Statistics

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Statistics
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Statistics may be defined as a body of methods
for making wise decisions in the face of
uncertainty.W. Allen WallisEconomist
Statistician
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Statistics Terms

• Statistics Procedures used to summarize and
  analyze quantitative data.
• Descriptive statistics Procedures used to
  summarize a set of numbers in terms of central
  tendency, variation, or relationships.
• Inferential statistics Procedures used to
  determine the error when estimating a value for a
  population based upon the measurement of the same
  value for a sample of that population.

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Types of Descriptive Statistics

• Central Tendency The typical score (best bet).
• Variability How different the scores are.
• Correlation Coefficient A measure of the
  relationship between two variables.
• z-Score The relationship of one score to the
  norm group in terms of standard units.
• Effect Size A measure of the magnitude and
  difference of the means of two groups.

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Descriptive Statistics

• Measures of Central Tendency
• - Mean The arithmetic average, sensitive to
  outliers
• - Median The middle score, reduces effect of
  outliers
• - Mode The most frequent score
• Measures of Variability
• - Range The difference between the largest and
  smallest.
• - Standard Deviation The average distance of
  all scores
• from the mean.
• Correlation Coefficient
• - How related two variables are, predictability.
• - Sensitive to outliers (moving R closer to
  zero).

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z-Score

• The quantity z represents the distance between
  the raw score (of an individuals score, for
  instance) and the group mean in units of the
  standard deviation. z is negative when the raw
  score is below the mean and positive when above.

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Effect Size

• The quantity ES represents the difference between
  the mean of the experimental group and the mean
  of the control group in units of the standard
  deviation.

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Inferential Statistics

• The purpose of inferential statistics is to
  make conclusions about some value of a population
  on the basis of that same value measured for a
  sample.
• Inferential statistics allow us to estimate the
  magnitude of our errorthe difference between the
  sample value and the population valueeven though
  we dont know what the population value is.
• One estimate of error is the confidence
  intervala range within which the true value is
  likely () to be. The wider the range, the higher
  the confidence level.

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Sampling Error

• Its always easier and quicker to measure a
  sample drawn from a population than it is to
  measure every person in the population (a
  census).
• Unfortunately, the value for the sample is
  never exactly equal to the true population value.
  This is called sampling error (error due to
  sampling).
• The larger the percentage of the population
  that is sampled, the smaller the sampling error.
  (Think about the increase in accuracy by moving
  from a sample of 50 of the population to 99.)

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Sampling Fluctuation

• Sampling fluctuation occurs when we measure a
  value for samples repeatedly drawn from the same
  population. The value for each sample is
  different from the others (and different from the
  true value of the population).

b 71
Population x 70


a 66
c 73

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Sampling Fluctuation Example

• Five people each grab a fistful of coins from a
  bucket.
• You would expect each to grab a different
  amount.
• When one persons amount is much different from
  anothers, you could say there is a statistically
  significant difference between their grab and
  the others grab.
• The difference between the two is larger than
  you would expect than from sampling fluctuation
  alone.

5.42
8.23
5.25
5.58
5.12
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Statistical Significance

• Statistical significance is a mathematical test
  that gives a yes/no answer to the question Are
  the differences we see larger than we would
  expect than from sampling fluctuation alone?
• - It doesnt tell us which value is larger.
• - It doesnt tell us how big the difference is.
• - It doesnt tell us how important the
  difference is.
• - And because statistical significance is based
  on the size of the sample, one experiment may
  have statistically significant results while
  another may not simply because the sample sizes
  were different.

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Practical Significance

• Practical significance answers the
  all-important question of So what?
• Statistical significance tells us whether the
  differences are larger than we would expect to
  see than from sampling fluctuation alone.
• Effect size tells us the magnitude and the
  direction of the differences.
• Practical significance tells us how important
  the differences are in terms of what people value.

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Restriction in the Range
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Do SAT Scores Predict College GPA?

• Based on your experience at Vanderbilt
• Does it seem like the students with the highest
  SAT scores have the highest GPA?
• Think about the kids you knew in high school
• Did you know smart kids with low SAT scores?
• Did you know kids that werent that bright who
  were able to achieve high SAT scores?
• Do you think that high school SAT scores
  predict college GPA?

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Vanderbilt Class of 2015
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What it Takes to Get In
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What it Takes to Stay In
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A Correlation between SAT GPA?
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A Correlation between SAT GPA?
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r 0.50
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Relationship between SAT GPA
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Minimum SAT of 1,000 to Enter
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Minimum SAT of 1,000 to Enter
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Minimum GPA of 2.0 to Remain
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SAT GPA of Vanderbilt Students
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What is the Correlation Coefficient?
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Restriction of the Range Conclusion