4 Statistical techiques this chapter covers... I this chapter we will explai how to calculate key statistical idicators which will help us to aalyse past data ad help us forecast what may happe i the future. We will start by examiig time series strigs of data that occur over time. We will see how a formula ca be used to represet a straight lie o a graph (regressio aalysis), ad how this ca be used to predict data at various poits. Next we will see how some data moves i regular cycles over time, ad how this, together with the uderlyig tred ca be used to develop forecasts. We will show how averagig techiques ca be used to detect the tred i give data. The third sectio cocers the use of idex umbers. These ca be used to compare umerical data over time for example prices of commodities or geeral price iflatio. We will see how to carry out various calculatios usig idex umbers that ca be useful. Fially we will briefly see how idex umbers ca be used to aalyse stadard costig variaces so that we ca determie performace more accurately.
s t a t i s t i c a l t e c h i q u e s 1 2 5 T I M E S E R I E S A N A LY S I S Time series aalysis ivolves aalysig umerical treds over a time period. It is ofte used to examie past ad preset treds so that future treds ca be forecast. The term tred aalysis is used to describe the techique that we will ow examie. At its simplest the cocept is based o the assumptio that data will cotiue to move i the same directio i the future as it has i the past. Usig the sales of a shoe shop as a example we will ow look at a rage of techiques for dealig with treds. a i d e t i c a l a u a l c h a g e A shoe shop Comfy Feet has sold the followig umbers of pairs of shoes aually over the last few years: 20-1 10,000 20-2 11,000 20-3 12,000 20-4 13,000 20-5 14,000 20-6 15,000 20-7 16,000 It does ot require a great deal of arithmetic to calculate that if the tred cotiues at the previous rate a icrease of 1,000 pairs a year the shoe sales could be forecast at 17,000 pairs i 20-8 ad 18,000 pairs i 20-9. Of course this is a very simple example, ad life is rarely this straightforward. For example, for how log ca this rate of icrease be sustaied? a v e r a g e a u a l c h a g e A slightly more complex techique could have bee used to arrive at the same aswer for the shoe shop. If we compare the umber of sales i 20-7 with the umber i 20-1, we ca see that it has rise by 6,000 pairs. By dividig that figure by the umber of times the year chaged i our data we ca arrive at a average chage per year. The umber of times that the year chages is 6, which is the same as the umber of spaces betwee the years (or alteratively the total umber of years mius 1). Show as a equatio this becomes: Average Aual Sales Chage = (Sales i Last Year Sales i First Year) (16,000 10,000) (Number of Years 1) = (7 1) = + 1,000, which is what we would expect.
1 2 6 m a a g e m e t a c c o u t i g : d e c i s i o a d c o t r o l t u t o r i a l The + 1,000 would the be added to the sales data i 20-7 of 16,000 (the last actual data) to arrive at a forecast of 17,000. This techique is useful whe all the icreases are ot quite idetical, yet we wat to use the average icrease to forecast the tred. A egative aswer would show that the average chage is a reductio, ot a icrease. We will use this techique whe estimatig the tred movemet i more complicated situatios. This is ot the oly way that we ca estimate the directio that data is movig over time, ad it does deped o the data (icludig especially the first ad last poits) fallig roughly ito a straight lie. We will ote alterative methods that ca be used later i this sectio. c o s t r u c t i g a g r a p h The same result ca be produced graphically. Usig the same shoe shop example we ca exted the graph based o the actual data to form a forecast lie. Comfy Feet: Sales of shoes forecast data 20-1 20-2 20-3 20-4 20-5 20-6 20-7 20-8 20-9 If i aother situatio the actual data does ot produce exactly equal icreases, the graph will produce the same aswer as the average aual chage provided the straight lie rus through the first ad last year s data poits.
