Measures of Dispersion

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• Measures of Dispersion

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• Series I 70 70 70 70 70 70 70 70 70 70
• Series II 66 67 68 69 70 70 71 72 73 74
• Series III 1 19 50 60 70 80 90 100 110 120
• n 10, ?X 700 Mean 700/10 70

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Measures of Variability

• A single summary figure that describes the spread
  of observations within a distribution.

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MEASURES OF DESPERSION

• RANGE
• INTERQUARTILE RANGE
• VARIANCE
• STANDARD DEVIATION

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Measures of Variability

• Range
• Difference between the smallest and largest
  observations.
• Interquartile Range
• Range of the middle half of scores.
• Variance
• Mean of all squared deviations from the mean.
• Standard Deviation
• Rough measure of the average amount by which
  observations deviate from the mean. The square
  root of the variance.

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Variability Example Range

• Hotel Rates
• 52, 76, 100, 136, 186, 196, 205, 150, 257, 264,
  264, 280, 282, 283, 303, 313, 317, 317, 325, 373,
  384, 384, 400, 402, 417, 422, 472, 480, 643, 693,
  732, 749, 750, 791, 891
• Range 891-52 839

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Pros and Cons of the Range

• Pros
• Very easy to compute.
• Scores exist in the data set.

• Cons
• Value depends only on two scores.
• Very sensitive to outliers.
• Influenced by sample size (the larger the sample,
  the larger the range).

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Inter quartile Range

• The inter quartile range is Q3-Q1
• 50 of the observations in the distribution are
  in the inter quartile range.
• The following figure shows the interaction
  between the quartiles, the median and the inter
  quartile range.

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Inter quartile Range
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Quartiles
Inter quartile IQR Q3
Q1
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Pros and Cons of the Interquartile Range

• Pros
• Fairly easy to compute.
• Scores exist in the data set.
• Eliminates influence of extreme scores.

• Cons
• Discards much of the data.

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Percentiles and Quartiles

• Maximum is 100th percentile 100 of values lie
  at or below the maximum
• Median is 50th percentile 50 of values lie at
  or below the median
• Any percentile can be calculated. But the most
  common are 25th (1st Quartile) and 75th (3rd
  Quartile)

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Locating Percentiles in a Frequency Distribution

• A percentile is a score below which a specific
  percentage of the distribution falls(the median
  is the 50th percentile.
• The 75th percentile is a score below which 75
  of the cases fall.
• The median is the 50th percentile 50 of the
  cases fall below it
• Another type of percentile The quartile lower
  quartile is 25th percentile and the upper
  quartile is the 75th percentile

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Locating Percentiles in a Frequency Distribution
25 included here
25th percentile
50 included here
50th percentile
80th percentile
80 included here
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Five Number Summary

• Minimum Value
• 1st Quartile
• Median
• 3rd Quartile
• Maximum Value

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• VARIANCE
• Deviations of each observation from
  the mean, then averaging the sum of squares of
  these deviations.
• STANDARD DEVIATION
• ROOT- MEANS-SQUARE-DEVIATIONS

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Variance

• The average amount that a score deviates from the
  typical score.
• Score Mean Difference Score
• Average of Difference Scores 0
• In order to make this number not 0, square the
  difference scores (no negatives to cancel out the
  positives).

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Variance Computational Formula

• Population

• Sample

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Variance

• Use the computational formula to calculate the
  variance.

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Variability Example Variance

• Hotel Rates

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Pros and Cons of Variance

• Pros
• Takes all data into account.
• Lends itself to computation of other stable
  measures (and is a prerequisite for many of
  them).

• Cons
• Hard to interpret.
• Can be influenced by extreme scores.

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Standard Deviation

• To undo the squaring of difference scores, take
  the square root of the variance.
• Return to original units rather than squared
  units.

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Quantifying Uncertainty

• Standard deviation measures the variation of a
  variable in the sample.
• Technically,

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Standard Deviation
Rough measure of the average amount by which
observations deviate on either side of the mean.
The square root of the variance.

• Population

• Sample

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Variability Example Standard Deviation
Mean 6 Standard Deviation 2
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Variability Example Standard Deviation
Mean 371.60 Standard Deviation
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Pros and Cons of Standard Deviation

• Cons
• Influenced by extreme scores.

• Pros
• Lends itself to computation of other stable
  measures (and is a prerequisite for many of
  them).
• Average of deviations around the mean.
• Majority of data within one standard deviation
  above or below the mean.

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Mean and Standard Deviation

• Using the mean and standard deviation together
• Is an efficient way to describe a distribution
  with just two numbers.
• Allows a direct comparison between distributions
  that are on different scales.

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WHICH MEASURE TO USE ?

• DISTRIBUTION OF DATA IS SYMMETRIC
• ---- USE MEAN S.D.,
• DISTRIBUTION OF DATA IS SKEWED
• ---- USE MEDIAN QUARTILES

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Mean, Median and Mode
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Distributions

• Bell-Shaped (also known as symmetric or
  normal)
• Skewed
• positively (skewed to the right) it tails off
  toward larger values
• negatively (skewed to the left) it tails off
  toward smaller values

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A 100 samples were selected. Each of the sample
contained 100 normal individuals. The mean
Systolic BP of each sample is presented
110, 120, 130, 90, 100, 140,
160, 100, 120, 120, 110,
130, etc
Mean 120 Sd., 10
Systolic BP level No. of samples 90 -
5 100 - 10 110 - 20
120 - 34 130 - 20
140 - 10 150 - 5 160 -
2
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Normal Distribution
Mean 120 SD 10
120
100
110
130
140
150
90

• The curve describes probability of getting any
  range of values ie., P(xgt120), P(xlt100), P(110
  ltXlt130)
• Area under the curve probability
• Area under the whole curve 1
• Probability of getting specific number 0, eg
  P(X120) 0

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• ANY QUESTIONS

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• THANK YOU