Measures of Central Tendency and Dispersion :Measures of Central Tendency and Dispersion Standard Deviation Geometric Mean Range


Components of this Module :Components of this Module Part A: Interpreting Curves for Dispersion Part B: Mean, Weighted Average, Median, Mode, Range, Geometric Mean, Standard Deviation, Variance, Coefficient of Variation


Real Life Scenario :Real Life Scenario A consumer electronics company has four salesmen. The manager of the company has the sales achieved by the four salesmen over the last six months. The manager would like to determine, based on this data, as to who is the most consistent salesmen amongst the four. At the end of this module, you should be in a position to help him out.


Dispersion :Dispersion Dispersion is the spread of data in a data set.


Example: Scores in a Tough Test :Example: Scores in a Tough Test The following are the scores (out of 100) of 20 students in a very tough test. Let’s draw a frequency distribution (having class intervals of width 20), observe the curve, and discuss its shape. 35, 38, 42, 45, 47, 49, 50, 50, 52, 53, 85, 28, 22, 44, 55, 50, 78, 52, 42, 58


Example: Scores in an Easy Test :Example: Scores in an Easy Test The following are the scores (out of 100) of 20 students in a very easy test. Let’s draw a frequency distribution (having class intervals of width 20), observe the curve, and discuss its shape. 85, 88, 92, 75, 47, 79, 70, 98, 52, 53, 71, 68, 22, 77, 68, 50, 93, 80, 42, 81


Example: Scores in a Test of Moderate Difficulty :Example: Scores in a Test of Moderate Difficulty The following are the scores of 20 students in a test of moderate difficulty. Let’s plot a frequency distribution for this and discuss the shape. 25, 45, 55, 60, 65, 70, 75, 85, 95, 55, 63, 44, 35, 55, 61, 52, 47, 81, 50, 49.


Exercise :Exercise Draw three curves, all symmetrical, but with different dispersions.


Check Your Understanding :Check Your Understanding Draw curves (distributions) for the following scenarios: Grades of students who are poorly prepared for a test; Grades of students who are well-prepared for a test; The ages of passengers in the Brindavan Express. Ages of attendees at a rock concert. Monthly incomes of passengers in the business class of a Bombay – New York flight.


Part B :Part B THE AVERAGE (OR MEAN)


The Arithmetic Mean (Average) :The Arithmetic Mean (Average) The most commonly used measure of central tendency. It represents the “typical” value in a data set. Excel formula for mean: average()


Exercise :Exercise The following are the ages of eight employees of the Math department. Find the average age of the department. 53, 32, 61, 27, 39, 44, 49, 57 Ans: 45.25


Exercise :Exercise A child care centre is eligible for social service grant from the government as long as the average age of the children stays below 9. If the data below represents the ages of all children currently at the centre, does the centre qualify for the grant? 8, 5, 9, 10, 9, 12, 7, 12, 13, 7, 8.


Exercise :Exercise A cosmetics manufacturer recently purchased a machine to fill 3-ounce cologne bottles. To test the accuracy of the machine’s volume setting, 18 trial bottles were run. The resulting volumes (in ounces) for the trials were as follows: 3.02, 2.89, 2.92, 2.84, 2.90, 2.97, 2.95, 2.94, 2.93, 3.01, 2.97, 2.95, 2.90, 2.94, 2.96, 2.99, 2.99, 2.97. The company does not normally recalibrate the filling machine for this cologne if the average volume is within 0.04 of 3.00 ounces. Should it re-calibrate based on the trial runs?


Part B :Part B THE WEIGHTED AVERAGE


Concept of Weighted Average :Concept of Weighted Average The weighted average enables us to calculate an average that takes into account the importance of each value to the overall total. Example: Weightage given to internals and final exam.


Formula for Weighted Average :Formula for Weighted Average Weighted Average = Σ (weight x value) _______________ Σ weights


Exercise :Exercise A management consulting firm has four types of professionals on its staff: managing consultant, senior associates, field staff, and office staff. Average rates charged to consulting clients for the work of each of these professional categories are $75/hour, $40/hour, $30/hour, and $15/hour. Office records indicate the following number of hours billed last year in each category: 8,000, 14,000, 24,000 and 35,000. What was the average billing rate / hour charged by the consultancy firm last year? [Ans: $29.69 / hour]


Tip :Tip How do you decide what the denominator should be when you calculate the weighted average?


