Probabilistic & Statistical Techniques

1. Probabilistic & Statistical Techniques Eng. Tamer Eshtawi First Semester 2007-2008

2. Lecture 5 Chapter 2 (part 3) Statistics for Describing Data Main Reference: Pearson Education, Inc Publishing as Pearson Addison-Wesley.

3. Section 3-4 Measures of position

4. Key Concept This section introduces measures that can be used to compare values from different data sets, or to compare values within the same data set. The most important of these is the concept of thez score.

5. Definition • z Score(or standardized value) the number of standard deviations that a given valuexis above or below the mean

6. Measures of Position z score Sample Population Round z to 2 decimal places

7. Interpreting Z Scores Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: z score between –2 and 2 Unusual Values: z score < -2 or z score > 2

8. Definition • Q1 (First Quartile)separates the bottom 25% of sorted values from the top 75%. • Q2 (Second Quartile)same as the median; separates the bottom 50% of sorted values from the top 50%. • Q1 (Third Quartile)separates the bottom 75% of sorted values from the top 25%.

9. 25% 25% 25% 25% Q1 Q2 Q3 (minimum) (maximum) (median) Quartiles Q1, Q2, Q3 divide ranked scores into four equal parts

10. Find lower & upper Quartile To fined Q1, first calculate one-quarter of n and add ½ to obtain ¼ n + ½ . Round this to nearest integer. Example 1 1 1 2 3 3 8 11 14 19 19 20 n = 11,then ¼ n + ½ = ¼ (11)+½ = 3.25 rounded off to 3 Q1 = 2 Q3 = 19 Example 2 2 5 5 6 7 10 15 21 21 23 23 25 n = 12,then ¼ n + ½ = ¼ (12)+½ = 3.5 then the Q1 in position 3 & 4 which is (5+6)/2=5.5 Q2 in position 9 & 10 which is (21+23)/2=22

11. number of values less than x Percentile of valuex=•100 total number of values Percentiles Just as there are three quartiles separating data into four parts, there are 99percentiles denoted P1, P2, . . . P99, which partition the data into 100 groups.

12. Converting from the kth Percentile to the Corresponding Data Value Notation n total number of values in the data set kpercentile being used

13. Example 1 Find the percentile corresponding the weight of 0.8143 & find P10, P25 Solution

14. Q3- Q1 Q3+ Q1 2 2 • Semi-interquartile Range: • Midquartile: Some Other Statistics • Interquartile Range (or IQR): Q3- Q1 • 10 - 90 Percentile Range: P90-P10

15. Recap In this section we have discussed: • z Scores • z Scores and unusual values • Quartiles • Percentiles • Other statistics

16. Section 3-5 Exploratory Data Analysis (EDA)

17. Key Concept This section discusses outliers, then introduces a new statistical graph called a boxplot, which is helpful for visualizing the distribution of data.

18. Important Principles • An outlier can have a dramatic effect on the mean. • An outlier can have a dramatic effect on the standard deviation. • An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured.

19. Definitions • For a set of data, the5-number summaryconsists of the minimum value; the first quartile Q1; the median (or second quartile Q2); the third quartile, Q3; and the maximum value. • Aboxplotis a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q1; the median; and the third quartile, Q3.

20. Boxplots

21. Boxplots – cont.

22. Boxplots – cont.

23. Boxplots – cont.

24. Boxplots - Example

25. Recap In this section we have looked at: • Exploratory Data Analysis • Effects of outliers • 5-number summary • Boxplots

26. General Examples

27. Example 1 Fine mean, median, mode, midrange Solution

28. Example 2 Fine Standard deviation, variance for each of the two sample

29. Example 3

30. Example 4 Fine the indicated quartile or percentile a) Q1, b) Q3, c) P80, d) P33 Q1 position = ¼ n + ½ = ¼ (36)+½ = 9.5 (between 9th – 10th) Q1= ( 0.8143+0.815 )/2=0.8147 Q3= ( 0.8207+0.8211 )/2=0.8209

32. Example 5 Draw the boxplot for the following data set Solution

34. Flash points

35. Which measure of center is the only one that can be used with data at the nominal level of measurement? • Mean • Median • Mode

36. Which of the following measures of center is not affected by outliers? • Mean • Median • Mode

37. Find the mode (s) for the given sample data. • 79, 25, 79, 13, 25, 29, 56, 79 • 79 • 48.1 • 42.5 • 25

38. Which is not true about the variance? • It is the square of the standard deviation. • It is a measure of the spread of data. • The units of the variance are different from the units of the original data set. • It is not affected by outliers.

39. Weekly sales for a company are $10,000 with a standard deviation of $450. Sales for the past week were $9050. This is • Unusually high. • Unusually low. • About right.

40. In a data set with a range of 55.1 to 102.8 and 300 observations, there are 207 data points with values less than 88.6. Find the percentile for 88.6. • 32 • 116.03 • 69 • 670

41. H.W 2 Fine mean, median, mode, midrange, range, standard deviation, variance, P30 Then draw the Boxplot Age of US President