1. (c) 2007 IUPUI SPEA K300 (4392) 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

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

3. (c) 2007 IUPUI SPEA K300 (4392) Central Tendency: Mean ClassMidpointFrequencyFrequency × Midpoint 90-98946564 99-10793222046 108-116112434816 117-125121283388 126-13413091170 Sum10810814 Question 12 on page 117 The mean is 100.12963 = 10814 / 108

4. (c) 2007 IUPUI SPEA K300 (4392) 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.

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

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

7. (c) 2007 IUPUI SPEA K300 (4392) 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)

8. (c) 2007 IUPUI SPEA K300 (4392) 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: ?

9. (c) 2007 IUPUI SPEA K300 (4392) 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 = 50 th percentile Table 3-3 on page 141 Figure 3-5 on page 142

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

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

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

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

14. (c) 2007 IUPUI SPEA K300 (4392) 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

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