s t a t i s t i c a l t e c h i q u e s 1 2 7 u s i g a f o r m u l a The data i the example could have bee expressed i the followig formula: y = mx + c where y is the forecast amout m is 1,000 (the amout by which the data icreases each year) x is the umber of years sice the start year (20-1) c is 10,000 (which is the sales figure i the start year of 20-1) If we wated a forecast for the year 20-9, we could calculate it as: Forecast = (1,000 x umber of years sice 20-1) + 10,000 y (the forecast) = (1,000 x 8) + 10,000 = 18,000, which is what we would expect. This formula works because the formula is based o the equatio of a straight lie. u s i g a f o r m u l a f o r m o r e c a l c u l a t i o s The formula of a straight lie (y = mx + c) that we have just used to calculate a forecast for y ca also be used to work out other iformatio. The formula always has the followig compoets: a fixed value ( c i the formula y = mx + c); this is the poit where the straight lie starts from a gradiet value ( m i the formula); this determies how steep the lie is, ad whether it is goig up (whe m is positive) or goig dow (whe m is egative) The formula ca be used (for example) to predict prices, costs or demad. Sometimes the formula is show i a slightly differet style (for example y = a + bx), but the compoets are still the same. The formula of a straight lie ties i with the calculatios that we carried out i Chapter 1 for the high-low method of calculatig cost behaviour. There the fixed value represeted the fixed costs, ad the gradiet value was the variable cost per uit. You will otice that the calculatio methods that we used for aalysig costs ca also be used for other situatios. We will ow use the formula to demostrate how differet elemets ca be calculated.
1 2 8 m a a g e m e t a c c o u t i g : d e c i s i o a d c o t r o l t u t o r i a l practical example For example, suppose we are told that the price of a compoet over time is believed to icrease based o the formula Y = a + bx, where Y is the price i, ad X is the year umber We are told that i year umber 4 the price was 68, ad i year umber 8 the price was 76. We would like to calculate a ad b i the formula, ad the use this iformatio to predict the price i year 11. We ca use a calculatio similar to the high-low method, to determie how much the price is movig by each year: Price Year 76 8 68 4 Differeces 8 divided by 4 = 2 per year This is the gradiet amout b, ad we ca use it to calculate the amout a by usig price iformatio from either of the years that we kow. For example, usig year 4 data ad puttig it ito the formula gives: 68 = a + ( 2 x 4 [the year umber]) 68 = a + 8 So a must be 60 Now we have the full formula that we ca use for ay year: Y = 60 + 2 x X I year 11, this would give a price of: y = 60 + ( 2 x 11) = 82 Just like i the high-low method, if you are provided with more tha two pairs of data, the usig the highest ad the lowest will probably give the most reliable aswer. We will ow use a example to illustrate the use of the formula to predict demad. Sales of a atioal daily ewspaper have bee decliig steadily for several years. The demad level is believed to follow the formula Y = a + bx, where Y is the demad i umbers of ewspapers, ad X is the year umber. Calculatios have already bee carried out to establish the values of a ad b, which are: a is 200,000 b is -2,500
s t a t i s t i c a l t e c h i q u e s 1 2 9 Note that b is a egative figure, so each year the demad decreases. You are asked to calculate the expected sales i year 14. If we isert the kow data ito the formula, we ca calculate the demad for year 14 as follows: Y = 200,000 (2,500 x 14) Y = 165,000 l i e a r r e g r e s s i o I the last sectio o time series aalysis we saw that whe some historical data moves i a cosistet ad regular way over time we ca use it to help estimate the future tred of that data. We also saw that i these circumstaces the data ca be represeted by a straight lie o a graph, ad / or a equatio of the lie i the form y = mx + c to help us develop the tred. Liear regressio is the term used for the techiques that ca be used to determie the lie that best replicates that give data. You should be aware of the techiques i geeral terms, ad be able to appreciate their usefuless. You may be give historical data or the equatio of a lie ad asked to use it to geerate a forecast. Where data exactly matches a straight lie (as with the Comfy Feet data) there is o eed to use ay special techiques. I other situatios the followig could be used: Average aual chage. This method was described earlier, ad is useful if we are cofidet that the first ad last poits (take chroologically sice we are lookig at data over time) are both represetative. It will smooth out ay mior fluctuatios of the data i betwee. We will see this method used i the Seasoal Compay case study later i this chapter. Lie of best fit. Where the data falls oly roughly ito a straight lie, but the first ad last poits do ot appear to be very represetative the average aual chage method would give a distorted solutio. Here a lie of best fit ca be draw oto the data poits o a graph that will form a better estimate of the movemet of the data. The graph o the ext page illustrates a situatio where the lie of best fit would provide a better solutio tha the average aual chage method.
