Statistics and Probability in Business - Dissertation Example

Application of Statistics and Probability in Business Executive Summary: Statistics and probability provides many useful tools for extracting useful information from myriad set of data and thus helps in making informed business decisions. These tools have been applied for a wide variety of data set pertaining to Handy Hydraulic Industries pertaining to different aspects of business like number of orders, average order size, total sales, sales across different profit centers, operation of frequently faltering copiers, numbers of calls per hour and active customers’ account size to name a few. Relevant tools from statistics and probability were applied to extract useful information from a vast data set, to identify trends and to take business decisions. Analysis of average order numbers and order size has provided the information that there is consistent increase in the number of daily orders and average daily order size over consequent quarters. Besides, there is wide variation in the performance of different profit centers and therefore, there is a need to strengthen the operation and revenue of center 2 and 3 so that these centers can stand on their own and do not become burden on center 1 in days to come. Analysis of the cost of different options for the frequently faltering copiers have helped in taking the decision that it is better to acquire a new copier which is much more reliable even though it service cost per call is much higher. Analysis of number of calls per hour has helped in deciding the number of sales representatives Stan needs to ensure that no representative has to handle more than eight calls per hour. Further analysis of the active customers’ account size has helped in better understanding of the same. Introduction: Handy Hydraulic Industries is doing well after many changes in the corporate and operating structure. However, the managers know that one needs to excel to sustain in the current environment of cut-throat competition. For excellence a business needs to evolve and execute a multipronged strategy targeting different aspects of performance and customer relationship. The aim is to become leaner on operating size to cut cost without compromising the expectations of customers. For this the managers of business need to take informed decisions and the information has to be derived from nowhere but from the data relevant to the business. It is here that statistics and probability comes to help them. Statistics is the branch of knowledge that processes data to discern hidden information, reveal patterns and extract executable results [1]. One can get lot of information by simply arranging data in frequency and relative frequency tables [2]. One can get a good idea of central tendency and dispersion of the data set. The organized dataset can also be represented pictorially as histograms, line charts etc. It is said that a picture is worth thousand words. This is because; even a person with common sense can see the patterns and trends in the data if the same is presented pictorially using suitable chart. Besides, statistics provides different parameters like mean, median, mode etc. to measure central tendency of data [3, 4]. But knowing merely central tendency of data is not good enough to describe a data set; one needs to quantify the dispersion of the data points around the central value [5]. This is done by means of different parameters that quantify dispersion of data points around the mean like variance, standard deviation, coefficient of variation etc. After describing a data set comparing different data set is also important. For this one needs to compare the values of mean and standard deviation etc. Using these values one can calculate value of a suitable statistics and then this value is compared with the critical value of statistics for a given significance level. This exercise helps in testing hypothesis on mean, variance etc. and hence in making decisions [6]. The concept of probability is very much the part and parcel of statistics. It relies on the assumption that a trend which has been established by a large number of observed data points can be extrapolated. For example, it has been our experience that a unbiased coin if tossed many times is likely to give equal number of head and tail so we say the probability of getting a head and a tail is ½ each, when an unbiased coin is tossed. Probability becomes very handy in calculating expected expenditure, calculating expected profit etc. as we seen subsequently in this report on analysis of problems associated with frequently faltering copiers of Handy Hydraulic