1. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D.1 Chapter 4. Describing Data: Displaying And Exploring Data http://statisticdescriptive.wordpress.com/

2. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 2 Stem And Leaf Displays Stem and leaf display: - a statistical technique to present a set of data. - Each numerical value is divided into two parts. - The leading digit becomes the stem, and the trailing digit the leaf. - The stems are located along the vertical axis, and the leaf values are stacked against each other along the horizontal axis.

3. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 3 Example Page 102: Data: 96, 93, 88, 117,127, 95, 113, 96, 108, 94, 148, 156, 139, 142, 94, 107, 125, 155, 155, ………………………………………………...... …................................................................... 124, 138 The smallest number is 88 The largest number is 156

4. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 4 SteamLeaf 88 9 93 4 4 5 6 6 7 103 3 4 6 7 8 111 2 2 3 3 7 7 8 9 120 0 4 5 5 5 7 7 132 4 5 6 8 9 9 142 3 8 155 5 6

5. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 5 Other Measures Of Dispersion The standard deviation is the most widely used measures of dispersion. Others are: - quartiles. - deciles. - percentiles.

6. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 6 Other Measures Of Dispersion (continued) Quartiles: divide a set of observations into four equal parts. Deciles: divide a set of observations into 10 equal parts. Percentiles: divide a set of observations into 100 equal parts.

7. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 7 Other Measures Of Dispersion (continued) Percentiles: - location of a percentile (equation 4-1, Page 107) L P = (n + 1) (P/100) L p = the location of a desired percentile n = the number of observation P = the desired percentile

8. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 8 If we want to find the 33 rd percentile we want to find L 33 If we want to find median, we want to find L 50

9. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 9 Example Page 107 The Data: 2038, 1758, 1721, 1637, 2097, 2047, 2205, 1787, 2287, 1940, 2311, 2054, 2406, 1471, 1460 n = 15 Sort the data from the smallest to the largest: 1460, 1471, 1637, 1721, 1758, 1787, 1940, 2038, 2047, 2054, 2097, 2205, 2287, 2311, 2406

10. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 10 Example Page 107 (continued) The median = L 50 = Second Quartile L 50 = (15+1)(50/100) = 8 The position number 8 is 2038 L 25 = First Quartile L 25 = (15+1)(25/100)= 4 The position number 4 is 1721

11. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 11 Example Page 107 (continued) L 75 = Third Quartile L 75 = (15+1)(75/100) = 12 The position number 12 is 2205

12. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 12 Example Page 108 The Data: 43, 61, 75, 91, 101, 104 n = 6 L 25 = (6+1)(25/100) = 1.75 First value is 43, second value is 61. The distance first value and second value is (61- 43) = 18. We need move 0.75 of the distance between first and second value: 0.75 (18) = 13.5, so L 25 = 43 + 13.5 = 56.5

13. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 13 Box Plots A graphical display based on quartiles. Needed: 1. The minimum value. 2. The first quartile. 3. The median. 4. The third quartile. 5. The maximum value.

14. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 14 Example Page 110 Alexander’s Pizza offers free delivery of its pizza within 15 miles. Alex, the owner, wants some information on the time it takes for delivery. - How long does a typical delivery take? - Within what range of times will most deliveries be completed?

15. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 15 Example Page 110 (continued) For a sample of 20 deliveries, he determined the following information: Minimum value is 13 minutes. Q1 (the first quartile) is 15 minutes. Median is 18 minutes. Q3 (the third quartile) is 22 minutes. Maximum value is 30 minutes. Develop a box plot for the delivery times. What conclusions can you make about delivery times?

16. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 16 Example Page 110 (continued) Step 1: Create an appropriate scale along the horizontal axis. Step 2: Draw a box that starts at Q1 and ends at Q3. Step 3: Inside the box, place a vertical line to represent the median. Step 4: Extend the horizontal lines from the box out to the minimum value and the maximum value. Look figure page 111 The box plot shows that the middle 50 percent of the deliveries take between 15 minutes and 22 minutes.

17. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 17 Skewness The shape Four shapes: 1. Symmetric. 2. Positively skewed (skewed to the right). 3. Negatively skewed (skewed to the left). 4. Bimodal. Chart 4-1 Page 114

18. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 18 Skewness (continued) To calculate skewness: - Pearson’s coefficient of skewness Equation 4-1 Page 114 sk = {3(X bar – Median)} / s s = standard deviation of sample **Skewness range: -3 up to 3 **Value near -3, indicates considerable negative skewness. **Value near 3, indicates considerable positive skewness. **Value of 0, indicates symmetrical and no skewness.

19. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 19 Example Page 115 The sample data (n=15): 0.09, 0.13, 0.41, 0.51, 1.12, 1.20, 1.49, 3.18, 3.50, 6.36, 7.83, 8.92, 10.13, 12.99, 16.40 - Compute the mean. - Compute the median. - Compute the standard deviation. - Find the coefficient of skewness using Pearson’s estimate. - What is your conclusion regarding the shape of the distribution?

20. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 20 Example Page 115 (continued) X bar is 74.26/15 = 4.95 The middle value is 3.18 S = root of {  (X – X bar) 2 / (n-1)} = 5.22 Sk = {3(4.95-3.18)}/5.22 = 1.017 Indicates there is moderate positive skewness.

21. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 21 Describing The relationship Between Two Variables Using “ A Scatter Diagram”: - need two variables. - One variable along the horizontal axis (X-axis). - The other variable along the vertical axis (Y-axis). Chart 4-1 page 119

22. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D.22 Chapter 4. Describing Data: Displaying And Exploring Data http://statisticdescriptive.wordpress.com/

23. Displaying & Exploring Data Ir.Muhril Ardiansyah,M.Sc.,Ph.D. 23 Homework No. 31 Page 124-125.