Introduction to Statistics Measures of Central Tendency and Dispersion.
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Transcript of Introduction to Statistics Measures of Central Tendency and Dispersion.
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Introduction to StatisticsIntroduction to Statistics
Measures of Central Tendency and Dispersion
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• 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.
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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
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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
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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.
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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
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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)
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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
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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.
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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.
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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.
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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)
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• 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.
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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.
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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.
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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?
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Normal curves and probability
Andre would be here Dr. Boehmer would be here
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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.
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=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
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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
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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%
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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%
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Organizing and Graphing Data
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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)
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What is the Distribution?
• Gives us a picture of the variability and central tendency.
• Can also show the amount of skewness and Kurtosis.
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Graphing Data: Types
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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.
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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
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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.
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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.
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Pie Charts:
Proportions of Donut-Eating Professors by Weight Class
130-150
151-185
186-210
211-240
241-270
271-310
311+
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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!!!!
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Line Graphs: A Time Series
0
10
20
30
40
50
60
70
80
90
100
Month
App
rova
l
Approval
Economic approval
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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