sum of all values n x = the number of values = i=1 x = n n. When finding the mean of a frequency distribution the mean is given by

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1 Statistics Module Revisio Sheet The S exam is hour 30 miutes log ad is i two sectios Sectio A 3 marks 5 questios worth o more tha 8 marks each Sectio B 3 marks questios worth about 8 marks each You are allowed a graphics calculator Before you go ito the exam make sureyou are fully aware of the cotets of theformula booklet you receive Also be sure ot to paic; it is ot ucommo to get stuck o a questio I ve bee there! Just cotiue with what you ca do ad retur at the ed to the questios you have foud hard If you have time check all your work, especially the first questio you attemptedalways a area proe to error J MS Explorig Data Measures of Cetral Tedecy The mea arithmetic mea of a set of data {x,x,x 3 x } is give by sum of all values x = the umber of values = i= x i x = Whe fidig the mea of a frequecy distributio the mea is give by xf xf = f If a set of umbers is arraged i ascedig or descedig order the media is the umber which lies half way alog the series It is the umber that lies at the positio Thus the media of {3,4,5,5} lies at the positio average of 4 ad 5 media = 45 The mode of a set of umbers is the umber which occurs the most frequetly Sometimes o mode exists; for example with the set {,4,7,8,9,} The set {,3,3,3,4,5,,,,7} has two modes 3 ad because each occurs three times Oe mode uimodal Two modes bimodal More tha two modes multimodal The mid-rage is give by the average of the miimum ad maximum values Mid-rage = x max x mi / wwwmathshelpercouk JMStoe

2 Advatages Disadvatages Mea The best kow average Greatly affected by extreme values Ca be calculated exactly Ca t be obtaied graphically Makes use of all the data Whe the data are discrete ca give a impossible figure 34 childre Ca be used i further statistical work Media Ca represet a actual value i the For grouped distributios its value ca data Ca be obtaied eve if some of the values i a distributio are ukow oly be estimated from a ogive Whe oly a few items available or whe distributio is irregular the media may ot be characteristic of the group Uaffected by irregular class widths ad uaffected by ope-eded classes Ca t be used i further statistical calculatios Not iflueced by extreme values Mode Uaffected by extreme values May exist more tha oe mode Easy to calculate Ca t be used for further statistical work Easy to obtai from a histogram Whe the data are grouped its value caot be determied exactly Measures of Spread The simplest measure of spread is the rage Rage = x max x mi The mea absolute deviatio from the mea is give by x x For example i the data set {4,5,7,8} the mea is, so the absolute deviatios are,,, so the mea absolute deviatio is 4 = 5 The sum of squares from the mea is called the sum of squares ad is deoted S xx = x x = x x For example give the data set {3,,7,8} the mea is ; x = = 58; so S xx = x x = 58 4 = 4 The mea square deviatio is defied: msd = S xx x = x The root mea square deviatio is defied: rmsd = Sxx x msd = = x The variace is defied: variace = S xx x = x The stadard deviatio s is defied: s = x Sxx variace = = x O graphical calculators from Casio the rmsd is give by xσ ad the sd by xσ Example: Give the set of data {5, 7, 8, 9, 0, 0, 4} calculate the stadard deviatio Firstly we ote that x = 9 x Sxx s = = x 5 = = = 8 = 884 Or we could have doe S xx = x x = = 4 wwwmathshelpercouk JMStoe

3 x Whe dealig with frequecy distributios such as, we could calculate the rmsd or the sd by writig out the data ad carryig out the calculatios as f above, but this is clearly slow ad iefficiet 3 To our rescue come formulae for rmsd ad sd that allow direct calculatio from the table They are rmsd = x f x x sd = f x Example: Calculate mea ad sd for the above frequecy distributio For easy calculatio we eed to add certai colums to the usual x ad f colums thus; x f xf x f = f = 5 xf = 75 x f = 7 So x = xf x = 75 5 = 3 ad s = f x = = 38 A item of data is a outlier if it is more tha two stadard deviatios from the mea ie outlier if x x > s It meas that some more ivestigatio is eeded to see if it eeds to be discarded 95% of the data lie withi two stadard deviatios ad 9975% lie withi three stadard deviatios assumig ormally distributed populatio Liear Codig Give the set of data {,3,4,5,} we ca see that x = 4 ad it ca be calculated that s = 58 3dp If we add 0 to all the data poits we ca see that the mea becomes 4 ad the stadard deviatio will be uchaged If the data set is multiplied by 3 we ca see that the mea becomes ad the stadard deviatio would become three times as large dp Combiig the above ideas we fid that give a data set x i ad we trasform it to create a ew data set y i = ax i b the the ew mea will be y = axb ad the ew stadard deviatio will be s y = as x This ca be used to make certai calculatios easier For example; x f y f Covert y = x 0 4, therefore x = 4y Oce we fid y ad s y we fid that x = 4y 0 ad s x = 4s y {,,,,,,,,,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5}!!! 3 Ideed, it would be early impossible if the frequecies were i the thousads wwwmathshelpercouk 3 JMStoe

