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# Statistics For Management

## Introduction

### (a) Analysing raw business data using a number of statistical methods:

In Statistics, raw data includes those figures or information that has not beet processed yet or converted into something more meaningful. This business data may be segregated into divisions based on the objectives it aims to achieve and utilized as information for many research and development purpose including population and sample. Population is referred to as a vast pool of data from which a sample is drawn (Al-Omari, 2016). It may include group of people, objects or units of measure. On the other hand, a sample is a smaller group of data reflecting all the characteristics of population it is drawn from. To achieve this, many researchers take help of data analysis or statistical methods to convert mass data into an insight.

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Under Data Analysis, a research may conduct either a quantitative analysis or a qualitative analysis. Even though both aim to serve the same purpose, Quantitative Analysis targets quantification of data whereas qualitative analysis targets the underlying reasons such as motivation factors and psychology of the consumer to gain an in-depth understanding from the collected data (Armstrong and Taylor, 2014). In the context of Leisure Centre Staff of London, a quantitative analysis has been carried out to derive meaningful inferences from the data collected.

### (i) Using Ogive to estimate Median and Quartiles for Leisure Centre Staff, London

An Ogive or Cumulative histogram, is a graphical representation of Cumulative Frequencies used to ascertain number of data points which lie below or above a certain value present in the data set. The Cumulative Frequency, here, is the addition of each frequency with that of next class interval (Boehm and Thomas, 2013). It is important to note that the Cumulative Frequency for the last interval would always be equal to the sum of all data values present in the data set. The following table shows the calculations regarding cumulative frequency for the purpose of generating Ogive:

 Hourly Earnings (£) Hourly Earnings (£) No. of Leisure Centre Staff Cumulative Frequency 0-10 5 4 4 10-20 15 23 27 20-30 25 13 40 30-40 35 7 47 40-50 45 3 50 Total 50

As per the table given above, information regarding a survey of 50 leisure centre staff members in London has revealed that the hourly earnings made by them range between £10 to £50. For facilitating better understanding, the given class limits have been converted into a singular unit by calculating average of each class interval (BrozoviÄ‡ and Schlenker, 2011). This has been done by taking the adding upper and lower-class limits and dividing the total by 2. Therefore, for Hourly Earnings falling between £0 to £10, the average will be £5 (=(0+10)/2). The cumulative frequency of first class interval will be 4 as its the first unit present in the table. For calculating cumulative frequency for second class intervals and so on, the preceding frequency has been added to the current frequency giving a total of 27 (=4+23). This process has been repeated for all the given intervals. Note here, the total of number of leisure centre staff is equal to the last cumulative frequency for £40 to £50 class interval.

The above graph depicts an Ogive for the hourly ratings earned by the Leisure Centre Staff present in the London area. Here, the X-Axis shows the Upper Class Boundaries from the previous table from £0 to £50 whereas the Y-Axis shows the Cumulative Frequency calculated in the previous table. This diagram shows the accumulation occurring in the data set. Here, the difference between two data points plotted on the Ogive renders the frequency. Also, there is a steep increase in the data points, thus, indicating that as the number of hours increases, the earnings made by the staff also go increasing. Just by observing the above image, one can estimate the Median as well as the Class interval in which it falls.

Median is referred to as the middle value of a given data set segregating higher data values from the lower ones. It is one of the most widely used measures of  central tendency after mean and mode. With the help of an ogive, median value can be ascertained by equally separating the values presented in the graph. As per the Ogive, the median class would likely to fall at £20 to £30 interval for hourly earnings. Also, the cumulative frequency would be near to 50% of 50 staff members. This gives a rough idea that the Median is likely to be 25 falling under £20 to £30 class interval.