1 3 0 m a a g e m e t a c c o u t i g : d e c i s i o a d c o t r o l t u t o r i a l liear regressio techiques uits lie of best fit lie usig average aual chage method time Least squares method. This is a mathematical techique for developig the equatio of the most appropriate lie. It is more accurate tha drawig a lie of best fit oto a graph by eye, but the calculatios ivolved are outside the scope of this book. I the followig example the regressio lie has already bee calculated, ad is used to forecast the cost of materials. practical example A colleague has calculated the least squares regressio lie (the lie of best fit) as y = 15.75 + 1.65x where y is the cost per kilogram i ad x is the period. April X5 is period 32. You are asked to forecast the cost per kilogram for July X5. The figures are iserted ito the formula as follows (July X5 is period 35) y = 15.75 + (1.65 x 35) Forecast cost per kilogram (y) = 73.50 All liear regressio techiques assume that a straight lie is a appropriate represetatio of the data. Whe lookig at time series this meas that we are assumig that the chages i the data that we are cosiderig (kow as the depedet variable) are i proportio to the movemet of time (the idepedet variable). This would mea that we are expectig (for example) the sales level to cotiually rise over time. Whe we use time series aalysis later i the book we must remember that sometimes data does ot travel forever i a straight lie, eve though they may do so for a short time. For example share prices o the stock market do ot cotiue to go up (or dow) steadily, but ofte move i a more erratic way.
s t a t i s t i c a l t e c h i q u e s 1 3 1 T I M E S E R I E S A N A LY S I S A N D S E A S O N A L VA R I AT I O N S There are four mai factors that ca ifluece data which is geerated over a period of time: The uderlyig tred This is the way that the data is geerally movig i the log term. For example the volume of traffic o our roads is geerally icreasig as time goes o. Log term cycles These are slow movig variatios that may be caused by ecoomic cycles or social treds. For example, whe ecoomic prosperity geerally icreases this may icrease the volume of traffic as more people ow cars ad fewer use buses. I times of ecoomic depressio there may be a decrease i car use as people caot afford to travel as much or may ot have employmet which requires them to travel. Seasoal variatios This term refers to regular, predictable cycles i the data. The cycles may or may ot be seasoal i the ormal use of the term (eg Sprig, Summer etc). For example traffic volumes are always higher i the daytime, especially o weekdays, ad lower at weekeds ad at ight. Radom variatios All data will be affected by iflueces that are upredictable. For example floodig of some roads may reduce traffic volume alog that route, but icrease it o alterative routes. Similarly the traffic volume may be iflueced by heavy sowfall. The type of umerical problems that you are most likely to face will ted to igore the effects of log-term cycles (which will effectively be cosidered as a part of the tred) ad radom variatios (which are impossible to forecast). We are therefore left with aalysig data ito uderlyig treds ad seasoal variatios, i order to create forecasts. The techique that we will use follows the process i the diagram o the ext page. The process is as follows: 1 The historical actual data is aalysed ito the historical tred ad the seasoal variatios. 2 The historical tred is used to forecast the future tred, usig the techiques examied i the last sectio.
1 3 2 m a a g e m e t a c c o u t i g : d e c i s i o a d c o t r o l t u t o r i a l 3 The seasoal variatios are icorporated with the forecast future tred to provide a forecast of where the actual data will be i the future. icorporatig seasoal variatios ito the tred historical actual data forecast of future data seasoal variatios historical tred forecast future tred f o r e c a s t i g u s i g d e s e a s o a l i s e d d a t a If we kow (or ca estimate fairly accurately) the seasoal variatios, the we ca use this iformatio together with actual data to work out what the tred is. The term deseasoalised data meas data from which the seasoal variatios have bee stripped away i other words the tred. We ca the extrapolate this tred. This meas forecastig how the tred will move i the future. The seasoal variatios for uit sales of a product have bee calculated to be the followig percetages of the uderlyig tred: Quarter 1 15% Quarter 2 +25% Quarter 3 +10% Quarter 4 20% I year 5 the actual uit sales results are as follows: Quarter 1 42,500 Quarter 2 75,000 Quarter 3 77,000 Quarter 4 64,000
