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Business Math Chapter 7: Business Statistics 1 7.1 Measures of Central Tendency Find the mean 9 8 Find the median 7 6 5 Find the mode 4 3 2 1 0 2 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Key Terms Data set: a collection of values or measurements that have a common characteristic. Statistic: a standardized, meaningful measure of a set of data that reveals a certain feature or characteristic of the data. 3 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Key Terms Mean: the arithmetic average of a set of data or sum of the values divided by the number of values. Median: the middle value of a data set when the values are arranged in order of size. Mode: the value or values that occur most frequently in a data set. 4 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 7.1.1 Find the mean Business records its daily sales. These values are an example of a data set. Data sets can be used to: Observe patterns Interpret information Make predictions about future activity 5 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Find the mean of a data set. 1. Find the sum of the values. 2. Divide the sum by the total number of values. Mean = sum of values number of values 6 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Here’s an example. Sales figures for the last week for the Western region have been as follows: Monday $4,200 Tuesday $3,980 Wednesday $2,400 Thursday $3,100 Friday $4,600 What is the average daily sales figure? $3,656 7 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Try these examples. Mileage for the new salesperson has been 243, 567, 766, 422 and 352 this week. What is the average number of miles traveled? 470 miles daily Prices from different suppliers of 500 sheets of copier paper are as follows: $3.99, $4.75, $3.75 and $4.25. What is the average price? $4.19 8 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 7.1.2 Find the median. Arrange the values in the data set from smallest to largest (or largest to smallest) and select the value in the middle. If the number of values is odd, it will be exactly in the middle. If the number of values is even, identify the two middle values. Add them together and divide by two. 9 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Here is an example. A recent survey of the used car market for the particular model John was looking for yielded several different prices. Find the median price. $9,400, $11,200, $5,900, $10,000, $4,700, $8,900, $7,800 and $9,200. Arrange from highest to lowest: $11,200, $10,000, $9,400, $9,200, $8,900, $7,800, $5,900 and $4,700. Calculate the average of the two middle values. $9050 is the median price. 10 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Try this example. Five local moving companies quoted the following prices to Bob’s Best Company: $4,900, $3800, $2,700, $4,400 and $3,300. Find the median price. $3,800 11 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 7.1.3 Find the mode. Find the mode in a data set by counting the number of times each value occurs. Identify the value or values that occur most frequently. There may be more than one mode if the same value occurs the same number of times as another value. If no one value appears more than once, there is no mode. 12 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Find the mode in this data set. Results of a placement test in mathematics included the following scores: 65, 80, 90, 85, 95, 85, 80, 70 and 80. Which score occurred the most frequently? 80 is the mode. It appeared three times. 13 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Use the mean, median and mode. A university recruiter is evaluating the number of community service hours performed by ten students who are applying for a job on campus. Find the mean, median and mode from this data set and determine which one or ones might help the recruiter the most in making a realistic assessment of the number of service hours performed last semester. (see next slide) 14 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved How many hours? Jack: Michelle: Bill: Jackie: Jason: Larissa: Tony: Melanie: Art: Sheila: 10 14 5 2 20 12 2 18 1 0 The mode is 2. The mean is 8.4. The median is 7.5 Of the three values, which one or one(s) would help you make a realistic assessment of the number of service hours? Why? 15 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 7.2 Frequency Distributions Make and interpret a frequency distribution. Interpret and draw a bar graph. Interpret and draw a line graph. Interpret and draw a circle graph. 16 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 7.2.1 Make and interpret a frequency distribution. Identify appropriate intervals for the data. Tally the data for the intervals. Count the number in each interval. 90 80 70 60 50 East West North 40 30 20 10 0 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr 17 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Key Terms Class intervals: special categories for grouping the values in a data set. Tally: a mark that is used to count data in class intervals. Class frequency: the number of tallies or values in a class interval. Grouped frequency distribution: a compilation of class intervals, tallies, and class frequencies of a data set. 18 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 7.2.2 Draw and interpret a bar graph Write an appropriate title. Make appropriate labels for bars and scale. The intervals should be equally spaced and include the smallest and largest values. Draw horizontal or vertical bars to represent the data. Bars should be of uniform width. Make additional notes as appropriate to aid interpretation. 