Applied Math 40S March 12, 2008

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Introduction to statistics: Why study statistics? Measures of central tendency (mean, median, mode), measures of dispersion (range, standard deviation (intro)).

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Applied Math 40S March 12, 2008

  1. 1. Introduction to Statistics Queen Victoria, Cunard's newest cruise ship by savannahgrandfather
  2. 2. Why Study Statistics? • Two students in two different schools each have marks of 95 percent. Which student should receive an award for getting the 'higher' mark? • How do doctors decide that teenagers should or should not get hepatitis vaccine? • Judith and Francine, both age 19, have decided to go on a Caribbean cruise, and they want to have an enjoyable time, which means that they want to travel with other people their own age. They buy tickets for a cruise where the average age of the other passengers is 20 years. Sounds like fun, no?
  3. 3. Can you imagine their surprise at the start of the cruise when they discover that all the other passengers are parents (average age 32) with children (average age 8)? big_girl_04_m1_screen by pntphoto
  4. 4. Statistics: the branch of mathematics that deals with collecting, organizing, displaying, and analyzing data. statistic: a number that describes one aspect of a group of data. EXAMPLE: mean, median, mode, range, standard deviation, etc... datum: one bit (piece) of information. data: many bits (pieces) of information. Types of Data quantitative data: data that is numeric (eg. height, weight, time..) There are two kinds of quantitative data: continuous and discrete continuous data: can be represented using real numbers (eg. height, weight, time, etc..) discrete data: can be represented by using ONLY intergers (eg. # of people, # of cars, # of animals, etc..) qualitative data: data that is non-numeric (eg. colours, flavours, etc...)
  5. 5. Measures of Central Tendency mean: ( A.K.A. 'the arithmetic meanquot;) the symbol for mean is quot;x barquot;. The arithmetic average of a set of values. where x is the mean where Σx means the sum of all data (x) in the set (Σ is called quot;sigmaquot;) where n is the number of data in a set EXAMPLE: find the average mark this set of 5 quizzes: 48,52,65,45,65.
  6. 6. Measures of Central Tendency median (med): 1) the middle value in an ordered (from smallest to largest) set of data. 2) if there are an even number of data, the median is the average of the middle pair in an ordered set of data. EXAMPLE: find the median of these quiz scores: 12,10,17,11,15 SOLUTION: 10, 11, 12, 15, 17 12 is the median. EXAMPLE: find the median of these scores: 12,10,17,11,15,11 SOLUTION: 10,11,11,12,15,17 the median is 11.5 mode (mo): the datum that occures most frequently in a set of data. EXAMPLE: find the mode in the set of quiz scores: 12,10,17,11,15,11 SOLUTION: the mode is 11 because it occurs more often that any other number in the set.
  7. 7. Mean, Median, Mode, ... A clerk in a men's clothing store keeps a weekly record of the number of pairs of pants sold. The following is her list for two weeks. Mon Tue Wed Thur Fri Sat Week1 34 40 36 36 38 38 Week 2 32 36 36 42 34 34 Calculate the mean, mode, and median for the data shown. Bimodal Distribution
  8. 8. Measures of Dispersion (Variability) Dave can drive to work using the downtown route or the perimeter route. The downtown route is shorter, but it has more traffic, and can become quite crowded. The driving times in minutes for each route (arranged in ascending order) for 5 days are shown on the table below. Downtown Route 15 26 30 39 45 Perimeter Route 29 30 31 32 33 The average driving time for each route is 31 minutes. Which route should he take?
  9. 9. Measures of Dispersion (Variability) determine how quot;spread outquot; or variedquot; a set of data is. Range: the difference between the largest and smallest value in a set of data. EXAMPLE: find the range of ages of people in our class highest value: lowest value: RANGE: with teacher MR K. 40 yrs old. highest value: 40 lowest value: RANGE:
  10. 10. Measures of Dispersion (Variability) Back to our example: Dave can drive to work using the downtown route or the perimeter route. The downtown route is shorter, but it has more traffic, and can become quite crowded. The driving times in minutes for each route (arranged in ascending order) for 5 days are shown on the table below. Downtown Route 15 26 30 39 45 Perimeter Route 29 30 31 32 33 Find the range associated with taking each route. Downtown Route Perimeter Route
  11. 11. Measures of Dispersion (Variability) determine how quot;spread outquot; or variedquot; a set of data is. Standard Deviation (σ): a measure that shows how the data are spread about the mean value. Every value in the data set is used in calculating the standard deviation. Find the standard deviation associated with taking each route to Dave's work using your calculator. Downtown Route 15 26 30 39 45 Perimeter Route 29 30 31 32 33 Downtown Route Perimeter Route
  12. 12. Measures of Dispersion (Variability) determine how quot;spread outquot; or variedquot; a set of data is. Standard Deviation (σ): How is the standard deviation calculated numerically? μ
  13. 13. Let's apply what we've learned ... HOMEWORK The mean math marks and standard deviation for two classes are shown below. Assume that 68 percent of the marks in each class are within one standard deviation of the mean mark. mean mark (μ) standard deviation (σ) Class A 74 4 Class B 72 8 (a) In which class is the set of marks more dispersed? (b) Bert in Class A and Beth in Class B each have a mark of 82%. How many standard deviations are they from their class means? Who appears to have the better mark?
  14. 14. HOMEWORK The following numbers represent the number of cars sold by Metro Motors in one week: Monday Tuesday Wednesday Thursday Friday Saturday 4 5 8 9 7 9 1. Determine the following statistics: (a) mean (b) mode (c) median (d) range 2. Which measure of central tendency may be the least significant? Explain.
  15. 15. HOMEWORK The two sets of data show the weights of potatoes in bags. There are six bags in each set. Set #1 49 51 48 52 47 53 Set #2 40 60 45 55 35 65 The mean weight of each set of bags is 50 pounds. Which set has the greater standard deviation? How do you know? (Do not do any calculations.)
  16. 16. HOMEWORK 78 92 62 52 65 59 A class of 30 students received the following marks in a mathematics examination. Calculate 53 63 68 73 71 63 the mean, median, range, and standard deviation. 69 74 73 81 55 71 75 81 84 77 80 75 41 57 91 62 65 49

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