s t a t i s t i c a l t e c h i q u e s 1 3 3 From these figures we ca calculate the deseasoalised data the tred figures. We eed to be careful because the seasoal variatios are calculated as percetages of the tred. Quarter 1 The tred must be greater tha 42,500 by 15% of the tred. Therefore 42,500 must equal 85% of the tred Tred = 42,500 x 100 / 85 = 50,000 Quarter 2 The tred must be lower tha 75,000 by 25% of the tred. Therefore 75,000 must equal 125% of the tred Tred = 75,000 x 100 / 125 = 60,000 Usig the same logic: Quarter 3 Tred = 77,000 x 100 / 110 = 70,000 Quarter 4 Tred = 64,000 x 100 / 80 = 80,000 Havig idetified the tred for year 5 as 50,000, 60,000, 70,000 ad 80,000 we ca see that it is risig by 10,000 uits per quarter. Therefore the forecast for year 6 ca be worked out as follows: Year 6 Quarter 1 Quarter 2 Quarter 3 Quarter 4 Extrapolated Tred 90,000 100,000 110,000 120,000 Seasoal Variatios 15% +25% +10% 20% Forecast 76,500 125,000 121,000 96,000 I a task the aalysis of actual data may have bee carried out already, or you may be asked to carry out the aalysis by usig movig averages. If you are usig movig averages it is importat that: your workigs are laid out accurately the umber of pieces of data that are averaged correspods with the umber of seasos i a cycle where there is a eve umber of seasos i a cycle a further averagig of each pair of averages takes place
1 3 4 m a a g e m e t a c c o u t i g : d e c i s i o a d c o t r o l t u t o r i a l m o v i g a v e r a g e s A movig average is the term used for a series of averages calculated from a stream of data so that: every average is based o the same umber of pieces of data, (eg four pieces of data i a four poit movig average ), ad each subsequet average moves alog that data stream by oe piece of data so that compared to the previous average it uses oe ew piece of data ad abados oe old piece of data. This is easier to calculate tha it souds! For example, suppose we had a list of six pieces of data relatig to the factory output over two days where a three-shift patter was worked as follows: Day 1 Morig Shift 14 uits Afteroo Shift 20 uits Night Shift 14 uits Day 2 Morig Shift 26 uits Afteroo Shift 32 uits Night Shift 26 uits If we thought that the shift beig worked might ifluece the output, we could calculate a three-poit movig average, the workigs would be as follows: First movig average: (14 + 20 + 14) 3 = 16 Secod movig average: (20 + 14 + 26) 3 = 20 Third movig average: (14 + 26 + 32) 3 = 24 Fourth movig average (26 + 32 + 26) 3 = 28 Notice how we move alog the list of data. I this simple example with six pieces of data we ca t work out ay more three-poit averages sice we have arrived at the ed of the umbers after oly four calculatios. Here we chose the umber of pieces of data to average each time so that it correspoded with the umber of poits i a full cycle. By choosig a threepoit movig average that correspoded with the umber of shifts we always had oe example of the output of every type of shift i our average. This meas that ay ifluece o the average by icludig a ight shift (for example) is cacelled out by also icludig data from a morig shift ad a afteroo shift.
s t a t i s t i c a l t e c h i q u e s 1 3 5 We must be careful to always work out movig averages so that exactly oe complete cycle is icluded i every average. The umber of poits is chose to suit the data. Whe determiig a tred lie, each average relates to the data from its mid poit, as the followig layout of the figures just calculated demostrates. Output Tred (Movig Average) Day 1 Morig Shift 14 uits Afteroo Shift 20 uits 16 uits Night Shift 14 uits 20 uits Day 2 Morig Shift 26 uits 24 uits Afteroo Shift 32 uits 28 uits Night Shift 26 uits This meas that the first average that we calculated (16 uits) ca be used as the tred poit of the afteroo shift o day 1, with the secod poit (20 uits) formig the tred poit of the ight shift o day 1. The result is that we: kow exactly where the tred lie is for each period of time, ad have a basis from which we ca calculate seasoal variatios Eve usig our limited data i this example we ca see how seasoal variatios ca be calculated. A seasoal variatio is simply the differece betwee the actual data at a poit ad the tred at the same poit. This gives us the seasoal variatios show i the followig table, usig the figures already calculated. Output Tred Seasoal Variatio Day 1 Morig Shift 14 uits Afteroo Shift 20 uits 16 uits + 4 uits Night Shift 14 uits 20 uits - 6 uits Day 2 Morig Shift 26 uits 24 uits + 2 uits Afteroo Shift 32 uits 28 uits + 4 uits Night Shift 26 uits The seasoal variatio for the afteroo shift, calculated o day 1, is based o the actual output beig 4 uits greater tha the tred at the same poit (20 mius 16 uits).