As far as quartiles are concerned, they are the intercept points which further separate the given data set into two main areas viz. 25% and 75%. Therefore, a quartile intercepting the graph at 25% level is known as First Quartile whereas when the area is separated at 75%  level, it is known as Third Quartile. Note that Median is also known as Second Quartile with a 50% level of area interception. From the above Ogive, the first quartile can be estimated near 25% of hourly earnings between £10 to £50 that comes to £12.50. On the other hand, 75% of the hourly earnings for same range gives a third quartile value of £37.50. In order to double check these values, the following calculations were carried out:

 Median (50%) = (2*(50+1)/4) £25.5 Quartile (25%) = (1*(50+1)/4) £12.75 Quartile (75%) =(3*(50+1)/4) £38.25

### (ii) Calculation of Mean and Standard Deviation for hourly earnings

 Hourly Earnings Hourly Earnings (x) No. of Leisure Centre Staff (f) Total Hourly Earnings of Staff (f)*(x) Squared Hourly Earnings (x2) Number of Staff * Squared hourly earnings (f)*(x2) 0-10 5 4 20 25 100 10-20 15 23 345 225 5175 20-30 25 13 325 625 8125 30-40 35 7 245 1225 8575 40-50 45 3 135 2025 6075 Total 50 1070 28050

Mean = Total Hourly Earnings of Staff/ Number of Staff = 1070/50 = £21.4

Standard Deviation = [(28050/50) – (21.4)]^ 0.5 = £10.15

Mean relates to the average of data values for a given sample. In the context of given scenario, the Mean Hourly Earnings for the London Staff is £21.4. This means that the 50 staff members earn a £21.4 hourly wage on an average basis. On the other hand, the standard deviation for the above table comes to £10.15 (Haimes, 2015). Since this measure of dispersion is not located nearby the mean, central tendency measure, it can be concluded that there is deviation of values from the central point of given data set.

### (b) Comparison of earnings between two regions

 London Manchester Median £25.5 £14 Mean £21.4 £16.5 Standard Deviation £10.15 £7 Inter-Quartile Range £25.5 £7.5

By taking into consideration two surveys conducted for the leisure centre staff of London and Manchester Area, it can be observed that the average hourly earnings of Manchester rank higher than that of London. However, the Manchester centre lacks in other measures as compared to that of London (Herrera and Schipp, 2014). Looking at the Median, it can be said that there is a higher gap between the two for London as compared to Manchester indicating that the average hourly earnings are closely held to the data set's middle value. The inter-quartile range for London Centre is higher as compared to the other signalling that the number of data sets falling within this range is high and lesser number of outliers are present in it as opposed to Manchester Centre v.

Economic Order Quantity or EOQ, is that level of quantity for a given product where the related ordering and carrying cost are minimal. It is one of the most widely used tool to ascertain volume and frequency of orders across organisations. Ordering Cost is the cost incurred when inventory is purchased from the suppliers. This cost tends to decline as the volume of order increases resulting in economies of scale (Kyriakarakos and et.al., 2013). On the other hand, Carrying Cost or the holding Cost is the one which is incurred for storing the bought inventory in warehouses until they are consumed for production of final goods. It is important to note that the larger the volume of inventory, the higher carrying cost is. This tool also helps in determining the most effective inventory policy an organisation must implement to avoid running out of stock.

### (a) Calculating Economic Order Quantity

Economic Order Quantity is calculated by using following formula:

EOQ= [(2*Annual Consumption* Ordering Cost)/ Carrying Cost]^0.5

Here, Annual Consumption is the total number of units consumed or demanded by customers regarding a particular product (Marchington and et.al., 2016). Order Cost tends to differ for purchased items and manufacturers as the former would include those costs that are incurred when purchasing or placing a requisition order while the latter would include the time incurred to initiate the work order including scheduling and inspection time.

In the given scenario, Jenny Jones wants to develop a new inventory policy that would enable her customers to purchase tee-shirt 95% of the time they asked. The following related information has been given regarding the tee-shirt:

 Number of Weeks Shop is open 50 Average Weekly demand (shirts) 30 Annual Demand for shirts (= 30*50) 1500 Delivery Cost (£) 5 Cost of Tee (£) 10 Holding Cost (£) (= 0.20*£10) 2

From the above table, Economic Order Quantity for Ms. Jones' Shop can be derived as follows:

EOQ = [(2*Annual Demand* Delivery Cost)/ Holding Cost]^0.5

= [(2*1500*5)/ 2]^0.5

= [15000/2]^0.5

= [7500]^0.5

= 86.60 or 87 units.

This means that at 87 units of tee-shirts, Ms. Jones is able to minimize her ordering as well as carrying cost to achieve economies of scale as well as revenues (Jiang and Pang, 2011). Thus, Jenny must order at least 87 units of tee-shirts from its suppliers that in order to minimize its total inventory cost, annually.

