Measuring Center

  • Median

    • The middle number

    • The median is resistant to outliers

  • Mean

    • The average number

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  • Calculation

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Measuring Position

  • Percentiles

    • Percentage of observations your value is above

    • 30th percentile is the value below which 30 percent of the observations may be found

    • Take the average if there are two values

  • Quartiles

    • Q1 = first quartile = 25th percentile

    • Q2 = median = 50 th percentile

    • Q3 = third quartile = 75th percentile

Measuring Spread

  • Range

    • Highest - Lowest
  • IQR (Interquartile Range)

    • Q3 - Q1
  • Variance / Standard Deviation

    Variance =

    Variance =

  • Use Sx in the TI-84 calculator

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Outliers

  • 1.5 * IQR rule

    • Q1 - 1.5 * IQR = Bottom fence

    • Q3 + 1.5 * IQR = Upper fence

    • Outside the "fence" = Outlier

  • Example

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Boxplots

  • Graph of the "5-number Summary"

  • Min Q1 Median Q3 Max

Lower Quarltile Ouliers Minimu 1.5 'QR Lower Limit Median (Q2) Data
Range Upper Quartile (Q2) Maximum 1.5 'QR Ouliers Upper Limit

Frequency Frequency Frequency Frequency 0 100 250 20 40 60 Y2 71 Y2 71

  • Example

    • 18, 18, 17, 27, 22, 19, 20, 21, 18, 35, 32, 40

    • Min: 17

    • Q1: 18

    • Median: 20.5

    • Q3: 29.5

    • Max: 40

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  • Calculator

    • Boxplot

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  • Modified Boxplot

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SOCS

  • Shape

    • Skewed left/right
  • Outlier

    • Q1 - 1.5 * IQR

    • Q3 + 1.5 * IQR

  • Center

    • Mean or Median
  • Spread

    • SD or IQR
  • Example

    • 18, 18, 17, 21, 26, 40, 23, 27, 22, 19, 20, 21, 18, 35, 32

    • Skewed right

    • No outliers

    • Centered at a median of 20.5

    • A spread of IQR = 11.5

The Effect of Changing Units on Summary Measures

  • Linear Transformations

    • Xnew = A + B * Xold

    • A: only affect mean and median

    • B: affect all

Examples

72 93 57 70 83 59 76 78 74 74 56 65 68 73 67 80 57 67 74 110

  • Construct a modified boxplot for this data set

    • Min = 56

    • Q1 = 66

    • Med = 72

    • Q3 = 77

    • Max = 100

    • IQR = Q3 - Q1 = 11

    • IQR * 1.5= 16.5

    • Bottom fence = Q1 - IQR * 1.5 = 49.5

    • Upper fence = Q3 + IQR * 1.5 = 93.5

I J SOI

  • Describe the distribution of test scores

    • Symmetric

    • One outlier at 110

    • Centered at a median of 72

    • With a spread of IQR of 11

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