Industries. This report presents solution of some of the cases of Handy Hydraulic Industries based on the analysis of data of this company. Solution of the Case Problems: Q1. (a) Frequency distribution and Relative Frequency Distribution of H H Industries average daily order size is presented in Table 1. This appears to be a unimodal distribution in with mode in $100 - $140 range. It can be seen that relative frequency of large size orders has come down while that of relatively small size orders have gone up in Q2 over Q1. This implies that customers are reducing their order size and increasing the order frequency instead. They might be doing so to reduce inventory cost. Table 1: Frequency Distribution and Relative Frequency Distribution of company’s daily average order size in Q1 and Q2 of 1991. Interval Frequency Relative Frequency Frequency Relative Frequency 0-20 0 0 0 0 20-40 0 0 0 0 40-60 0 0 0 0 60-80 2 0.032258 1 0.015625 80-100 6 0.096774 10 0.15625 100-120 19 0.306452 23 0.359375 120-140 18 0.290323 20 0.3125 140-160 10 0.16129 5 0.078125 160-180 5 0.080645 3 0.046875 180-200 1 0.016129 2 0.03125 200-220 1 0.016129 0 0 = 62  = 64 The data presented in Table 1 is pictorially represented in Figure 1 as histogram. The data of Q1 and Q2 both are presented on the same histogram for the ease of comparison. It can be seen that frequency of orders of different order sizes is more in Q2 than the same in Q1. This means performance of the company has improved in Q2 over Q1. Q1. (b) Table 2 shows frequency distribution and relative frequency distribution of number of daily orders of received by the company. This is also a unimodal distribution with mode occurring around 180 orders per day. Besides, number of orders per day is much more in Q2 than that in Q1. This also confirms improved performance of the company in Q2 over the same in Q1. Table 2: Table 1: Frequency Distribution and Relative Frequency Distribution of number of daily orders size in Q1 and Q2 of 1991. Interval Frequency Relative Frequency Frequency Relative Frequency 100-110 1 0.016129 1 0.015625 110-120 1 0.016129 0 0 120-130 1 0.016129 0 0 130-140 2 0.032258 0 0 140-150 3 0.048387 1 0.015625 150-160 9 0.145161 7 0.109375 160-170 11 0.177419 8 0.125 170-180 14 0.225806 17 0.265625 180-190 8 0.129032 14 0.21875 190-200 4 0.064516 9 0.140625 200-210 6 0.096774 5 0.078125 210-220 1 0.016129 1 0.015625 220-230 1 0.016129 1 0.015625  = 62  = 64 The data presented in Table 2 is pictorially represented in Figure 2 as histogram. The data of Q1 and Q2 both are presented on the same histogram for the ease of comparison. It can be seen that frequency of number of orders is more in Q2 than the same in Q1. This means performance of the company has improved in Q2 over Q1. Q1. (c) Combining the information in Fig. 1 and Fig. 2 it can be seen that both the number of orders per day as well as the size of the orders has increased in Q2 over the same in Q1. This means total sales, which is product of these two quantities has also increased in Q2 over Q1. In other words performance of the company has increased in Q2 over Q1. Q2. (a) Mean, median and mode of the number of daily orders and average order size for different quarters is presented in the following table. It can be seen in this table that the number of orders has consistently increased over different quarters while the opposite trend was witnessed for the order size. This clearly points towards the changing pattern of procurement from the customers. It could be that they are preferring to go for smaller order at increased frequency as that cuts inventory cost. Another reason can be that number of customers is increasing and they are buying in smaller lots as trail. Another possible scenario that the large firms using these heavy equipments are outsourcing the maintenance job to small workshops instead of doing it on their own to cut cost is also very much possible. All these reasons suggested by Ms. Laura are supported by these findings. However, seasonality cannot explain this trend as the trend is unidirectional. Mean = Arithmetic Average Median = Middle value Mode = 3*Median – 2*Mean Number of Orders Order Size ($) Mean Median Mode Mean Median Mode Q3, 1990 155.80 156 156.40 149.50 133 100 Q4, 1990 171.70 168.50 162.10 120.70 120 118.60 Q1, 1991 171.26 171.50 171.98 126 126 126 Q2, 1991 177.80 177.0 175.40 119.90 117 111.20 Q2. (b) Average Number of daily orders and average daily order size for profit centre 3 is presented in the following table. The number of daily order as well as the average daily order size shows the trend as that of the company. This means the Pennsylvania centre of the company is also showing