1 3 6 m a a g e m e t a c c o u t i g : d e c i s i o a d c o t r o l t u t o r i a l Case Study T H E AV E R A G E C O M PA N Y M O V I N G AV E R A G E S A N D F O R E C A S T S The Average Compay has sales data that follows a 3 period cycle. The sales uits show i the table below have bee compiled from actual data i periods 11 to 19. Period Actual Data 3 Poit Movig Seasoal Averages (Tred) Variatios 11 7,650 12 7,505 13 7,285 14 7,590 15 7,445 16 7,225 17 7,530 18 7,385 19 7,165 r e q u i r e d (a) Usig a 3 poit movig average, calculate the tred figures ad the seasoal variatios (b) Extrapolate the tred to periods 20 to 25, ad usig the seasoal variatios forecast the sales uits for those periods s o l u t i o (a) The 3 poit movig averages ca be calculated for each period except the first ad the last. The seasoal variatios are calculated as (actual data tred). Period Actual Data 3 Poit Movig Seasoal Averages (Tred) Variatios 11 7,650 12 7,505 7,480 +25 13 7,285 7,460 175 14 7,590 7,440 +150 15 7,445 7,420 +25 16 7,225 7,400 175 17 7,530 7,380 +150 18 7,385 7,360 +25 19 7,165
s t a t i s t i c a l t e c h i q u e s 1 3 7 (b) The tred calculated from 3 poit movig averages ca be see to be reducig by 20 each period, ad so ca be easily extrapolated. The seasoal variatios operate o a 3 period repeatig cycle, so ca be iserted. The forecast data is the calculated as (extrapolated tred + seasoal variatios). Period Forecast Data Extrapolated Tred Seasoal Variatios 20 7,470 7,320 +150 21 7,325 7,300 +25 22 7,105 7,280-175 23 7,410 7,260 +150 24 7,265 7,240 +25 25 7,045 7,220-175 I N D E X N U M B E R S Idex umbers are used to assist i the compariso of umerical data over time. A commoly used idex is the Retail Price Idex that gives a idicatio of iflatio by comparig the cost of a group of expeses typically icurred by households i the UK from year-to-year. There are may other types of idex umbers that have bee created for specific purposes, for example: the average wage rate for a particular job, or for all employmet the average house price either by regio or throughout the UK the market price of shares (eg the FTSE 100 idex) the quatities of specific items that are sold or used (eg litres of uleaded petrol) the quatities of a group of items that are sold or used (eg litres of all motor fuel) the maufactured cost of specific items or a rage of items (sometimes called factory gate prices) May govermet idices ad other idicators are available at www.gov.uk/govermet/statistics. If you have the opportuity, have a look at the eormous rage of data that ca either be dowloaded free, or ca be purchased i govermet publicatios. Whe usig published statistics it is importat to make sure that they are specific eough to be useful for your purpose. For example, data o the
1 3 8 m a a g e m e t a c c o u t i g : d e c i s i o a d c o t r o l t u t o r i a l growth i the populatio of the West of Eglad will be of limited use if you are tryig to forecast the sales i a bookshop i Tauto. Of far more use would be details of proposed housig developmets withi the immediate area, icludig the umbers of ew homes ad the type of households that form the developers target market. l e a d i g a d l a g g i g i d i c a t o r s Some idicators ca be classified as leadig idicators, whilst others are kow as laggig idicators. This meas that some idicators aturally give advace warig of chages that may take place later i other idicators. For example a idex that moitors the prices of maufactured goods ( factory gate prices) will react to chages before they have filtered through to retail price idices. The idex of factory gate prices ca therefore be cosidered to be a leadig idicator of retail prices, ad give early warig of implicatios to idustrial situatios. I a similar way, a idex recordig the volume of maufactured output from factories will lag behid a idex measurig the volume of purchases of raw materials made by idustrial buyig departmets. w e i g h t i g s o f i d i c e s Those idices that are based o iformatio from more tha oe item will use some form of weightig to make the results meaigful. For example while a idex measurig the retail price of premium grade uleaded petrol is based o a sigle product ad therefore eeds o weightig, this would ot be true for a price idex for all vehicle fuel. I this case it will require a decisio about how much weight (or importace) is to be placed o each compoet of the idex. Here the relative quatities sold of types of fuel (for example uleaded petrol ad diesel) would be a logical way to weight the idex. This would esure that if petrol sales were double those for diesel, ay price chages i petrol would have twice the impact o the idex tha a price chage i diesel. As the purchasig habits of cosumers chage, the the weightig ad compositio of complicated idices like the Retail Price Idex ad the Cosumer Price Idex are ofte chaged to reflect this. This will iclude chages to the weightig of certai items, for example due to the icrease i the proportio of household expediture o holidays. It ca also ivolve the additio or deletio of certai items etirely (for example the iclusio of certai fast foods). You may have see ews items from time to time about the revisio of items cotaied withi the RPI or CPI as cosumers tastes chage.
