Data Analysis Essentials

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Data Analysis Essentials

• Lecture Notes
• Probability and Statistics I, R. Sinn
• October 20, 2005

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Praxis Ideas from Prob-Stat

• Probability
• Simple probabilities
• Binomial Theorem
• 2 Urn Problems

• Stats
• Mean is heavily influenced by outliers
• Empirical Rule and estimates
• Integration w/ constants (answer given)

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Data Analysis

• Why do we analyze data?
• To determine the distribution the sample was
  drawn from (guesswork)
• Assumptions of major statistical tests
• Independence
• Normality
• Homogeneity of Sample Variances
• The last 2 assumptions are verified by analyzing
• Histograms (bell-shaped?)
• Box Plots (outliers?)

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Ex Stem-and-Leaf Data
USA
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Ex Stem-and-Leaf Plot
1. Is the data sample from a normal
distribution?2. Find the median.
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Lying with Statistics

• There are nearly as many ways to lie with
  statistics are there are people who dont
  understand statistics.
• What is the flaw in the following reasoning?
• Half of all marriages (in the U.S.) end in
  divorce.
• If you graduate from college, marry your college
  sweetheart, is it really just a coin flip if
  youll still be married in 20 years?

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Histograms

• Bar Graphs with special properties.
• Must be flat, 2-D bar graph
• All bars must be same width
• Bars must be touching
• Axes must include origin (no gaps!)
• Provides standard look for all research tables
  and graphs not deceptive
• Standard Excel Bar Graphs are often misleading!

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Central Tendencies

• Mean or Average
• VERY sensitive to OUTLIERS
• Good if data is
• Nearly symmetric (no skew)
• Bad if data is
• Skewed (has outliers on only one side)
• Median
• Compare to mean
• If mean median, data is likely symmetric
• If not, check for outliers/skew
• Misleading to quote it as the average
• Mode
• Limited usefulness

Excel Example
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Skewed Distributions

• Left Test Scores
• Cant get higher than 100
• Outliers only lie to leftdrag mean left
• Right Income Levels
• Cant earn less than 0 per year
• Outliers only lie to rightdrag mean right

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Example Skewed Right
Median
Mean
Mean moves right.
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Example Skewed Left
Mean moves left.
Mean
Median
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Hypothesis Testing

• There are 3 basic formats for 1-Sample hypothesis
  testing

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2 Sample Tests 2 Types

• Dependent Samples
• Data Lists Related
• Pretest-Posttest
• Twins, parent-child, husband-wife
• Called Matched-Pairs Data
• Data List MUST BE SAME SIZE

• Independent Samples
• Data Lists Unrelated
• Demographic Comparisons
• Treatment vs. Control
• Samples MAY BE DIFFERENT SIZE

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Example 1

• Is the following research data dependent samples
  or independent samples?
• A pediatrician measured cholesterol in her young
  patients discovering surprisingly high levels.
  Ten such patients were randomly selected to
  participate in a research study. A treatment was
  performed for 2 months. The cholesterol was then
  re-measured for all ten participants.

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Example 2

• Dependent samples or independent samples?
• A researcher is comparing the average number of
  miles driven by households. A sample of 14
  Midwestern households (with an average of 16,229
  miles) is compared to a sample of 15 Southern
  households (with an average of 17,689).

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Example 3

• Dependent samples or independent samples?
• Dr. Sinn is comparing the mathematics
  self-confidence of White teenagers (n 61) with
  that of Black teenagers (n 43).

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Example 4

• Dependent samples or independent samples?
• A general contractor wishes to compare the
  lifetimes of two major brands of water heaters,
  Eagle and National.

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Example 5

• Dependent samples or independent samples?
• Ten married couples are selected at random to
  participate in a study about relative ages of
  married men and women. The data lists are the
  ages of husbands (list 1) and the ages of their
  wives (list 2). Do the data suggest married men
  are older than their wives?

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Hypothesis Test Steps

• Hypothesis Symbolic Set-Up
• Sample (n lt 25?)
• Histogram (Normal?)
• Box-and-Whisker Plot (No outliers?)
• Set a (Analyze Type I II Error)
• Select Test (use t-test)
• 1 Sample?
• 2 Samples Independent or Dependent?
• Run Test (get p-value)
• Is p lt a ? (reject null)
• State research conclusion in real-world context