ࡱ> CEBlg bjbjVV 7rr<r<+|HHH\\\8t\Uj"yyyTTTTTTT$V8YZTHyW"yyyT9Tyj8HTyTK0Ow)WN-M,TT0UYMlY1YXOOnYH3STyyyyyyyTTyyyUyyyyYyyyyyyyyy : Class notes Chapter 3: STATISTICS FOR DESCRIBING, EXPLORING, AND COMPARING DATA Notation: ( denotes the sum of a set of values. x is the variable usually used to represent the individual data values. n represents the number of values in a sample (sample size). N represents the number of values in a population (population size). Section 3.2: Measures of Center Definitions Measure of Center the value at the center or middle of a data set Mean Arithmetic average of all data values  EMBED Equation.3  is pronounced x-bar and denotes the mean of a set of sample values is pronounced mu and denotes the mean of all values in a population  EMBED Equation.3   EMBED Equation.3  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude denoted by  EMBED Equation.3  (pronounced x-tilde) is not affected by an extreme value Mode the value that occurs most frequently EXAMPLE: You are given 3 data sets: SAMPLE 1SAMPLE 2SAMPLE 31121221441661626 Find the Mean ( EMBED Equation.3 ) Median ( EMBED Equation.3 ) mode midrange EXAMPLE: Find a median of the following data set: 2, 2, 10, 5 ,6, 7, 8, 5 #17 p.89. Waiting times (in minutes) of customers at the Jefferson Valley Bank (where all customers enter a single waiting line) and the Bank of Providence (where customers wait in individual lines at three different teller windows) are listed below. Find the mean and median for each of the two samples, then compare the two sets of results. Jefferson Valley (single line): 6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7 Providence (individual lines): 4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0 Set 1 Set 2 Mean ( EMBED Equation.3 ) Median ( EMBED Equation.3 ) Determine whether there is a difference between the two data sets that is not apparent from a comparison of the measures of center. If so, what is it? EXAMPLE: The given frequency distribution summarizes a sample of Mesa College students heights. Find the mean of the sample using your calculator. How does the mean compare to the value of 68, which is the value assumed to be the mean by most people? MidpointHeightsFrequencies60 - 62363 - 65366 - 68569 - 71272 - 741 Section 3.3: Measures of Variation A dietician obtains the amounts of sugar (in centigrams) from 1 gram in each of 6 different cereals, including Cheerios, Corn Flakes, Fruit Loops, and 3 others (Note: this is not a real data, used as an example for teaching purposes only.) Those values are listed below. 1 2 10 12 17 18 Find the Range Variance ( EMBED Equation.3  2) Standard deviation ( EMBED Equation.3 ) Is the standard deviation of those values likely to be a good estimate of the standard deviation of the amounts of sugar in each gram of cereal consumed by the population of all Americans who eat cereal? Why or why not? #17 p.107. Waiting times (in minutes) of customers at the Jefferson Valley Bank (where all customers enter a single waiting line) and the Bank of Providence (where customers wait in individual lines at three different teller windows) are listed below. Find the range, variance, and standard deviation for each of two samples, then compare the two sets of results Jefferson Valley (single line): 6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7 Providence (individual lines): 4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0 Set 1 Set 2 Range St. Deviation ( EMBED Equation.3 ) Variance( EMBED Equation.3 ) Coefficient of Variation Standard deviations of the data sets that use different scale and units and/or have significantly different means should not be compared. In such a case we use Coefficient of Variation (CV)  EMBED Equation.3  or  EMBED Equation.3  Example: A sample of 100 guppy fish had a mean lengths of 22 mm with a standard deviation of 8 mm. A sample of 100 humpback whales had an average length of 42.5 ft. with a standard deviation of 3.8 ft. Which sample was more variable? Empirical Rule ( 68-95-99.7) For data sets having a distribution that is approximately bell shaped, the following properties apply: About 68% of all values fall within 1 standard deviation of the mean. About 95% of all values fall within 2 standard deviations of the mean. About 99.7% of all values fall within 3 standard deviations of the mean. Estimation of Standard Deviation: Range Rule of Thumb For interpreting a known value of the standard deviation s, find rough estimates of the minimum and maximum usual sample values by using:  #32 p.109. Aluminum cans with a thickness of 0.0111 in. have axial loads with a mean of 281.8 lb and a standard deviation of 27.8 lb. The axial load is measured by applying pressure to the top of the can until it collapses, Use the range rule of thumb to find the minimum and maximum usual axial loads. One particular can had an axial load of 504lb. Is that unusual? Section 3.4: Measures of Relative Standing #14 p.118. Three students take equivalent stress tests. Which is the highest relative score? 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