s t a t i s t i c a l t e c h i q u e s 1 3 9 c a l c u l a t i o s u s i g i d e x u m b e r s Whatever type of idex we eed to use, the priciple is the same. The idex umbers represet a coveiet way of comparig figures. For example, the RPI was 82.61 i Jauary 1983, ad 245.8 i Jauary 2013. This meas that average household costs had early tripled i the 30 years betwee. We could also calculate that if somethig that cost 5.00 i Jauary 1983 had rise exactly i lie with iflatio, it would have cost 14.88 i Jauary 2013. This calculatio is carried out by: historical price x idex of time covertig to idex of time covertig from ie 5.00 x (245.8 82.61) = 14.88 p r a c t i c a l e x a m p l e This is a icrease of ( 14.88 5.00) x 100 = 197.6% 5.00 You may be told that the base year for a particular idex is a certai poit i time. This is whe the particular idex was 100. For example the curret RPI idex was 100 i Jauary 1987. Idex umbers referrig to costs or prices are the most commoly used oes referred to i the uit studied i this book. If we wat to use cost idex umbers to moitor past costs or forecast future oes, the it is best to use as specific a idex as possible. This will the provide greater accuracy tha a more geeral idex. For example, if we were operatig i the food idustry, ad wated to compare our coffee cost movemets with the average that the idustry had experieced, we should use a idex that aalyses coffee costs i the food idustry. This would be much more accurate tha the RPI, ad also better tha a geeral cost idex for the food idustry. The followig table shows the actual material costs for Jauary for years 20X2 to 20X5, together with the relevat price idex. Period Actual costs ( ) Cost idex Costs at Jauary 20X2 prices Jauary 20X2 129,300 471 Jauary 20X3 131,230 482 Jauary 20X4 135,100 490 Jauary 20X5 136,250 495 Required: Restate all the actual costs at Jauary 20X2 prices, to the earest.
1 4 0 m a a g e m e t a c c o u t i g : d e c i s i o a d c o t r o l t u t o r i a l Solutio Period Actual costs ( ) Cost idex Costs at Jauary 20X2 prices Jauary 20X2 129,300 471 129,300 Jauary 20X3 131,230 482 128,235 Jauary 20X4 135,100 490 129,861 Jauary 20X5 136,250 495 129,644 Case Study T U R N E R L I M I T E D : A D J U S T I N G T O R E A L T E R M S Sales reveue ad Net Profit figures are give for Turer Ltd for the five years eded 31 December 20-1 to 20-5. A suitable idex for Turer Ltd s idustry is also give. 20-1 20-2 20-3 20-4 20-5 Sales reveue ( 000s) 435 450 464 468 475 Net Profit ( 000s) 65 70 72 75 78 Idustry Idex 133 135 138 140 143 r e q u i r e d Calculate the sales reveue ad profit i terms of year 20-5 values ad commet o the results. s o l u t i o To put each figure ito 20-5 terms, it is divided by the idex for its ow year ad multiplied by the idex for 20-5, ie 143. For example: Sales reveue Year 20-1 435 x 143 = 467.7 133 Sales reveue Year 20-2 450 x 143 = 476.7 ad so o. 135 I year 20-5 terms: 20-1 20-2 20-3 20-4 20-5 Sales reveue ( 000s) 467.7 476.7 480.8 478.0 475.0 Net Profit ( 000s) 69.9 74.1 74.6 76.6 78.0 The adjusted figures compare like with like i terms of the value of the poud, ad the Net Profit still shows a icreasig tred throughout, but the sales reveue decreases i the last two years.
