Topic2 basic statistics_student

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Topic2 basic statistics_student

  1. 1. Measures of Central Tendency Quantitative Skills Find the mean Topic 2 Find the median 9 8 Find the mode 7 Business Statistics 6 5 Make and interpret a 4 3 frequency distribution 2 1 Find the mean of 0 grouped data Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Key Terms Key Terms Mean: the arithmetic average of a set of data or Data set: a collection of values or measurements that sum of the values divided by the number of values. have a common characteristic. Median: the middle value of a data set when the values are arranged in order of size. Statistic: a standardized, meaningful measure of a set of data that reveals a certain feature or Mode: the value or values that occur most frequently characteristic of the data. in a data set.Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJCleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Find the Mean Find the mean of a data set A business records its daily sales. These values are 1. Find the sum of the values. an example of a data set. 2. Divide the sum by the total number of values. Data sets can be used to: Observe patterns sum of values Mean Interpret information number of values Make predictions about future activityBusiness Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJCleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved
  2. 2. Find the mean of a data set Find the mean of a data setExample: Exercise 1:Sales figures for the last week for the Western Region have Mileage (in km) for the new salesperson has been 243, 567, been as follows: 766, 422 and 352 this week. What is the average number Monday $4,200 of km traveled? Tuesday $3,980 Wednesday $2,400 Average = Thursday $3,100 Friday $4,600What is the average daily sales figure? Average =(4,200 + 3,980 + 2,400 + 3,100 + 4,600) ÷ 5 = $3,656 Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Find the mean of a data set Find the MedianExercise 2: Arrange the values in the data set from smallest toPrices from different suppliers of 500 sheets of copier paper largest (or largest to smallest) and select the value in are as follows: $3.99, $4.75, $3.75 and $4.25. What is the middle. the average price? If the number of values is odd, it will be exactly in the middle. Average = e.g. 4 6 10 25 30 If the number of values is even, identify the two middle values. Add them together and divide by two. e.g. 1 4 6 10 25 30 Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Find the Median Find the MedianExample: Exercise 1:A recent survey of the used car market for the particular Five local moving companies quoted the following prices model John was looking for yielded several different prices: to Bob’s Best Company: $4,900, $3800, $2,700, $9,400, $11,200, $5,900, $10,000, $4,700, $8,900, $4,400 and $3,300. Find the median price. $7,800 and $9,200. Find the median price. Arrange from lowest to highest : Arrange from lowest to highest : 4,700, 5,900, 7,800, 8,900, 9,200, 9,400, 10,000, 11,200 Identify the middle value. Calculate the average of the two middle values. Median = (9,200 + 8,900) ÷ 2 = $9,050 (the median price) Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved
  3. 3. Find the Median Find the ModeExercise 2: Find the mode in a data set by counting the number ofEight local moving companies quoted the following prices times each value occurs. to Bob’s Best Company: $4,900, $3800, $2,700, Identify the value or values that occur most frequently. $4,400, $1,500, $1,500, $5,500 and $3,300. Find the median price. There may be more than one mode if the same value occurs the same number of times as another value. Arrange from lowest to highest : e.g. 2 3 6 3 2 If no one value appears more than once, there is no Calculate the average of the two middle values. mode. Median = e.g. 2 3 6 1 9 Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Find the mode in this data set Find the ModeExample: Exercise :Results of a placement test in mathematics included the Eight local moving companies quoted the following prices to following scores. Which score occurred the most Bob’s Best Company: $4,900, $3800, $2,700, $4,400, $1,500, $1,500, $5,500 and $3,300. Find the mode. frequently? The mode is as it appears . 65, 80, 90, 85, 95, 85, 80, 70 and 80. 80 is the mode (it appears three times). Other numbers appear only once. Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Use the mean, median and mode How many hours?Exercise: The mean is ¨Jack: 10 ¨Michelle: 14 The median is A university recruiter is evaluating the number of ¨Bill: 5 The mode is community service hours performed by ten students who ¨Jackie: 2 Of the 3 values, which one(s) would help are applying for a job on campus. you make a realistic assessment of the ¨Jason: 20 Observe the mean, median and mode from this data number of service hours? ¨Larissa: 12 set and determine which one or ones might help the ¨Tony: 2 recruiter the most in making a realistic assessment of the Why? number of service hours performed last semester. ¨Melanie: 18 ¨Art: 1 (see next slide) ¨Sheila: 0 Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved
  4. 4. Make and interpret a frequency distribution Make and interpret a frequency distribution Example: Test scores on the last math test were as follows: Class Class How many students get 90 and above? 78 84 95 88 99 92 87 94 90 77 Interval Frequency Make a relative frequency distribution using intervals of 75 - 79 2 75-79, 80-84, 85-89, 90-94, and 95-99. 80 - 84 1 85 - 89 2 What is the percentage of students 90 - 94 3 getting below 80? 95 - 99 2 Total 10 Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved How to Find the Mean of Grouped Data How to Find the Mean of Grouped Data Class Class Product of Make a frequency distribution. Midpoint Interval Frequency Midpoint & Freq Find the products of the midpoint of the interval and the 75 - 79 2 frequency for each interval for all intervals. 80 - 84 1 Find the sum of the frequencies. 85 - 89 2 Find the sum of the products from step 2. 90 - 94 3 95 - 99 2 Divide the sum of the products by the sum of the Total 10 frequencies. Mean of the grouped data: ÷ = Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved How to Find the Mean of Grouped Data How to Find the Mean of Grouped DataExercise: Solution:Find the grouped mean to the nearest whole number of the data in the frequency distribution. Round to hundredths. Class Class Product of Midpoint Interval Frequency Midpoint & Freq Class 0 - 4 3 Credit Hour Frequency 5 - 9 7 0-4 3 10 - 14 4 5-9 7 15 - 19 2 10 - 14 4 Total 16 15 - 19 2 Total 16 Mean of the grouped data: Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved
  5. 5. Interpret and draw a bar graph Interpret and draw a bar graph Maths Test Scores Example: Exercise: 4 Which grade interval has the Draw a bar graph for the table below: highest number of scores? 3 ClassFrequency Credit Hour 2 Frequency How many students get 95- 0-4 3 1 99? 5-9 7 10 - 14 4 0 15 - 19 2 75 - 79 80 - 84 85 - 89 90 - 94 95 - 99 Total 16 Scores Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Interpret and draw a bar graph Interpret and draw a line graph Exercise: Draw a line graph for the table below: Month Personal Income In 2003 (Billions of dollars) Mar 9102 Apr 9119.8 May 9155.4 Jun 9192.9 Jul 9219.4 Aug 9248.9 Sep 9277.5 Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Interpret and draw a line graph Interpret and draw a line graph Personal Income for Year 2003 (Billions of dollars) Questions: 9300 Is the graph increasing, decreasing or fluctuating? Income (Billions of dollars) 9250 9200 9150 Which month showed the highest personal income? 9100 9050 9000 Mar Apr May Jun Jul Aug Sep Month Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved
  6. 6. Interpret and draw a circle graph Interpret and draw a circle graphExercise: Personal Income for Year 2003 (Billions of dollars)Construct a circle graph showing the distribution of market share for the data shown below: Marvel Comics 20% Market Share (%) Publisher 35% DC Comics (Comic & Magazine Sales for Sep) Marvel Comics 35% 4% Image Comics DC Comics 32% 4% Image Comics 5% 5% Dark Horse Comics Dark Horse Comics 4% Dreamweave ProductionsDreamweave Productions 4% Others 20% 32% Others Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Interpret and draw a circle graph Interpret and draw a circle graphQuestions: Questions: What percent of market share is held by the largest If the total market had $80,000.000 in comics and three companies? magazine sales for September, what were the sales for Marvel Comics? What was Image Comic’s sales for September if the total market was $80,000,000? Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights ReservedMeasures of Dispersion Key Terms Measures of central tendency: statistical Find the range. measurements such as the mean, median or mode that indicate how data groups toward the center. Find the standard deviation. Measures of variation or dispersion: statistical measurement such as the range and standard deviation that indicate how data is dispersed or spread. Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved
  7. 7. Key Terms Key Terms Range: the difference between the highest and Variance: a statistical measurement that is the average lowest values in a data set. (also called the spread) of the squared deviations of data from the mean. The square root of the variance is the standard deviation. Deviation from the mean: the difference between a value of a data set and the mean. Square root: the quotient of number which is the product of that number multiplied by itself. The square root of Standard variation: a statistical measurement that 81 is 9. (9 x 9 = 81) shows how data is spread above and below the mean. Normal distribution: a characteristic of many data sets that shows that data graphs into a bell-shaped curve around the mean.Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJCleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Find the Range in a Data Set Find the Standard Deviation Find the highest and lowest values. Find the difference between the two. The deviation from the mean of a data value is the Example: difference between the value and the mean. The grades on the last exam were: Get a clearer picture of the data set by examining how much each data point differs or deviates from 78, 99, 87, 84, 60, 77, 80, 88, 92, 94 the mean. The highest value is 99. The lowest value is 60. The difference or the range is 39.Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJCleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights ReservedDeviations from the mean Find the deviation from the mean Find the mean of a set of data.When the value is smaller than the mean, the sum of valuesdifference is represented by a negative number Meanindicating it is below or less than the mean. number of values Find the amount that each data value deviates or isConversely, if the value is greater than the mean, the different from the mean.difference is represented by a positive numberindicating it is above or greater than the mean. Deviation from the mean = Data value - MeanBusiness Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJCleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved
  8. 8. Find the deviation from the mean Average deviationExample: One value is below the mean and its deviation is -4.5. Data set: 38, 43, 45, 44 Three values are above the mean and the sum of those deviations is 4.5. Mean = 42.5 The sum of all deviations from the mean is zero. This is true 1st value: 38 – 42.5 = -4.5 below the mean of all data sets. Sum of deviations 4.5 0.5 2.5 1.5 2nd value:43 – 42.5 = 0.5 above the mean Average deviation Number of values 4 0 3rd value: 45 – 42.5 = 2.5 above the mean We have not gained any statistical insight or new 4th value: 44 – 42.5 = 1.5 above the mean information by analyzing the sum of the deviations from the mean. Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Find the standard deviation Find the standard deviation A statistical measure called the standard deviation The square root is taken of the variance so that the uses the square of each deviation from the mean. result can be interpreted within the context of the problem. The square of a negative value is always positive. This formula averages the values by dividing by one The squared deviations are averaged (mean) and less than the number of values (n-1). the result is called the variance. Several calculations are necessary and are best organized in a table. Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Find the standard deviation Find the standard deviation Example:1. Find the mean. Find the standard deviation for 45, 63, 87, 91.2. Find the deviation of each value from the mean. Deviation Squares of3. Square each deviation. Value Mean from Mean Deviation4. Find the sum of the squared deviations. 45 71.5 45 - 72 = -27 -27 x -27 = 702.35. Divide the sum of the squared deviations by one 63 71.5 63 - 72 = -9 -9 x -9 = 72.25 less than the number of values in the data set (n-1). 87 71.5 87 - 72 = 16 16 x 16 = 240.3 This amount is called the variance. 91 71.5 91 - 72 = 20 20 x 20 = 380.3 Total 286 Sum = 13956. Find the standard deviation by taking the square root of the variance. Mean = 71.5 Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Cleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved
  9. 9. Find the standard deviation Find the standard deviation Exercise: Sum of squared deviationsVariance Find the standard deviation for 18, 22, 29, 27. n 1 1,395 Deviation Squares of Value Mean from Mean Deviation 3 18 24 465 22 24 29 24Standard deviation Variance 27 24 265 Total 96 Sum = Mean = 24 21.56Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJCleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved Find the standard deviation Time for More Exercises Sum of squared deviationsVariance n 1Standard deviation VarianceBusiness Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJ Business Math, Eighth Edition © 2009 Pearson Education, Inc. Upper Saddle River, NJCleaves/Hobbs 07458 All Rights Reserved Cleaves/Hobbs 07458 All Rights Reserved

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