Central Tendency and Variability

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Central Tendency and Variability

• The two most essential features of a distribution

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Questions

• Define
• Mean
• Median
• Mode
• What is the effect of distribution shape on
  measures of central tendency?
• When might we prefer one measure of central
  tendency to another?

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Questions (2)

• Define
• Range
• Average Deviation
• Variance
• Standard Deviation
• When might we prefer one measure of variability
  to another?
• What is a z score?
• What is the point of Tchebycheffs inequality?

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Variables have distributions

• A variable is something that changes or has
  different values (e.g., anger).
• A distribution is a collection of measures,
  usually across people.
• Distributions of numbers can be summarized with
  numbers (called statistics or parameters).

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Central Tendency refers to the Middle of the
Distribution
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Variability is about the Spread
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1. Central Tendency Mode, Median, Mean

• The mode the most frequently occurring score.
  Midpoint of most populous class interval. Can
  have bimodal and multimodal distributions.

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Median

• Score that separates top 50 from bottom 50
• Even number of scores, median is half way between
  two middle scores.
• 1 2 3 4 5 6 7 8 Median is 4.5
• Odd number of scores, median is the middle number
• 1 2 3 4 5 6 7 Median is 4

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Mean

• Sum of scores divided by the number of people.
  Population mean is (mu) and sample mean is
  (X-bar).
• We calculate the sample mean by
• We calculate the population mean by

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Deviation from the mean

• x X . Deviations sum to zero.
• Deviation score deviation from the mean
• Raw scores
• Deviation scores

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8 9 10
7 8 9 10 11
0
-1 0 1
-2 -1 0 1 2
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Comparison of mean, median and mode

• Mode
• Good for nominal variables
• Good if you need to know most frequent
  observation
• Quick and easy
• Median
• Good for bad distributions
• Good for distributions with arbitrary ceiling or
  floor

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Comparison of mean, median mode

• Mean
• Used for inference as well as description best
  estimator of the parameter
• Based on all data in the distribution
• Generally preferred except for bad
  distribution. Most commonly used statistic for
  central tendency.

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Best Guess interpretations

• Mean average of signed error will be zero.
• Mode will be absolutely right with greatest
  frequency
• Median smallest absolute error

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Expectation

• Discrete and continuous variables
• Mean is expected value either way
• Discrete
• Continuous
• (The integral looks bad but just means take the
  average)

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Influence of Distribution Shape
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Review

• What is central tendency?
• Mode
• Median
• Mean

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2. Variability aka Dispersion

• 4 Statistics Range, Average Deviation,
  Variance, Standard Deviation
• Range high score minus low score.
• 12 14 14 16 16 18 20 range20-128
• Average Deviation mean of absolute deviations
  from the median

Note difference between this definition
undergrad text- deviation from Median vs. Mean
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Variance

• Population Variance
• Where means population variance,
• means population mean, and the other terms
  have their usual meaning.
• The variance is equal to the average squared
  deviation from the mean.
• To compute, take each score and subtract the
  mean. Square the result. Find the average over
  scores. Ta da! The variance.

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Computing the Variance
(N5)
5 15 -10 100
10 15 -5 25
15 15 0 0
20 15 5 25
25 15 10 100
Total 75 0 250
Mean Variance Is ? 50
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Standard Deviation

• Variance is average squared deviation from the
  mean.
• To return to original, unsquared units, we just
  take the square root of the variance. This is
  the standard deviation.
• Population formula

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Standard Deviation

• Sometimes called the root-mean-square deviation
  from the mean. This name says how to compute it
  from the inside out.
• Find the deviation (difference between the score
  and the mean).
• Find the deviations squared.
• Find their mean.
• Take the square root.

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Computing the Standard Deviation
(N5)
5 15 -10 100
10 15 -5 25
15 15 0 0
20 15 5 25
25 15 10 100
Total 75 0 250
Mean Variance Is ? 50
Sqrt SD Is ?
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Example Age Distribution
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Review

• Range
• Average deviation
• Variance
• Standard Deviation

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Standard or z score

• A z score indicates distance from the mean in
  standard deviation units. Formula
• Converting to standard or z scores does not
  change the shape of the distribution. Z-scores
  are not normalized.

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Tchebycheffs Inequality (1)

• General form

Suppose we know mean height in inches is 66 and
SD is 4 inches. We assume nothing about the
shape of the distribution of height. What is the
probability of finding people taller than 74
inches? (Note that b is a deviation from the
mean in this case 74-668.). Also 74 inches is
2 SDs above the mean therefore, z 2.
If we assume height is normally distributed, p
is much smaller. But we will get to that later.
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Tchebycheff (2)

• Z-score form
• Probability of z score from any distribution
  being more than k SDs from mean is at most 1/k2.
• Z-scores from the worst distributions are rarely
  more than 5 or less than -5.
• For symmetric, unimodal distributions, z is
  rarely more than 3.

For the problem in the previous slide
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Review

• Z-score in words
• Z-score in symbols
• Meaning of Tchebycheffs theorem

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Median House Price Data

• Find data
• Show Univariate
• Show plots