s t a t i s t i c a l t e c h i q u e s 1 4 1 c r e a t i o o f a i d e x You may be required to create a idex from give historical data, ad we will ow see how this is carried out. Suppose that you are provided with the followig prices for oe uit of a certai material over a period of time: Moth Ja Feb March April May Jue Price 29.70 30.00 28.30 30.09 31.00 31.25 The first thig to do is to decide which poit i time is to be the base poit the price at this poit will be 100 i our ew idex. I this example we will first use Jauary as our base poit, but later we will see how aother date could have bee chose. Next, the price of aother date (we ll use February) is divided by the price at the base poit. The result is the multiplied by 100 to give the idex at that poit (ie February): ( 30.00 / 29.70) x 100 = 101.01 Note that the idex umber is ot a amout i s, it is just a umber used for compariso purposes. I this example we ve rouded to 2 decimal places ad we will eed to be cosistet for the other figures. If we carry out the same calculatio for the March price we get the followig: ( 28.30 / 29.70) x 100 = 95.29 Notice that here the aswer is less tha 100, which makes sese because the price i March is lower tha the price i Jauary. Checkig that each idex umber is the expected side of 100 (ie higher or lower) is a good idea ad will help you to detect some arithmetical errors. The full list of idex umbers is as follows make sure that you ca arrive at the same figures. Moth Ja Feb March April May Jue Price 29.70 30.00 28.30 30.09 31.00 31.25 Idex 100.00 101.01 95.29 101.31 104.38 105.22 We could have chose a differet date to act as our base poit if we chose March, the the calculatio for Jauary would have bee: ( 29.70 / 28.30) x 100 = 104.95 The the full list of idex umbers would have bee as follows:
1 4 2 m a a g e m e t a c c o u t i g : d e c i s i o a d c o t r o l t u t o r i a l Moth Ja Feb March April May Jue Price 29.70 30.00 28.30 30.09 31.00 31.25 Idex 104.95 106.01 100.00 106.33 109.54 110.42 Agai, make sure that you could arrive at the same figures. Do t forget that although we have used the creatio of a price idex i the above example, you could also be asked to create a idex from ay suitable historical data. Whatever the type of data, the arithmetic required is the same. U S I N G I N D E X N U M B E R S T O A N A LY S E VA R I A N C E S Whe stadard costig is beig used, stadards for material prices will have bee set based o expected costs. This will ofte be based o a expected level of a price idex for that material (if there is oe available). If the price idex for the material chages sigificatly the we ca calculate what the stadard would be if it was based o the idex (the revised stadard price). We ca the see how that impacts o ay price variace that has bee calculated. Oce we have worked out what the revised stadard price would be, we ca the calculate the part of a price variace explaied by the idex chage as: the origial stadard cost of the actual material mius the revised stadadr cost of the actual material The remaider of the origial variace would be the part ot explaied by the chage i price idex. The followig case study illustrates the process. Case Study A N A LY S I S L I M I T E D R E V I S E D S TA N D A R D P R I C E Aalysis Limited operates a stadard costig system ad uses a raw material that is a global commodity. The stadard price was set based upo a market price of 950 per kilo whe the relevat price idex was 315. I April the price idex was 330. The quatity of material purchased ad used was 128 kilos, which cost 125,000.
s t a t i s t i c a l t e c h i q u e s 1 4 3 r e q u i r e d Calculate the material price variace, based o the origial stadard Calculate what the revised stadard price per kilo would be, based o the chage i the idex, to the earest Aalyse the material price variace ito the part explaied by the chage i the idex, ad the remaider. s o l u t i o The material price variace is ( 950 x 128 kilos) - 125,000 = 3,400 adverse The revised stadard price per kilo would be: 950 x 330 / 315 = 995 to the earest The part of the price variace explaied by the idex chage: ( 950 x 128 kilos) ( 995 x 128 kilos) = 5,760 adverse The remaider of the variace is therefore: 3,400 adverse mius 5,760 adverse = 2,260 favourable I this situatio the price actually paid for material is lower tha would be expected from the chage i the idex. Chapter Summary A time series is formed by data that occurs over time. If the data icreases or decreases regularly (i a straight lie ) the it ca be represeted by a formula. The formula ca the be used to predict the data at various poits. Some data moves i regular cycles over time, ad the distaces that the data is from the uderlyig tred are kow as seasoal variatios. Iformatio about the uderlyig tred ad the seasoal variatios ca be used to forecast data. Movig averages ca be used to split data ito the tred ad the seasoal variatios. Idex umbers ca be used to compare umerical data over time. Examples of the use of idex umbers are for prices of commodities or geeral price iflatio. Idex umbers ca be used to aalyse stadard costig variaces so that we ca determie performace more accurately.
