Educational Statistics

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Educational Statistics

• Measures of Central Tendency and Variability
  (Dispersion)

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Over the Counter Drug Sales (in Millions)
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Measures of Central Tendency

• Measures of central tendency tell us something
  about the typicalness of a set of data.
• Tell us what the typical score is in a
  distribution of scores.
• Three measures of central tendency
• Mode
• Median
• Mean

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Measures of central tendencythe mode

• The mode is the score that occurs most frequently
  in a distribution.
• Sometimes more than one score occurs at
  frequencies distinctively higher than other
  scores, in which case there is(are) more than one
  mode
• Bi-modal distributions.
• Multi-modal distributions.
• Only appropriate measure of central tendency for
  nominal data.

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Over the Counter Drug Sales(in Millions)
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Measures of central tendencythe median (Md)

• The median is the middle score in an ordered
  distribution of scores.
• It is the score that divides a distribution in
  half.
• It is also the score at the 50th percentile rank.
• The median can be found by computing the median
  location (N 1)/2.
• The median is the most appropriate measure of
  central tendency for ordinal data.

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Measures of central tendencythe median (Md)

• In the distribution given to the right, find the
  median.
• What is the percentile rank (PR) of a score of
  13?
• What is the score cor-responding to a percentile
  rank of 29?

• Score Freq Cum. F c
• 6 2 2 3.8
• 8 5 7 13.5
• 9 0 7 13.5
• 10 8 15 28.8
• 11 11 26 50.0
• 12 9 35 67.3
• 13 6 41 78.8
• 14 4 45 86.6
• 15 5 50 96.2
• 16 2 52 100.0

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Over the Counter Drug Sales (in Millions)
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Measures of central tendencythe Mean (µ,M, or
)

• The most widely used measure of central tendency
• In words, the mean ( ) is the sum (S) of the
  scores (the Xs) divided by the number of scores
  (N).

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Over the Counter Drug Sales (in Millions)
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Measures of Central Tendency with special
Distributions

• The mode and bimodal distributions.
• For distributions with more than one mode, the
  other measures of central tendency are
  misleading.
• The Median and skewed distributions.
• When a distribution is skewed the use of the mean
  may be misleading
• Skew can be determined by the relative positions
  of the mean, median, and mode.

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Measures of Variability

• How would you describe the variability in the
  distribution of SAT-V scores given at the right?
• In other words, how spread-out are the scores?
• Think about it.
• Write these values down.

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Measures of Variability

• Three common measures of variability are
• The Range.
• The Variance.
• The Standard deviation.
• Other measures of variability are
• The interquartile range.
• The quartile deviation or semi-interquartile
  range.

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Measures of VariabilityThe Range (R)

• R The difference between the largest value in
  the distribution and the smallest value in the
  distribution.
• I.e. R Xlargest Xsmallest.
• Compute the Range for the distribution given.
• R 175.

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Measures of VariabilityThe Variance (Var)

• The variance is more computationally comples.
• Defined as the average squared deviation from the
  mean of the distribution.
• In symbols

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Computing the Variance

• First, compute the sum

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Computing the Variance

• First, compute the sum
• Then, divide by N

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Computing the Variance

• Next, subtract the mean from each score (call
  these deviations from the mean, or, d )

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Computing the Variance

• Next, subtract the mean from each score (call
  these deviations from the mean, or, d )

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Computing the Variance

• Next, Square the deviations from the mean

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Computing the Variance

• Next, Square the deviations from the mean

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Computing the Variance

• Now, sum the squared deviations
• And divide by N

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Measures of VariabilityThe Standard Deviation

• Generally, we would prefer a measure of
  variability that tells us something about how
  far, on average, scores deviate from the mean.
• This is what the standard deviation tells us.
• Since the variance is the average squared
  deviation from the mean, the standard deviation,
  computed as the square root of the variance gives
  us the average deviation from the mean.

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Measures of VariabilityThe Coefficient of
Variation (CV)

• Distributions with larger means tend to have
  larger variances (and SDs) than distributions
  with smaller means.
• The CV provides convenient way to compare the
  variances of two or more distributions.

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Using Statistics as Estimators

• We are rarely interested in sample statisticswe
  are interested in population parameters.
• Statistics are used to estimate (or make
  inferences about) parameters.
• The best statistics are sufficient, unbiased,
  efficient, and robust (or resistant)

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• End of Presentation