Chapter 5 : Describing Distributions Numerically
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Chapter 5 : Describing Distributions Numerically I. Finding the Center: The Medianmidrange_ - (highest + lowest) / 2 sensitive to outlying valuesmedian the middle value that divides the histogram into 2 equal areas (include units)After you find it ask yourself how well it actually summaries the dataIf odd number of values ; if n is even there is 2 middles so Find the median of the values:12, 15, 38, 25, 12, 15, 16, 22, 13, 33, 11, 25, 16, 18, 23, 18, 19, 13, 14 Median: _______12, 15, 38, 25, 12, 15, 16, 22, 13, 33, 11, 25, 16, 18, 23, 18, 19, 13, 14, 16 Median: _______


  • Spread: Home on the Range

  • The more the data vary, the less the median alone can tell us. So, you should always report a measure of spread.

  • Range: max – min (single number, not an interval, also sensitive to outliers)

  • Spread: The Interquartile Range

  • Concentrate on the middle . (ignore extremes)

  • Quartiles – divides data into 4 equal parts

    Lower Quartile (Q1) Median (Q2) Upper Quartile (Q3)

  • Interquartile Range(IQR): Upper Quartile – Lower Quartile

    • Textbook includes median in each half, graphing calculator does not)

  • Lower Quartile 25th percentile); Upper Quartile (75thpercentile)


  • 5 Number Summary

  • Reports a distributions median, quartiles, and extremes (min, Q1, median, Q3, max)

  • Making Boxplots

  • Box plot– displays the 5 number summary as a central box with whiskers that extend to the non-outlying data values

  • Particularly effective for comparing distributions.

  • Fences - used to identify outliers

  • (help with construction, but never include in your boxplot)

    • If a data value falls outside one of the fences, we do not connect it with whiskers

    • Lower Fence: Q1 – 1.5IQR

    • Upper Fence: Q3+ 1.5IQR


  • Mean or Median?

  • Mean cuts the data into 2 halves not taking into account their size

  • Median takes their size into account (the point at which the histogram would balance)

    • Left Skewed → mean to the left of the median

    • Right Skewed → mean to the right of the median

  • If data is skewed better to use themedian_.


  • What about the Spread? The Standard Deviation

  • IQR is good but ignores individual data

  • Standard deviation– takes into account how far each value is from the mean

    • Only appropriate for symmetric data

    • Deviation– distance a value is from the mean

    • Could average them but the + and – would cancel each other out, so we square them

    • Standard Deviation_ – the average (almost) of the deviations


  • Shape, Center, and Spread

  • So…

  • Skewed →IQR & MEDIAN

  • Symmetric → MEAN & STANDARD DEVIATION

  • Outliers → median / IQR_ OR Mean / standard deviation without outliers

  • Read page 87 (What Can Go Wrong) and 88- 89 (Terms)


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