### (b) Calculating Frequency of Orders

After determining the quantity that Jenny to order for developing an effective inventory policy, it is imperative to also calculate how often would she need to place such orders. This can be found out by dividing annual demand by the economic order quantity for the tee-shirts.

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Thus,

Number of Orders required to be placed = 1500 tee-shirts/ 87 = 17.24 or 17 orders.

This indicates that, Ms. Jones would require to place 17 orders annually to minimize ordering as well as carrying costs incurred by her.

### (c) Ascertaining Inventory Policy Cost

For the given case scenario, Jenny owns a store which is open for 50 weeks in a year. One of her tee-shirts is popular among customers who only visit her shop to make purchase of that product. Jenny has been facing a difficult time dealing with holding an adequate amount of stock for everyone, especially between placement and receipt of an order (Qiu, Qin and Zhou, 2016). This has led to her losing customers to other competing stores located in the shopping centre. For this purpose, she needs to minimize her ordering and carrying costs for prevention of losses as well as maintenance of competitive advantage.

In order to calculate Inventory Policy Cost, she would need to consider three main costs viz. Cost of Product, Annual Ordering Cost and Annual Carrying Cost. Inventory Policy Cost or Total Cost is the minimum outlay incurred for implementation of policy. She would have to expend a minimum of following ordering, purchase and carrying costs:

 Delivery Cost (£) 5 Cost of Tee (£) 10 Holding Cost (£) (= 0.20*£10) 2 Economic Order Quantity (tee-shirts) 87 Total Annual Holding Cost = [(87 units/ 2)* 2] (A) £87 Total Annual Ordering Cost = [(1500/87)*5] (B) £86.21 Total Annual Inventory Cost [(C) = (A) + (B)] £173.21

Hence, the total inventory cost per annum for the business is £173.21. This means that if Ms. Jenny wants to implement an inventory policy where her desired service level is 95% along with a minimum purchasing and holding costs, then, she would need to incur a minimum outlay of £173.21.

(d) Current Service Level to Customers:

 Desired Probability to purchase the tee-shirts 95.00% Selling Price per tee-shirt (£) 20 Standard Deviation 15 Safety Stock 150 Shirts

Current Level of service = Weekly Demand * Availability of t-shirt

=   30*95%

=   28.5 units

1. e) Work out the re-order level to achieve desired service levels

Re-order level (ROQ) = (Lead time*daily average usage)+safety stock

= (28*2)+150

= 206 units

### References

• Al-Omari, A. I., 2016. Time truncated acceptance sampling plans for Generalized Inverse Weibull Distribution. Journal of Statistics and Management Systems. 19(1). pp.1-19.
• Armstrong, M. and Taylor, S., 2014. Armstrong's handbook of human resource management practice. Kogan Page Publishers.
• Boehm, M. and Thomas, O., 2013. Looking beyond the rim of one's teacup: a multidisciplinary literature review of Product-Service Systems in Information Systems, Business Management, and Engineering & Design. Journal of Cleaner Production. 51. pp.245-260.
• BrozoviÄ‡, N. and Schlenker, W., 2011. Optimal management of an ecosystem with an unknown threshold. Ecological economics. 70(4). pp.627-640.
• Embrechts, P. and Hofert, M., 2014. Statistics and quantitative risk management for banking and insurance. Annual Review of Statistics and Its Application. 1. pp.493-514.
• Haimes, Y. Y., 2015. Risk modeling, assessment, and management. John Wiley & Sons.
• Herrera, R. and Schipp, B., 2014. Statistics of extreme events in risk management: The impact of the subprime and global financial crisis on the German stock market. The North American Journal of Economics and Finance. 29. pp.218-238.
• Jiang, H. and Pang, Z., 2011. Network capacity management under competition. Computational Optimization and Applications. 50(2). pp.287-326.
• Kyriakarakos, G. and et. al., 2013. Intelligent demand side energy management system for autonomous polygeneration microgrids. Applied Energy. 103. pp.39-51.
• Marchington, M. and et. al., 2016. Human resource management at work. Kogan Page Publishers.
• Qiu, Z., Qin, J. and Zhou, Y., 2016. Composite Estimating Equation Method for the Accelerated Failure Time Model with Lengthâ€biased Sampling Data. Scandinavian Journal of Statistics. 43(2). pp.396-415.
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