the same trend as that of the entire company. Yes, the idea of investigating the performance of each profit centre is a good idea to ensure that all the three centers continue to remain healthy and any intervention can be done in case health of any of the centers starts faltering. Average Number of Daily Orders Average Daily Order Size Q3, 1990 28.2 98 Q4, 1990 30.7 101.6 Q1, 1991 31.2 100.7 Q2, 1991 35.1 95.3 Q3. (a) The inter quartile range of the average order size of the company for each quarter is shown in the following table. For Q3 the outlier data point $1114 has been ignored to get correct picture. The range of the order size for each quarter is also presented in the same table. It can be seen that inter quartile range is shrinking with respect to the total range in subsequent quarters. This means that order size is becoming less dispersed. Q1 Q2 Q3 Q4 Inter Quartile Range (IQR) 31 20 39 26 Range 126 120 147 121 Q3. (b) Quarterly sample variance, standard deviation and coefficient of variation for number of orders for different quarters is shown in the following table. It can be seen that the coefficient of variation for number of orders shows a increasing trend except in the drop in Q2, 1991. Similarly for the average order size, it decreases and increases alternatively. This could be due to seasonality. Another reason could be the fact that once a customer buys a larger quantity, he buys a smaller lot next time and vice versa. Number of Orders Average Order Size ($) Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Sample Variance 548.40 336.95 281.34 467.72 702.68 597.08 786.64 582.41 Standard Deviation 23.42 18.36 16.77 21.63 26.51 24.44 28.05 24.13 Coefficient of Variation 3.20 1.90 1.80 2.72 5.58 5.0 5.87 4.83 Q3. (c) The coefficient of variation for number of orders and average order size for the three centers is presented in the following table. It can be seen that there is no significant difference in the coefficient of variation in the average number of orders across the profit centers. However, the coefficient of variation of the average order size is much higher at profit center 1 as compared to the same at the other profit centers. This could be due to the fact that center 1 is the oldest and well established one and therefore, it has a diverse set of customers and hence large coefficient of variation in the average order size. Number of Orders Average Order Size Centre 1 Centre 2 Centre 3 Centre 1 Centre 2 Centre 3 Mean 95.41897 45.58893 31.67556 155.8455 88.2921 99.0127956 Variance 250.8317 99.34623 47.48802 7746.375 1108.196 1040.21171 Coeff. of Variation 2.62874 2.179174 1.499201 49.70549 12.55147 10.5058311 Q3. (d) First of all the staff will be explained the concept of mean and dispersion in data. Then they will be explained that large dispersion in data means we are handling customers of a diverse profile, while small variance means we are handling customers of more or less similar profile. Handling customers of similar profile makes life much easier and hence the staff will be oriented accordingly. Q4. Probability of an event (here a copier being down) is the ratio of number of favorable events (here number of days a copier was down) to the total number of events in the sample space (here total number of working days in a year i.e. 250). Copier 1 was down on 27 days out of 250 working days. Therefore, probability of copier 1 (C1) being down will be Similarly, copier 2 was down on 27 days out of 250 working days. Therefore, probability of copier 1 (C2) being down will be Both the copiers 1 and 2 were down on one out of 250 working days. Therefore, Q4. (a) The probability that a machine will be down on any given day will be Q4. (b) (i) Number of days one machine will be down in 250 working days will be *250 = 53 (ii) Number of days both machines will be down in 250 working days will be *250 = 1 Q5. Total expected yearly cost for the current situation will comprise of the cost incurred on servicing of the copiers plus the cost of number of copies lost due to the machine being down. The cost of service of the copiers will be 53*$68 + 1*$100 = $3704 The number of copies lost due to down copiers will be 53*150 + 1*300 = 8250 Therefore, cost of the downtime of the copier will be 8250 * $0.05 = $412.5 Hence, yearly cost of the current situation will be $3704 + $412.5 =$4116.5 Q6. Option I Cost Of the current situation over three years period will be 3*$4116.5 = $12349.50 Option II: Hiring two Copiers: Annual Rent = 12*$350 = $4200 Down time = 0.05*2*250 days = 25 days Cost of downtime = 25*150*$0.05 = $187.5 Cost of repair is included in the rent Total cost for one year will be $4200 + $187.5 = $4387.50 Cost of hiring the copiers for three years will be 3 * $4387.50 = $13162.50 Option III: Buying a New Copier Cost of the copier is $8750 inclusive of first year’s service Down time of the copier = 0.017*250 days = 4.25 days Cost of down time per year = 4.25*300*$0.05 = $63.75 (300 pages has been taken as number of lost copies as this new copier is replacing two existing copiers) Therefore, cost of down time for three years will be 3*$63.75 = $191.25 Cost of servicing for the first year = Nil (included in the cost price) Cost of service for 2nd and 3rd years will be 2*4.25*$175 = $1487.50 Total cost over three years of operation of acquiring a new copier will be $8750 + $1487.50 + $191.25 = $10428.75 It can be seen that buying a new copier is less expensive over three year period than hiring two copiers or continuing with current situation. So the best option will be to acquire the new copier. Q7. (a) From the given data average number of calls per hour is 27.5 Q7. (b) First we need to calculate 98% confidence interval around average number of calls per hour. Average calls per hour Standard Deviation z = 9.34 For 98% Confidence Interval z = 2.33 Therefore, 98% Confidence Interval around mean will be Hence one can be 98% sure that maximum calls per hr will not be more than 49.26 i.e. 50 (rounded off to the higher integer). Now if she has to be 98% sure that no sales representative takes more than 8 calls per hour then she should recommend Stan to keep Seven sales representatives, including Stan if handles customers’ calls. Q7. (c) Even if Stan handles just two calls per hour it is not going to alter the number of representative needed, because out of maximum number of 50 calls expected within an hour Stan can handle two calls and remaining 48 will be handled by remaining Six representatives. Q8. (a) From the given data, customers’ purchases appear to be normally distributed. Q8. (b) For the active customers’ purchases Mean Median Standard Deviation Q8. (c) Assuming normal distribution of active customers’ purchases with the mean and standard deviation calculated in Q8. (b) (i) For, X > $20000 Therefore, reading from normal distribution table [1] P(X > $20000) = 0.0455 or 4.55% i.e. Only 4.55% of active customers have account greater than $20,000. Similarly, (ii) For, X < $10000 Therefore, reading from normal distribution table [1] P(X < $10000) = 0.3192 or 31.92% i.e. Approximately 31.92% of active customers have account bigger than $10,000. (iii) The proportion of customers that fall between these ranges will be 100% – (31.92 + 4.55)% = 63.53% i.e. Approximately 63.53% of active customers have account size between $10,000 and $20,000. Conclusions: Statistical analysis was performed on the data pertaining to the number of daily orders and daily sales of the company in the recent four quarters across the three profit centers of the company. It was found that there is increase in the number of orders and simultaneously decrease in the average order size. This could be because of changing customer preference to buy smaller lot to reduce inventory cost. It was also found that there is large coefficient of variation at center 1 as compared to that at center 2 and 3. This means the customer profile is much diverse at center 1 and relatively compact at center 2 and 3. This means it will be much easier to run center 2 and 3 as customer profile is less dispersed in terms of order size. Further, analysis of the cost associated with consistently faltering copiers and that of the other two options of hiring two copiers or buying a brand new copier led to the conclusion that it is best option to procure the brand new copier which is much less prone to breakdowns. References: [1] Kundu S. “An Introduction to Business Statistics” Retrieved on March 24, 2011 from http:// 210.212.119.186/studymaterial/mcom/mc-106.pdf [2] Waner S. “Business Statistics I: QM I” Retrieved on March 24, 2011 from http://people.hofstra.edu/Stefan_Waner/RealWorld/pdfs/QM1Notes.pdf [3] “Measures of Central Tendency” Retrieved on March 24, 2011 from http://faculty.uncfsu.edu/dwallace/lesson%204.pdf [4] “Introduction to Measures of Central Tendency”, Retrieved on March 24, 2011 from http://www.diacritech.com/samples/school/Math_4colour.pdf [5] “Measures of Central Tendency and Dispersion”, Retrieved on March 24, 2011 from www.cios.org/readbook/rmcs/ch08.pdf [6] Grewal B. S. “Statistical Methods” In “Higher Engineering Mathematics”, Khanna Publishers, Delhi, India [7] Kreyszig E. “Probability and Statistics” In “Advanced Engineering Mathematics”, JOHN WILEY & SONS Inc., New York [8] http://www.statsoft.com/textbook/distribution-tables/#z Read More