After studying this chapter you will be able to: define and calculate the mean, explain and calculate a weighted mean, define and calculate the median, define and identify the mode, prepare a frequency distribution, prepare bar, line, and circle graphs, calculate price relatives and cost comparisons, explain and calculate the range, define and calculate the standard deviation, estimate percentage of data by using standard deviations.
BUSINESS STATISTICS Chapter Twenty-Two McGrawHill/Irwin Copyright © 2014 by The McGrawHill Companies, Inc. All rights reserved Learning unit objectives LU22-1: Mean, Median, and Mode Define and calculate the mean Explain and calculate a weighted mean Define and calculate the median Define and identify the mode LU 22-2: Frequency Distributions and Graphs Prepare a frequency distribution Prepare bar, line, and circle graphs Calculate price relatives and cost comparisons LU 22-3: Measures of Dispersion (Optional) Explain and calculate the range Define and calculate the standard deviation Estimate percentage of data by using standard deviations 222 Terminology Mean – The average used to indicate a single value that represents an entire group of numbers Median - A measurement that indicates the center of the data (average) Mode - A measurement that records values; the value that occurs most often 223 Mean of all values Mean = Sum Number of values Example: The accountant of Bill’s Sport Shop told Bill, the owner, that the average daily sales for the week were $150.14 The accountant stressed that $150.14 was an average and did not represent specific daily sales Bill wanted to know how the accountant arrived at $150.14 Sun Mon Tues Wed $400 $100 $68 $115 Thur $120 Fri Sat $68 $180 Mean = $400 + $100 + $68 + $115 + $120 +$68 + $180 = $150.14 224 Weighted Mean Weighted mean = Sum of products Sum of frequencies Example: How Jill Rivers calculated her GPA to the nearest tenth: Courses Intro to Comp Psychology English Comp Business Law Business Math Credit Attempted 3 3 16 Grade Received A B B C B 49 = 3.1 16 Points (Credits x Grade) 16 (4 x 4) (3 x 3) (3 x 3) (2 x 3) (3 x 3) 49 225 Finding the Median of a Group of Values Look below at the following yearly salaries of the employees of Rusty’s Clothing Shop Alice Knight $95,000 Jane Hess $27,000 Joel Floyd $32,000 Jane Wang Bill Joy $67,000 $40,000 Find the median value of all employees $95,000 + $27,000 + $32,000 + $67,000 + $40,000 = $52,200 226 Finding the Median of a Group of Values Step Orderly arrange values from the smallest to the largest Step Find the middle value 95, 27, 32, 67, 40 27, 32, 40 , 67, 95 40 a Odd number of values: Median is the middle value Divide the total number of numbers by (5/2 = ½) The next-higher number is the median b Even number of values: Median is the average of the two middle values 95, 32, 67, 40 32, 40, 67, 95 40 + 67 = 53.5 227 Mode Mode is the value that occurs most often If two or more numbers appear most often, you may have two or more modes If all the values are different, there is no mode 3, 4, 5, 6, 3, 8, 9, 3, 5, 3 is the mode since it is listed times 228 Frequency Distribution Frequency Distribution A way of collecting and organizing raw data Computer costs 1000 7000 4000 1000 5000 1000 3000 5000 2000 3000 3000 3000 8000 9000 3000 6000 6000 1000 10000 1000 Price of Tally Frequency Computer $1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 llll l llll l ll ll l l l l 5 2 1 1 Frequency distribution table 229 Bar Graph Frequency of purchase 4.5 3.5 2.5 1.5 0.5 1000 3000 2000 5000 4000 7000 6000 9000 10000 8000 Price of Computers 2210 Bar Graph $1000 $3001 $5001 $8001 $9001 Frequency - $ 3,000.99 5,000.99 7,000.99 9,000.99 - 11,000.99 11 3 15 Frequency of purchase Class 10 $1,000 $5,001 $9,001 $3,000.99 $7,000.99 $11,000.99 $3,001$7,001$5,000.99 $9,000.99 2211 Circle Graph 12.9% 12.9% 17.3% 56.9% 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr Revenues 1st Qtr $20,400 2nd Qtr $27,400 3rd Qtr $90,000 4th Qtr $20,400 2212 Measure of Dispersion Measure of Dispersion – A number that describes how the numbers of a set of data are spread out or dispersed Range – The difference between the two extreme values (highest and lowest) in a group of values or a set of data Range = Highest value – Lowest value Find the range of the following values: 83.6, 77.3, 69.2, 93.1, 85.4, 71.6 Range = 93.1 – 69.2 = 23.9 2213 Index Numbers Price relative = Current price x 100 Base year’s price Example: A calculator may cost $9 today relative to a cost of $75 some 30 years ago What is the relative price? $9 $75 x 100 = 12 = 12% 2214 Consumer Price Index (in percent) 2215 Standard Deviation Step Find the mean of the set of data Step Subtract the mean from each piece of data to find each deviation Step Square each deviation (multiply the deviation by itself) Step Sum all squared deviations Step Divide the sum of the squared deviations by n - 1, where n equals the number of pieces of data Step Find the square root of the number obtained in Step This is the standard deviation Standard deviation is intended to measure the spread of data around the mean 2216 Standard Deviation Data Set A xx x x x 10 11 12 13 Step (1 + + + 10 + 12) = (Mean) Step Step Data Data Mean (Data Mean) 1- = -5 25 2 - = -4 16 5 - = -1 10 10 - = 16 12 12 - = 36 Total 94 (Step 4) Step Divide by n-1: 94 = 94 = 23.5 5-1 Step The square root of 23.5 is 4.8 The standard deviation of data set A is 4.8 2217 Standard Deviation Data Set B x xx xx 10 11 12 13 Step (4 + + + + 9) = (Mean) Step Step Data Data Mean (Data Mean) 4- = -2 4 - = -2 5 - = -1 10 - = 12 - = Total 22 (Step 4) Step Divide by n-1: 22 = 22 = 5.5 5-1 Step The square root of The standard deviation of data set B is 5.5 is 2.3 2218 ... Step Step Data Data Mean (Data Mean) 1- = -5 25 2 - = -4 16 5 - = -1 10 10 - = 16 12 12 - = 36 Total 94 (Step 4) Step Divide by n-1: 94 = 94 = 23.5 5-1 Step The square root of 23.5 is 4.8 The... (Mean) Step Step Data Data Mean (Data Mean) 4- = -2 4 - = -2 5 - = -1 10 - = 12 - = Total 22 (Step 4) Step Divide by n-1: 22 = 22 = 5.5 5-1 Step The square root of The standard deviation... calculated her GPA to the nearest tenth: Courses Intro to Comp Psychology English Comp Business Law Business Math Credit Attempted 3 3 16 Grade Received A B B C B 49 = 3.1 16 Points (Credits x Grade)