Exercise :Exercise The Economics course at an MBA college is a 36-hour course with one test and one final exam. The test has a weightage of 30% and the final exam a weightage of 70%. A student gets 45 out of 50 in the test, and 70 out of 100 in the final exam. What is his average percentage score? [Ans: 76%]


Exercise :Exercise A sample of 500 subscribers of Fortune magazine were asked: “How many of the last four issues have you read or looked through?”. The following frequency distribution resulted. What is the average number of issues read by a Fortune subscriber? [Ans: 3.49]


Part B :Part B THE GEOMETRIC MEAN (GM)


Concept of GM :Concept of GM The GM is used to compute the average rate of increase (or decrease) when the rate of change itself is changing. Example: The average growth in money when the interest rate each year changes.


Example :Example The geometric mean of three number 4, 16 and 28 is third root of (4 x 16 x 28) Likewise, the geometric mean of 4 numbers a, b, c and d is the fourth root of (a x b x c x d)


Excel Function for GM :Excel Function for GM geomean()


Illustration of GM :Illustration of GM Rs. 100 deposited at start of first year


Slide 27:The GM of the Growth Factors is geomean(1.07, 1.08, 1.1, 1.12, 1.18) = 1.1093 = an average growth of 10.93% Key Point: When Rs. 100 increases at a steady rate of 10.93% each year, the amount after 5 years will be Rs. 168. Let’s verify this.


Exercise :Exercise The growth in bad-debt expense for an organization over the last few years follows. Calculate the average percentage increase in bad-debt expense over the indicated time period. [Ans: 9.5%]


Exercise :Exercise A manufacturer of electrical circuit boards has manufactured the following number of units over the last 5 years. Calculate the average rate of increase in production (in %) over the years. Assuming the same growth rate holds, use this figure to estimate the production for 2009. [Ans: 8.977%, 19,213 units]


Decreasing Trend :Decreasing Trend Given here are the prices of an item that costs Rs. 100 but whose price has been declining each year. Find the average decrease in price (in %) over the period shown.


Part B :Part B THE MEDIAN


What is the Median :What is the Median The median represents the “middle” number in a set of numbers.


Excel Formula :Excel Formula The Excel formula for median is median()


Example :Example Given below are the annual salaries (in Rs. Lakh) for seven employees. Find their mean salary and median salary. What do you observe? 2, 3, 6, 8, 12, 40, 75


Use of Median :Use of Median The median is used as a measure of central tendency when the data set has values which vary a lot in magnitude. Example: Magazines report the median salaries of CEOs


Test Your Understanding :Test Your Understanding Indicate whether you would report the mean or the median for the following variables: The typical price of tomatoes in Bangalore The typical price of a 1000 sq. ft. house in Bangalore The typical waiting time at a railway reservation counter


Test Your Understanding (contd…) :Test Your Understanding (contd…) The typical salary of a car driver The typical salary of a CEO in the world The typical price of a digital wrist watch in the world


Exercise :Exercise Please open the file titled “Beer”. It contains the media expenditure in ($ millions) for 10 major brands of beer (Source: Superbrands, 20 October 1997 issue). For the media expenditure, should you report the average expenditure or the median expenditure?


Revival of Real Estate? :Revival of Real Estate? Given below (on next slide) are some large property deals in Chennai in 2009 (Source: Times of India, Chennai, 13 July 2009). If you were a journalist reporting this story based on this data, would you report the mean value of the real estate area transacted or the median value of the real estate area?


Fundamentals :Fundamentals The mean is affected by extreme values whereas the median is not.


Part B :Part B THE MODE


Think About It :Think About It Why was the Maruti 800 designed as a 4-seater car? Why do newspaper vendors stock maximum copies of the Times of India in Bangalore?


Concept of Mode :Concept of Mode The mode is the most frequently occurring value in a data set. The modal value of the data set below is 2. 2, 4, 6, 2, 2, 25, 35, 42, 2


Excel Formula :Excel Formula mode( )


Exercise :Exercise A librarian polled a random 10 students as they left the library, and recorded the number of books they checked out. The data is shown below. What is the modal number of books checked out by a student? 1, 2, 2, 2, 3, 1, 3, 2, 2, 1


Fundamentals! :Fundamentals! A data set can have only one mean and one median, but it can have multiple modes.


Remember! :Remember! The median value divides a data set into two halves; one half of the values are less than the median and one half of the values are more than the median value.