4 3 Probability A idepedet evet is oe which has o effect o subsequet evets The evets of spiig a coi ad the cuttig a pack of cards are idepedet because the way i which the coi lads has o effect o the cut For two idepedet evets A & B PA ad B = PA PB For example a fair coi is tossed ad a card is the draw from a pack of 5 playig cards Fid the probability that a head ad a ace will result Phead =, Pace = 4 5 = 3, so Phead ad ace = 3 = Mutually Exclusive Evets Two evets which caot occur at the same time are called mutually exclusive The evets of throwig a 3 or a 4 i a sigle roll of a fair die are mutually exclusive For ay two mutually exclusive evets PA or B = PAPB For example a fair die with faces of to is rolled oce What is the probability of obtaiig either a 5 or a? P5 =, P =, so P5 or = = 3 No-Mutually Exclusive Evets Whe two evets ca both happe they are called omutually exclusive evets For example studyig Eglish ad studyig Maths at A Level are o-mutually exclusive By cosiderig a Ve diagram of two evets A & B we fid PA or B = PAPB PA ad B, PA B = PAPB PA B Tree Diagrams These may be used to help solve probability problems whe more tha oe evet is beig cosidered The probabilities o ay brach sectio must sum to oe You multiply alog the braches to discover the probability of that brach occurrig For example a box cotais 4 black ad red pes A pe is draw from the box ad it is ot replaced A secod pe is the draw Fid the probability of i two red pes beig obtaied ii two black pes beig obtaied iii oe pe of each colour beig obtaied iv two red pes give that they are the same colour Draw tree diagram to discover: R = 5/9 R = /0 B = 4/0 B = 4/9 R = /9 i Ptwo red pes = = = 3 ii Ptwo black pes = = 90 = 5 iii Poe of each colour = = 8 5 iv Ptwo reds same colour = /3 /3/5 = 5 7 B = 3/9 wwwmathshelpercouk 4 JMStoe

5 Coditioal Probability I the above example we see that the probability of two red pes is 3, but the probability of two red pes give that both pes are the same colour is 5 7 This is kow as coditioal probability PA B mea the probability of A give that B has happeed It is govered by PA B = PA ad B PB = PA B PB For example if there are 0 studets i a year ad 0 study Maths, 40 study Eglish ad 0 study both the Pstudy Eglish study Maths = Pstudy Maths & Eglish Pstudy Maths = 0/0 0/0 = A is idepedet of B if PA = PA B = PA B ie whatever happes i B the probability of A remais uchaged For example flickig a coi ad the cuttig a deck of cards to try ad fid a ace are idepedet because Pcuttig ace = Pcuttig ace flick head = Pcuttig ace flick tail = 3 4 Discrete Radom Variables The table below shows the probability distributio for the outcome X of a die PX = r r I geeral for ay evet, the probability distributio is of the form r r r r 3 r 4 r 5 r PX = r p p p 3 p 4 p 5 p The expected value of the evet is deoted EX or µ It is defied For example for a fair die EX = = 3 EX = µ = rpx = r The variace of a evet is deoted VarX or σ ad is defied VarX = σ = EX [EX] = EX µ = r PX = r µ So for the biased die with distributio r PX = r wwwmathshelpercouk 5 JMStoe

6 we fid that ad EX = 3 VarX = r PX = r µ = 3 = = 7 3 = 4 The other way of calculatig these quatities is by usig a table We will cosider the example of the bias die above r PX = r rpx = r r PX = r 3 3 = 3 3 = 3 = 3 = = = = = = 5 5 = = 3 = rpx = r = 3 So, as before EX = 3 ad VarX = 7 3 = 4 5 Further Probability r PX = r = 7 Factorials are defied! = May expressios ivolvig factorials simplify with a bit of thought For example!/! = Also there is a covetio that 0! = The umber of ways of arragig differet objects i a lie is! For example how may differet arragemets are there if 4 differet books are to be placed o a bookshelf? There are 4 ways i which to select the first book, 3 ways i which to choose the secod book, ways to pick the third book ad way left for the fial book The total umber of differet ways is 4 3 = 4! Several evets If there are 3 roads from A to B ad roads from B to C How may routes are there from A to C? A x y z B u v C The solutio to our problem is 3 = because the set of possible routes is x u y u z u x v y v z v I geeral if there are a ways for trial A to result, b ways for trial B to result ad c ways for trial C to result the there are a b c differet possible outcomes wwwmathshelpercouk JMStoe

7 Permutatios The umber of ways of selectig r objects from whe the order of the selectio matters is P r It ca be calculated by P r =! r! For example i how may ways ca the gold, silver ad broze medals be awarded i a race of te people? The order i which the medals are awarded matters, so the umber of ways is give by 0 P 3 = 70 I aother example how may words of four letters ca be made from the word CON- SIDER? This is a arragemet of four out of eight differet objects where the order matters so there are 8 P 4 = 8!/4! = 80 differet words Combiatios The umber of ways of selectig r objects from whe the order of the selectio does ot matter is C r It ca be calculated by C r =! r! r! For example i how may ways ca a committee of 5 people be chose from 8 applicats? Solutio is give by 8 C 5 = 8!/5! 3! = 5 I aother example how may ways are there of selectig your lottery umbers where oe selects umbers from 49? It does ot matter which order you choose your umbers, so there are 49 C = possible selectios wwwmathshelpercouk 7 JMStoe

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