Descriptive Statistics

Transcription

1 Chapter 2 Descriptive Statistics 2-1 Overview 2-2 Summarizing Data 2-3 Pictures of Data 2-4 Measures of Central Tendency 2-5 Measures of Variation 2-6 Measures of Position 2-7 Exploratory Data Analysis Review and Projects 1

2 2-1 Overview Descriptive Statistics summarizes or describes the important characteristics of a known set of population data Inferential Statistics uses sample data to make inferences about a population 2

3 Important Characteristics of Data 1. Nature or shape of the distribution, such as bell-shaped, uniform, or skewed 2. Representative score, such as an average 3. Measure of scattering or variation 3

4 2-2 Summarizing Data With Frequency Tables Frequency Table lists categories (or classes) of scores, along with counts (or frequencies) of the number of scores that fall into each category 4

5 5 Axial Loads of in. Cans Table

6 Table 2-2 Frequency Table of Axial Loads of Aluminum Cans Axial Load Frequency

7 Class: An interval. Frequency Table Definitions Lower Class Limit: The left endpoint of a class. Upper Class Limit: The upper endpoint of a class. Class Mark: The midpoint of the class. Class width: the difference between the two consecutive lower class limits. 7

8 Score Definition values for the example Table 2-2 Frequency Lower Class Limits: 200, 210, Upper class limits: 209,219 Class Marks: 204.5=( )/2,, 214.5, Class width: =10. 8

9 Determine the Definition Values for this Frequency Table Quiz Scores Frequency Classes Lower Class Limits Upper Class Limits Class Marks Class Width 9

10 Constructing A Frequency Table 1. Decide on the number of classes. 2. Determine the class width by dividing the range by the number of classes (range = highest score lowest score) and round up. class width = round up of range number of classes Select for the first lower limit either the lowest score or a convenient value slightly less than the lowest score. Add the class width to the starting point to get the second lower class limit. List the lower class limits in a vertical column and enter the upper class limits. Represent each score by a tally mark in the appropriate class. Total tally marks to find the total frequency for each class. 10

11 Guidelines For Frequency Tables 1. Classes should be mutually exclusive. 2. Include all classes, even if the frequency is zero. 3. Try to use the same width for all classes. 4. Select convenient numbers for class limits. 5. Use between 5 and 20 classes. 6. The sum of the class frequencies must equal the number of original data values. 11

12 Relative Frequency Table relative frequency = class frequency sum of all frequencies 12

13 Relative Frequency Table Table 2-2 Table 2-3 Score Frequency 9 3 Axial Load Relative Frequency = = =

14 Cumulative Frequency Table Score Table 2-2 Frequency Axial Load Table 2-4 Cumulative Frequency Less than 210 Less than 220 Less than 230 Less than 240 Less than 250 Less than 260 Less than 270 Less than 280 Less than 290 Less than Cumulative Frequencies 14

15 Frequency Tables Table 2-2 Score Frequency Table 2-3 Axial Load Relative Frequency Table 2-4 Axial Load Cumulative Frequency Less than 210 Less than 220 Less than 230 Less than 240 Less than 250 Less than 260 Less than 270 Less than 280 Less than 290 Less than

16 Mean as a Balance Point Mean FIGURE

17 Notation S denotes the summation of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population x is pronounced x-bar and denotes the mean of a set of sample values µ is pronounced mu and denotes the mean of all values in a population 17

18 Mean Definitions the value obtained by adding the scores and dividing the total by the number of scores Sample x = S x n Population µ = S x N Calculators can calculate the mean of data 18

19 Median Definitions the middle value when scores are arranged in (ascending or descending) order often denoted by x (pronounced x-tilde ) is not affected by an extreme value ~ 19

20 (in order) exact middle MEDIAN is no exact middle -- shared by two numbers = 4.5 MEDIAN is

21 Mode Definitions the score that occurs most frequently Bimodal Multimodal No Mode the only measure of central tendency that can be used with nominal data 21

22 Examples a b c Mode is 5 Bimodal No Mode 22

23 Examples a b c Mode is 5 Bimodal No Mode d e Mode is 3 No Mode 23

24 Definitions Midrange the value halfway between the highest and lowest scores Midrange = highest score + lowest score 2 24

25 Round-off rule for measures of central tendency Carry one more decimal place than is present in the orignal set of data 25

26 Frequency Frequency Frequency An Example of Skewness 3 Dataset 1: 3, 4, 4, 5, 5, 5, 6, 6, Symmetric Mean = 5, Median = C1 Dataset 2: 3, 4, 4, 5, 5, 5, 7, 7,9. Mean=5.444, Median = Skewed right C2 Dataset 3: 2, 3, 3, 5, 5, 5, 6, 6, 7. 3 Mean = 4.667, Median = Skewed left C3 26

27 Skewness Figure 2-8 (b) Mode = Mean = Median SYMMETRIC Figure 2-8 (a) Mean Median Mode SKEWED LEFT (negatively) Mode Median Mean SKEWED RIGHT (positively) Figure 2-8 (c) 27

28 Best Measure of Central Tendency Table 2-6 Advantages - Disadvantages 28

29 Mean from a Frequency Table use class mark of classes for variable x S (f x) x = Formula 2-2 S f x = class mark f = frequency S f = n 29

30 Quiz Scores Frequency Class Marks Mean of this frequency table =

31 Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank Bank of Providence Mean Jefferson Valley Bank 7.15 Bank of Providence 7.15 Median Mode Midrang

