EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 1- ALGEBRAIC TECHNIQUES TUTORIAL 3 - STATISTICAL TECHNIQUES

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1 EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 1- ALGEBRAIC TECHNIQUES TUTORIAL 3 - STATISTICAL TECHNIQUES CONTENTS Be able to apply algebraic techiques Arithmetic progressio (AP): irst term (a), commo dierece (d), th term e.g. a +( 1)d ; arithmetic series e.g. sum to terms, S a 1) d Geometric progressio (GP): irst term (a), commo ratio (r), th term e.g. ar -1 ; ar 1 a geometric series e.g. sum to terms, S, sum to iiity S r 1 1 r solutio o practical problems e.g. compoud iterest, rage o speeds o a drillig machie Complex umbers: additio, subtractio, multiplicatio o a complex umber i Cartesia orm, vector represetatio o complex umbers, modulus ad argumet, polar represetatio o complex umbers, multiplicatio ad divisio o complex umbers i polar orm, polar to Cartesia orm ad vice versa, use o calculator Statistical techiques: review o measure o cetral tedecy, mea, stadard deviatio or ugrouped ad grouped data (equal itervals oly), variace It is assumed that the studet has completed the module MATHEMATICS FOR TECHNICIANS. D.J.Du 1

2 1. VARIANCE AND STANDARD DEVIATION Stadard deviatio is a more diicult cocept tha the others we've covered. To uderstad this cocept, it ca help to lear about what statisticias call ormal distributio o data. Most statistical samples produce a lot o results aroud the mea. Further away rom the mea there are ot so may results. The requecy distributio graph produces a bell shaped curve called the ormal distributio. However, some are more ormal tha others ad some times there are a lot more results close to the mea tha others. Cosider the three cases show o the diagram. The distributio show by graph B has most o the examples i the set o data close to the "average," while those i graph A are more widely spread rom the average. I reality ew examples ted to oe extreme or the other. The stadard deviatio is a statistic that tells you how tightly all the various examples are clustered aroud the mea i a set o data. Whe the examples are tightly buched together ad the bell-shaped curve is steep, the stadard deviatio is small (graph B). Whe the examples are spread apart ad the bell curve is relatively lat, that tells you that you have a relatively large stadard deviatio (graph A). Let s irst try to uderstad graphically what a stadard deviatio represets... Oe stadard deviatio away rom the mea i either directio o the horizotal axis () accouts or somewhere aroud 68 percet o the samples. Two stadard deviatios away rom the mea (4), accout or roughly 95 percet o the samples. Three stadard deviatios (6) accout or about 99 percet o the samples. I this curve were latter ad more spread out, the stadard deviatio would have to be larger i order to accout or the 68 percet so that's why the stadard deviatio ca tell you how spread out the examples i a set are rom the mea. This is useul i mauacturig as it tells us a lot about the quality o what you are makig ad how the equipmet used i the process is behavig. So i or example you were moitorig the values o electrical resistors or the diameter o pistos beig produced by machiery, a small stadard deviatio will tell us that they are beig produced very accurately ad close to the mea. Suppose all the samples betwee 4 ad 6 are rejects. The larger the value o, the more rejects. Also i the mea o the sample is movig as time goes o, it meas that more samples will be rejected o oe side tha the other ad idicates that the somethig i the machie (e.g. the gridig wheel) is wearig away. D.J.Du

3 . CALCULATION OF STANDARD DEVIATION or UNGROUPED DATA Ugrouped data is preseted i a table listig the value o each sample. I the umber o samples is large, this becomes a large table but it is probably best to use this method with small umbers o samples. Stadard deviatio = Square root o the mea o the variables squared. The VARIANCE is deoted S = x x S σ = umber o samples 1 You might id it better to arrage your tables i colums rather tha rows. Let s look at aother example. WORKED EXAMPLE No. 1 The ollowig is a table o Resistor values oud i a batch. Calculate the mea ad the stadard deviatio. Sample Resistace (Ohms) Dierece rom mea Diereces squared Totals Mea = 108/10 = 10.8 The sum o the squares o the diereces (or deviatios) rom the mea, 3.6, is ow divided by the total umber o observatio mius oe, to give the variace. VARIANCE x x S σ Fially, the square root o the variace provides the stadard deviatio: =.6 ½ = Ohms D.J.Du 3

4 SELF ASSESSMENT EXERCISE No The hardess o te steel samples was measured ad the results were as ollows. Sample Hardess Calculate the mea ad the stadard deviatio. Aswer 95.1 ad 3.9. The thickess o 0 steel strips was measured i mm ad tabulated as show. Sample Thickess Calculate the mea ad the stadard deviatio. Aswer 19.9 ad D.J.Du 4

5 3. CALCULATION OF STANDARD DEVIATION or GROUPED DATA Grouped data is preseted i tables showig the bads ad the requecy ad is more likely to be used with large umbers o samples. Cosider a ormal distributio curve. The mea occurs at the middle. The deviatio rom the mea at ay poit is d. Next cosider the graph o d plotted agaist ad urther the graph o d plotted agaist. O this last graph we id the mea d as ollows. 1 3 ) The area o the graph may be oud rom the mid-ordiates. A w(d d d...d ) w(d d d...d The mea height o the graph is the variaces w is the width o each strip = / ad is the umber o strips. The mea height is the (d1 d d3...d ) (d1 d d3...d ) S d I geeral S For reasos ot explaied here, -1 is ote used istead o o the bottom lie. d is the deviatio d = x - x x x S σ is the stadard deviatio. 1 It ca be show that the ormula simpliies to σ 1 x 3 x D.J.Du 5

6 WORKED EXAMPLE No. The ollowig is a grouped set o data or visits made to the doctor by a sample o childre. Visit to Doctor No. o Childre Total Visits Cumulative x x x x Totals = = 140 x = 455 x =1697 Mea umber o visits = 455/140 = 3.5. σ σ 1.5 x x x x Some texts give the ormula as σ σ 1.5 This does ot make much dierece so log as the total umber o samples is very small. D.J.Du 6

7 WORKED EXAMPLE No. 3 The hardess o 143 samples o steel is measured ad grouped ito bads as show. Calculate the mea ad stadard deviatio. The igures o 17.5 ad 1.5 result rom oe sample beig exactly 91 uits ad so hal is allocated to each bad. Rage mid poit req. x acc x x x Totals Mea = 13575/143 = σ σ.99 x x It is o iterest to ote that i this populatio, we get a very dieret aswer usig the other ormula. x σ 1 σ 8.51 x D.J.Du 7

8 SELF ASSESSMENT EXERCISE No. 1. The accuracy o 100 istrumets was measured as a percetage ad the results were grouped ito bads o 1% as show. Calculate the mea ad the stadard deviatio. Rage Mid req Aswers ad.74%. The breakig stregths o 150 spot welds was measured i Newto ad grouped ito bads o 0 N as show. Rage Calculate the mea ad the stadard deviatio. (Aswers N ad 9.04 N) D.J.Du 8