1 4 4 m a a g e m e t a c c o u t i g : d e c i s i o a d c o t r o l t u t o r i a l Key Terms time series aalysis tred seasoal variatios extrapolatio of data deseasoalised data liear regressio idex umbers the examiatio of historical data that occurs over time, ofte with the itetio of usig the data to forecast future data the uderlyig movemet i the data, oce seasoal ad radom movemets have bee stripped away regular variatios i data that occur i a repeatig patter usig iformatio about a kow rage of data to predict data outside the rage (for example i the future) data that has had seasoal variatios stripped away usig a mathematical formula to demostrate the movemet of data over time. This techique is sometimes used to help forecast the movemet of the tred a sequece of umbers used to compare data, usually over a time period Activities 4.1 Sales (i uits) of a product are chagig at a steady rate ad do t seem to be affected by ay seasoal variatios. Use the data give for the first three periods to forecast the sales for periods 4 ad 5. Period 1 2 3 4 5 Sales (uits) 212,800 210,600 208,400
s t a t i s t i c a l t e c h i q u e s 1 4 5 4.2 Sales (i uits) of a product are chagig at a broadly steady rate ad do t seem to be affected by ay seasoal variatios. Use the average chage i the data give for the first five periods to forecast the sales for periods 6 ad 7. Period 1 2 3 4 5 6 7 Sales (uits) 123,400 123,970 124,525 125,085 125,640 4.3 The table below shows the last three moths cost per kilo for material Beta, together with estimated seasoal variatios: Moth Ja Feb March April May Actual Price 6.80 6.40 7.00 Seasoal Variatio +0.40 0.10 +0.40 0.30 0.40 Tred Calculate the tred figures for Jauary to March, ad extrapolate them to April ad May. Forecast the actual prices i April ad May 4.4 Computer modellig has bee used to idetify the regressio formula for the mothly total of a specific idirect cost as Y = 13.20 x + 480.00 Where y is total mothly cost Ad x is mothly productio i uits Calculate the total mothly cost whe output is 500 uits, ad 800 uits State whether the total cost behaves as a Fixed cost, or Variable cost, or Semi-variable cost, or Stepped cost.
1 4 6 m a a g e m e t a c c o u t i g : d e c i s i o a d c o t r o l t u t o r i a l 4.5 The regressio formula for mothly sales of a certai product (i uits) has bee idetified as: Y = 1,200 + 13X Where Y is total mothly sales, ad X is the moth umber. Jauary 20X9 was moth 30 Forecast the mothly sales i August 20X9 4.6 A compay has sales data that follows a 3 period cycle. The sales uits show i the table below have bee compiled from actual data i periods 30 to 36. Period Actual Data 3 Poit Movig Seasoal Averages (Tred) Variatios 30 3,500 31 3,430 32 3,450 33 3,530 34 3,460 35 3,480 36 3,560 37 38 39 40 Complete the table to show your resposes to the followig: (a) (b) Usig a 3 poit movig average, calculate the tred figures ad the seasoal variatios for periods 31 to 35 Extrapolate the tred to periods 37 to 40, ad usig the seasoal variatios forecast the sales uits for those periods
s t a t i s t i c a l t e c h i q u e s 1 4 7 4.7 The table below shows details of 5 urelated materials. Complete the blak parts of the table. Show all figures to 2 decimal places. Material Old Price New Price New price as idex % icrease umber with old price as base i price A 2.13 2.16 B 10.25 11.00 C 3.60 3.75 D 240.00 105.00 E 68.00 124.00 4.8 The table below shows details regardig purchases of a specific material. Complete the table to show the actual cost per kilo (to the earest pey) ad create a idex based o the cost per kilo with Jauary as the base, to the earest whole umber. Jauary February March Total cost 20,000 24,000 25,000 Total quatity 2,000 kilos 2,200 kilos 2,140 kilos Cost per kilo Cost idex 4.9 The stadard price for material M was set at 3.00 per kilo, based o a price idex of 155.3. Durig April, 15,000 kilos of material M costig 58,500 was purchased ad used. The price idex i April was 179.7 Complete the followig table, showig amouts to the earest : Amout Adverse / Favourable Material price variace Part of variace explaied by chage i idex