19 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Here’s an example. Sales Volume 2001-2004 Product 3 2004 2003 2002 2001 Product 2 Product 1 0 10 20 30 40 50 Thousands of Units 20 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 7.2.3 Interpret and draw a line graph Write an appropriate title. Make and label appropriate horizontal and vertical scales, each with equally spaced intervals. Often, the horizontal scale represents time. Use points to locate data on the graph. Connect data points with line segments or a smooth curve. 21 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Here’s an example. Thousands of $ First Semester Sales 100 80 60 40 20 0 Jan Feb Judy Mar Denise Apr May Linda Jun Wally 22 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 7.2.4 Interpret and draw a circle graph. Write an appropriate title. Find the sum of values in the data set. Represent each value as a fraction or decimal part of the sum of values. For each fraction, find the number of degrees in the sector of the circle to be represented by the fraction or decimal. (100% = 360°) Label each sector of the circle as appropriate. 23 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Here’s an example. Local Daycare Market Share 6% 16% 43% Teddy Bear La La Land Little Gems Other 35% 24 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 7.3 Measures of dispersion Find the range. Find the standard deviation. From here to there... 25 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Key Terms Measures of central tendency: statistical measurements such as the mean, median or mode that indicate how data groups toward the center. Measures of variation or dispersion: statistical measurement such as the range and standard deviation that indicate how data is dispersed or spread. 26 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Key Terms Range: the difference between the highest and lowest values in a data set. (also called the spread) Deviation from the mean: the difference between a value of a data set and the mean. Standard variation: a statistical measurement that shows how data is spread above and below the mean. 27 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Key Terms Variance: a statistical measurement that is the average of the squared deviations of data from the mean. The square root of the variance is the standard deviation. Square root: the quotient of number which is the product of that number multiplied by itself. The square root of 81 is 9. (9 x 9 = 81) Normal distribution: a characteristic of many data sets that shows that data graphs into a bellshaped curve around the mean. 28 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 7.3.1 Find the range in a data set Find the highest and lowest values. Find the difference between the two. Example: The grades on the last exam were 78, 99, 87, 84, 60, 77, 80, 88, 92, and 94. The highest value is 99. The lowest value is 60. The difference or the range is 39. 29 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved 7.3.2 Find the standard deviation The deviation from the mean of a data value is the difference between the value and the mean. Get a clearer picture of the data set by examining how much each data point differs or deviates from the mean. 30 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Deviations from the mean When the value is smaller than the mean, the difference is represented by a negative number indicating it is below or less than the mean. Conversely, if the value is greater than the mean, the difference is represented by a positive number indicating it is above or greater than the mean. 31 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Find the deviation from the mean. Find the mean of a set of data. Mean = Sum of data values Number of values Find the amount that each data value deviates or is different from the mean. Deviation from the mean = Data value - Mean 32 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Here’s an example. Data set: 38, 43, 45, 44 Mean = 42.5 First value: 38 – 42.5 = Second value: 43 – 42.5 = 0.5 above the mean Third value: 45 – 42.5 = 2.5 above the mean Fourth value: 44 – 42.5 = 1.5 above the mean -4.5 below the mean 33 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Interpret the information One value is below the mean and its deviation is -4.5. Three values are above the mean and the sum of those deviations is 4.5. The sum of all deviations from the mean is zero. This is true of all data sets. We have not gained any statistical insight or new information by analyzing the sum of the deviations from the mean. 34 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Average deviation Average deviation = Sum of deviations Number of values =0 =0 n 35 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Find the standard deviation of a set of data. A statistical measure called the standard deviation uses the square of each deviation from the mean. The square of a negative value is always positive. The squared deviations are averaged (mean) and the result is called the variance. 36 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Find the standard deviation of a set of data. The square root is taken of the variance so that the result can be interpreted within the context of the problem. This formula averages the values by dividing by one less than the number of values (n-1). Several calculations are necessary and are best organized in a table. 37 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved Find the standard deviation of a set of data. 1. Find the mean. 2. Find the deviation of each value from the mean. 3. Square each deviation. 4. Find the sum of the squared deviations. 5. Divide the sum of the squared deviations by one less than the number of values in the data set. This amount is called the variance. 6. Find the standard deviation by taking the square root of the variance. 38 Cleaves/Hobbs: Business Math, 7e Copyright 2005 by Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

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