CHAPTER 3 Data Description Student Notes Stat 200 Elementary Statistics Introduction Section 3 1 Measures of Central Tendency Statistic Parameter Data Array Mean Weighted Mean Summation Variables X Example The data represent the annual chocolate sales in billions of dollars for a sample of seven countries in the world Find the mean 2 0 4 9 6 5 2 1 5 1 3 2 16 6 Example Weight of wild bears Weight 0 49 50 99 100 149 150 199 200 249 250 299 300 349 350 399 400 449 450 499 500 549 Frequency 6 10 10 7 8 2 4 3 3 0 1 Midpoints 24 5 74 5 124 5 174 5 224 5 274 5 324 5 374 5 424 5 474 5 524 5 f Xm X f X n Weighted Mean X Example Find the student s grade point average if he received the following grades Composition I Intro to Psychology Biology Physical Ed 2 3 3 4 A 4 points C 2 points B 3 points D 1 point Median Mode Modal Class Example Find the median for the data 34 23 33 36 48 34 26 45 29 m Find the median for the data 34 23 33 36 48 34 26 45 Example Find the mode for the data 34 23 33 36 48 34 26 45 29 Example The annual salaries for different positions are listed Find the mean median and mode for the data Staff Owner Manager Salesperson Technician Technician Salary 50 000 20 000 12 000 9 000 9 000 Midrange Distribution Shapes Section 3 2 Measures of Variation Example Find the mean of each group Group A Group B 10 35 60 45 50 30 30 35 40 40 20 25 Range Population Variance and Standard Deviation Example Find the variance and standard deviation for the data set for Group A 10 60 50 30 40 20 Find the variance and standard deviation for the data set for Group B 35 45 30 35 40 25 Sample Variance and Standard Deviation s2 s s2 Find the variance and standard deviation for the data set 11 2 11 9 12 0 12 8 13 4 14 3 Variance and Standard Deviation for Grouped Data s2 n f X m2 f X m 2 n n 1 Coefficient of Variation Rule of Thumb Chebyshev s Theorem The proportion of values from a data set that will fall within k standard deviations of the mean will be at least 1 1 k2 where k is a number greater than 1 Empirical Rule for Normal Distributions 3 4 Measures of Position z score A student scored 65 on a calculus test that had a mean of 50 and a standard deviation of 10 she scored 30 on a history test with a mean of 25 and a standard deviation of 5 Compare her relative positions on the two tests Interpreting Z Scores Example Compare Joe s height of 78 in to Susan s height of 76 in Use the following information Men have heights with a mean of 69 0 in and st dev of 2 8 in women have heights with mean of 63 6 in and a st dev of 2 5 in Percentiles Quartiles and Deciles Percentile Formula Example A teacher gives a 20 point test to 10 students The scores are shown below Find the percentile rank of a score of 12 18 15 12 6 8 2 3 5 20 10 Example Find the value corresponding to the 25th percentile 18 15 12 6 8 2 3 5 20 10 Finding Data Values Corresponding to Q1 Q2 and Q3 Step 1 Step 2 Step 3 Step 4 Example Find Q1 Q2 and Q3 for the data set 15 13 6 5 12 50 22 18 Outliers Example Check the following data set for outliers 5 6 12 13 15 18 22 50 3 4 Exploratory Data Analysis Boxplots and Five Number Summaries Example Find 5 number summary and construct a boxplot 32 39 40 47 37 76 43 45 36 37 62 60 32 42 43 46 51 40 42 40 53 32 44 36 33 60 41 61 38 56 35 56 39 45 48 46 55 48 31
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