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Outline: Numerical Methods

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  1. Outline: Numerical Methods • Measures of Central Tendency • Representative value • Mean • Median, mode, midrange • Measures of Dispersion (Variability) • How are data points are deviated from the mean? • Range • variance, standard deviation (c) 2007 IUPUI SPEA K300 (4392)

  2. Central Tendency: Mean • Arithmetic average • Sum divided by N (# of observations) • Key statistic in data analysis (c) 2007 IUPUI SPEA K300 (4392)

  3. Central Tendency: Mean Question 12 on page 117 The mean is 100.12963 = 10814 / 108 (c) 2007 IUPUI SPEA K300 (4392)

  4. Central Tendency: Median • Midpoint of data arranged in order • When even number of observations, mean of the two data points in the middle • Useful when data are skewed to the right or left substantially. (c) 2007 IUPUI SPEA K300 (4392)

  5. Central Tendency: Mode • Value that occurs most often • Peak in the histogram • Bimodal with two peaks (modes) • Figure 3-1 on page 115 (c) 2007 IUPUI SPEA K300 (4392)

  6. Central Tendency: Midrange • Mean of minimum and maximum values • (minimum + maximum)/2 (c) 2007 IUPUI SPEA K300 (4392)

  7. Central Tendency: Others • Weighted mean when individual data points should be weighted differently (other than 1) • Trimmed mean in the presence of outliers (extremely large or small data points) (c) 2007 IUPUI SPEA K300 (4392)

  8. Quantiles • Quantiles are points taken at equal intervals from CDF (cumulative density function) • 100 quantiles: percentiles • 10 quantiles: dociles • 5 quantiles: quintiles • 4 quantiles: quartiles • 2 quantiles: ? (c) 2007 IUPUI SPEA K300 (4392)

  9. Percentiles • Percentiles divide data into 100 groups with an equal interval • 100 quantiles • Nth percentile is located at nth from the smallest in data w/ 100 observations • Median = 50th percentile • Table 3-3 on page 141 • Figure 3-5 on page 142 (c) 2007 IUPUI SPEA K300 (4392)

  10. Quartiles • 4-quantiles • 1st quartile (25th percentile) • 2st (50th percentile or median) • 3st (75th percentile) • IQR (interquartile range) = 3Q-1Q • Box plot include 1Q, 2Q, 3Q, minimum, maximum (c) 2007 IUPUI SPEA K300 (4392)

  11. Quartiles in a Box Plots (c) 2007 IUPUI SPEA K300 (4392)

  12. Quartiles in Histograms (c) 2007 IUPUI SPEA K300 (4392)

  13. Example 1: Example 3-38, p159 (c) 2007 IUPUI SPEA K300 (4392)

  14. Example 2: SAS Output SAS includes mean in the box plot Histogram # Boxplot 2.75+* 1 | .** 4 | .******** 23 | .**************** 46 | .*********************** 68 +-----+ .*************************** 80 | | .*************************************** 116 *--+--* .********************** 64 +-----+ .******************* 56 | .********* 27 | .***** 13 | -2.75+* 2 | ----+----+----+----+----+----+----+---- * may represent up to 3 counts (c) 2007 IUPUI SPEA K300 (4392)

  15. Example 3: Box Plots | 9 + | | | | | | | | | | | 8 + | *-----* | | | | | | | + | | | | | 7 + +-----+ +-----+ | | | | | | | | | | | | 6 + | | | | | | | | | | | | | | | | 5 + | | +-----+ 0 | | + | | | | | | | | | | | | | 4 + *-----* | | | | | | | | | | | + | | | | | | 3 + +-----+ *-----* | | | | | | | | | | | | 2 + | +-----+ | | | | | | | | | 1 + | | ------------+-----------+-----------+----------- • Site 102 134 137 (c) 2007 IUPUI SPEA K300 (4392)

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