise)

N Minimum Maximum Mean Std. Deviation

What is the value of Urban population one std. deviation above and below the mean?

The answer is 83.86509 and 48.36811

What percentage of all the cases fall within 83.86 and 48.36?

Roughly 68%

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)

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

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.

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.

Pie Charts:

Proportions of Donut-Eating Professors by Weight Class

130-150

151-185

186-210

211-240

241-270

271-310

311+

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!!!!

Line Graphs: A Time Series

0

10

20

30

40

50

60

70

80

90

100

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

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Approve