Part B :Part B THE RANGE


Concept of Range :Concept of Range The range of a data set is the maximum value minus the minimum value. Example: The range of 2, 3 and 4 is 4 – 2 = 2


Excel Formula :Excel Formula Max() – min()


Exercise :Exercise The Hawaii Visitors Bureau collects data on the number of visitors to the islands. The following data are a representative sample of visitors from mainland USA, Canada and Europe (in thousands) for several days in November. Find the range for this data. 108.7, 112.25, 94.01, 144.03, 162.44, 161.61, 76.20, 102.11, 110.87, 79.36, 129.04, 95.16, 114.16, 121.88.


Limitations of Range :Limitations of Range The range is decided by only two values in a data set. Hence it is sparingly used.


Percentile :Percentile The percentile divides a data set into 100 equal parts. Example: Competitive examinations announce percentile scores. Why?


Part B :Part B STANDARD DEVIATION (SD)


Concept of SD :Concept of SD The standard deviation of a data set is an indication of how closely the data points are clustered about the mean. Greater the SD of a data set, greater is the dispersion or spread in the data set.


Key Features of SD :Key Features of SD A data set that has all values the same, will have zero SD. SD can never be negative. The SD represents the “average” distance of a data point from the mean of the data set.


Excel Formula :Excel Formula The Excel formula for computing SD is stdev()


Example :Example Given below are the number of hours of electricity supply in my house in Bangalore and my house in Chennai, on 6 randomly selected days. Find the standard deviation of the two data sets. What is your conclusion BLR: 22, 24, 21, 22, 23.5, 22.5 CHN: 24, 24, 24, 24, 23.95, 24


Exercise :Exercise Given below are the time taken (in mts.) by two tellers at a bank to process 8 random customers. Who is the more predictable (consistent) teller Anup or Gauri? Anup: 2, 1.5, 1.75, 1.8, 2.2, 3, 1, 2.8, 2 Gauri: 2, 3, 2.75, 3.25, 1, 3, 2.8, 2.95,


Exercise :Exercise Students’ ages in the regular daytime MBA programme and the evening programmes in a popular Management institute are shown below as two samples. If homogeneity of the class (as regards age) is a positive factor in learning, using SD, determine which class will be easier to teach. Regular MBA: 23, 25, 27, 22, 24, 21, 25, 20, 24, 25 Evening MBA: 27, 29, 34, 31, 35, 28, 32, 26, 25, 35


Exercise :Exercise There are a number of possible measures sales performance, including how consistent a salesperson is in meeting established sales goals. The data that follow represent the percentage of goal met by each of three sales people over the last 5 years. Which salesperson is the most consistent and who is the least consistent? Ashok: 88, 68, 89, 92, 103 Priya: 76, 88, 90, 86, 79 Gautam: 104, 88, 118, 88, 123.


Practical Tip :Practical Tip Always examine the mean and the SD of a data set before coming to any conclusion.


Exercise :Exercise You are employed as a statistician for a company that sells electronic equipment by salespersons. The company has four salespersons (A, B, C, and D) employed in a small town. The sales records (in thousands of rupees) for the past six months for these four salespersons is shown on the next slide. The MD of the company wants to reward the salesperson who is most consistent and also meets/surpasses the average 6-monthly target of Rs. 1,80,000. Who is the star salesperson?


Data… :Data…


Part B :Part B COEFFICIENT OF VARIATION (CV)


Concept of CV :Concept of CV Note: Explain with suitable example


Formula for CV :Formula for CV CV = SD ------ Mean CV is also called relative variation.


Fundamentals! :Fundamentals! Compare the SDs of two data sets directly only if their means are similar. Else, use the CV as a measure of relative variation.


Variance :Variance Please remember that variance is the square of SD.


Exercise :Exercise A manufacturing company is considering employing one of two training programmes. Two groups were trained for the same task. Group 1 was trained by Program A; Group 2 by Program B. For Group 1 the time required to train the employees had an average of 32.11 hours and a variance of 68.09. For Group 2, the average was 19.75 hours and the variance was 71.14. Which training programme has less relative variation (i.e. CV)?


Exercise :Exercise Paramount is a wholesaler selling auto components, and is contemplating becoming the supplier to three retailers. But of late, inventory shortages have forced the wholesaler to select only one retailer. The credit manager of Paramount is evaluating the credit record of these three retailers. Over the past 5 years, these retailers’ accounts receivable have been outstanding for the following average number of days. The credit manager feels that consistency, in addition to lowest average, is important. Based on this logic, which retailer would make the best customer?


Putting It All Together :Putting It All Together CASE STUDY - B-School Education in the Asia Pacific Ref.: The file titled “Asian”