32 Measure of Variation Range highest score lowest score 32

33 Measure of Variation Standard Deviation a measure of variation of the scores about the mean (average deviation from the mean) 33

34 Sample Standard Deviation Formula S (x x) 2 S = n 1 Formula 2-4 calculators can calculate sample standard deviation of data 34

35 Find the standard deviation of the sample data: 2, 3, 4, 5, 5, 5. S 2 = 8/5=1.6, S=1.26. Use the shortcut formula to find the standard deviations of the above data, and the waiting times at the two banks. 1) S x 2 =104, 2) Jefferson Valley Bank: S x 2 =513.27, S x =71.5, s= ) Bank of Providence: S x 2 =541.09, S x =71.5, s=

36 Population Standard Deviation s = S (x µ) N 2 calculators can calculate the population standard deviation of data 36

37 Symbols for Standard Deviation Textbook Sample s Population s Book Some graphics calculators Some nongraphics calculators Sx xs n 1 s x xs n Some graphics calculators Some nongraphics calculators 37

38 Measure of Variation Variance standard deviation squared Notation } s s 2 2 use square key on calculator 38

39 Variance s 2 = S (x x) 2 n 1 Sample Variance s2 = S (x µ) 2 N Population Variance 39

40 Round-off Rule for measures of variation Carry one more decimal place than was present in the original data 40

41 Standard Deviation Shortcut Formula s = n (S x 2 ) (S x) 2 n (n 1) Formula

42 Frequency IGURE 2-10 Same Means (x = 4) Different Standard Deviations s = s = 0.8 s = 1.0 s = Standard deviation gets larger as spread of data increases. 42

43 FIGURE 2-10 The Empirical Rule (applies to bell shaped distributions) 68% within 1 standard deviation x s x x + s 43

44 FIGURE 2-10 The Empirical Rule (applies to bell shaped distributions) 95% within 2 standard deviations 68% within 1 standard deviation x 2s x s x x + s x + 2s 44

45 FIGURE 2-10 The Empirical Rule (applies to bell shaped distributions) 99.7% of data are within 3 standard deviations of the mean 95% within 2 standard deviations 68% within 1 standard deviation x 3s x 2s x s x x + s x + 2s x + 3s 45

46 Range Rule of Thumb (minimum) x 2s x Range 4s x + 2s (maximum) or s Range 4 46

47 Chebyshev s Theorem applies to distributions of any shape the proportion (or fraction) of any set of data lying within k standard deviations of the mean is always at least 1 1/k 2, where k is any positive number greater than 1. 47

48 Measures of Variation Summary For typical data sets, it is unusual for a score to differ from the mean by more than 2 or 3 standard deviations. 48

49 An application of measure of variation There are two brands, A, B or car tires. Both have a mean life time of 60,000 miles, but brand A has a standard deviation on lifetime of 1000 miles and Brand B has a standard deviation on lifetime of 3000 miles. Which brand would you prefer? 49

50 Quartiles Q 1, Q 2, Q 3 divides ranked scores into four equal parts 25% 25% 25% 25% Q 1 Q 2 Q 3 50

51 Percentiles 99 Percentiles 51

52 Finding the Percentile of a Given Score number of scores less than x Percentile of score x = 100 total number of scores Sorted Axial Loads of 175 Aluminum Cans [1] [16] [31] [46] [61] [76] [91] [106] [121] [136] [151] [166]

53 Start Rank the data. (Arrange the data in order of lowest to highest.) Compute L = ( k ) n where 100 n = number of scores k = percentile in question Is L a whole number? No Change L by rounding it up to the next larger whole number. Yes Finding the Value of the kth Percentile The value of the kth percentile is midway between the Lth score and the highest score in the original set of data. Find P k by adding the L th score and the next higher score and dividing the total by 2. The value of P k is the Lth score, counting from the lowest 53

54 Sorted Axial Loads of 175 Aluminum Cans [1] [16] [31] [46] [61] [76] [91] [106] [121] [136] [151] [166] The 10th percentile: L=175*10/100=17.5, round up to 18. So the 10th percentile is the 18th one in the sorted data, i.e., 230. The 25th percentile: L=175*25/100=43.52, rounded up to 44. The 25th percentile is the 44th one in the sorted data, I.ei

55 Interquartile Range: Q 3 Q 1 Semi-interquartile Range: Midquartile: Q 1 + Q 3 2 Q 3 Q

56 Exploratory Data Analysis Used to explore data at a preliminary level Few or no assumptions are made about the data Tends to evolve relatively simple calculations and graphs 56

57 Exploratory Data Analysis Used to explore data at a preliminary level Few or no assumptions are made about the data Tends to evolve relatively simple calculations and graphs Traditional Statistics Used to confirm final conclusions about data Typically requires some very important assumptions about the data Calculations are often complex, and graphs are often unnecessary 57

58 Boxplots Box-and-Whisker Diagram 5 - number summary Minimum first quartile Q1 Median third quartile Q3 Maximum 58

59 Boxplots Box-and-Whisker Diagram Figure 2-13 Boxplot of Pulse Rates (Beats per minute) of Smokers 59

60 Figure 2-14 Boxplots Normal Uniform Skewed 60

61 Axial Load Outliers Values that are very far away from most of the data

62 Height Class Survey Data n Bone y Boxplots for the heights of those who never broke a bone and those who did 62

63 PULSE When comparing two or more boxplots, it is necessary to use the same scale (yes) SMOKE (No) 63