AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if the data values are very varied Mea : importat as it uses all the data values Disadvatage affected by extreme values If the data is grouped use the mid-poit of each group as your x Media : the middle value whe the data are i order For data values the media is the + 1 th value 2 Not affected by extreme values For 10 values the media will be the 5½ th value halfway betwee the 5 th ad the 6 th values Rage biggest value smallest value - greatly affected by extreme values Iterquartile Rage Upper quartile Lower quartile - measures the spread of the middle 50% of the data ad is ot affected by extreme values 3 4 5 7 8 9 9 LQ M UQ IQR = 9-4 = 5 2 3 4 4 5 7 8 9 LQ M UQ IQR = 7.5 3.5 = 4 Stadard Deviatio Deviatio from the mea is the differece from a value from the mea value The stadard deviatio is the average of all of these deviatios Formulas to work out stadard deviatio are give i the q SCALING DATA Additio if you add a to each umber i the list of data : New mea = old mea + a New media = old media + a New mode = old mode + a Stadard Deviatio is UNCHANGED Multiplicatio - If you multiply each umber i the list of data by b : New mea = old mea b New media = old media b New mode = old mode b New Stadard Deviatio = old stadard deviatio b
PROBABILITY Outcome : each thig that ca happe i a experimet Sample Space : list of all the possible values q NOTATION A Ç B A ad B both happe A È B either A or B or both happe C' C does t happe C P(C ) = 1 P(C) P(A È B) = P(A) + P(B) P(A Ç B) Mutually Exclusive Evets two or more evets that caot happe at the same time P(A È B) = P(A) + P(B) Idepedet Evets the outcome of oe evet does ot affect the outcome of aother P( B ) = P(A) P(B) A Ç Coditioal Probability : whe the outcome of the first evet affects the outcome of a secod evet, the probability of the secod evet depeds o what has happeed P(B/A) meas the coditioal probability of B give A P(B/A) = P(A Ç B) P(A) A Ç so P( B )= P(A) P(B/A) q If the questio states that the evets are idepedet the a tree diagram might be a good idea multiply alog the braches the add the appropriate combiatios together q Probability of at least 1 = 1 Probability of oe q If you are asked to fid probabilities usig data i a table work out the row/colum totals before you start
BINOMIAL DISTRIBUTION q A questio is biomial if: Probability of a evet happeig is give (p) Number of people/trials/objects chose give () q EQUALS or EXACTLY use the formula P(x=r) Make sure you write it out with the values substituted i C r p r (1 p) r Check that your powers add to make (umber of trials) from your calculator make sure you write dow this value q MORE/LESS THAN/AT LEAST use tables Remember tables give less tha or equal to Make sure you list the umbers ad idetify which oes you eed to iclude P(X>5) 0 1 2 3 4 5 6 7 8 9 = 1 P(X 5) P(X<5) 0 1 2 3 4 5 6 7 8 = P(X 4) q MEAN ad VARIANCE Mea = p Variace = p(1-p) stadard deviatio = p(1 p) q ASSUMPTIONS Idepedet evets with a fixed probability of success Radomly selected q COMPARISONS You may be asked to calculate the mea ad stadard deviatio of a biomial distributio ad compare them to the mea ad stadard deviatio of a sample (table of results) - if both the meas are approximately the same AND both stadard deviatios (or variaces) are approximately equal the you ca say that biomial model appears to fit the data ad that it must be idepedet, radom observatios.
NORMAL DISTRIBUTION FINDING PROBABILITIES q State the mea ad variace (stadard deviatio) q Stadardise to fid the z value z = x mea stadard deviatio q Sketch a graph ad shade i the area to represet the probability P(z <1.2) P(z >-0.8) 1.2-0.8 q Use the table to fid the probability - take care with egative z-values P(z > -0.8) = P(z < 0.8) -0.8 0.8 P(z < -0.8) = 1 P(z < 0.8) -0.8 0.8 q ALWAYS CHECK YOUR ANSWER WITH YOUR GRAPH if your shaded area is more that ½ ad your aswer is 0.4 (for example) you kow you have goe wrog somewhere!! WORKING BACKWARDS to fid the mea/stadard deviatio or both (simultaeous equatios) q State the probability you kow ad sketch a graph P(X < 34) = 0.95 0.05 Stadard deviatio = 8 q Use tables to fid the appropriate z value q Write dow the equatio used to stadardise with all of the kow values substituted q Rearrage to fid the mea 34 1.6445 = 34 mea 8
SAMPLES ad PROBABILITIES If you are asked to calculate probabilities ivolvig samples remember to divide the (populatio) stadard deviatio by the square root of the sample size whe you stadardise. z = x mea s or z = x mea variace ESTIMATION estimatig the populatio mea from a sample mea ad fidig a cofidece iterval If a radom sample of size is take from a ormal populatio ad the sample mea x is foud, the the 95% cofidece iterval of the populatio mea is give by where 2 Z value x 1.96 www.mathsbox.org.uk s is either the populatio variace (if give) or a ubiased estimate of the populatio variace (foud from the sample see below) s 2, x + 1.96 s 2 q CASE 1 : Stadard deviatio or Variace of the populatio is stated i the questio - from the data you oly eed to calculate the mea - use your tables to fid the appropriate z value - write out the above expressios for the cofidece itervals with all your values substituted i - calculate the two values for your cofidece itervals ad state clearly (3 sf) CHECK your aswer add the two aswers together ad divide by 2 this should be the sample mea!!! q CASE 2 : Stadard deviatio or Variace of the populatio is UNKNOWN - you will eed to use the data to calculate the variace of the sample ad the a ubiased estimate of the populatio variace - your calculator will give you value of the stadard deviatio for the sample you have etered s x - square this to calculate the sample variace A ubiased estimate of the populatio variace is Sample variace 1 Use this as the value of s 2 i your cofidece iterval whe substitutig the values i q INTERPRETATION a 95% cofidece iterval tells us that if we took the same size sample 100 times the 95 of the cofidece itervals we would calculate should cotai the TRUE populatio mea.
CENTRAL LIMIT THEOREM you oly eed to use the cetral limit theorem if you are ot told that the sample is selected from a populatio which is Normally Distributed The theorem cocers the distributio of the sample meas ad as log as the sample size is large eough (greater tha 30) the the sample meas will be ormally distributed ad so we ca calculate cofidece itervals REGRESSION fidig the equatio of the lie of best fit least squares y = a + bx Gradiet the chage i y for each uit chage i x e.g for every 1 degree rise i temperature sales icrease by b Itercept - the value of y whe x is zero. e.g. Whe the temperature is 0 C ice cream sales are 10 q If you are asked to iterpret the values of a ad b, make sure you discuss it i the cotext of the questio NOT i terms of x ad y (see examples above) q If you are give a table of values use your calculator to fid a ad b I your workigs state a =. b =.. ad show your values substituted ito y = a + bx q If you are calculatig a ad b usig the formulae make sure you use the formula book showig how you substituted the values i Always work out b first Use b ad the meas of x ad y to work out a a = (mea of y) - b(mea of x) q TO PLOT THE REGRESSION LINE choose 2 differet values of x use your equatio y=a + bx to work out the predicted y-values - plot the two poits ad joi with a straight lie q RESIDUAL = OBSERVED(actual value) PREDICTED(usig equatio y= a + bx) - the smaller the residuals the greater the accuracy of the lie of best fit i predictig values - sometimes a average residual ca be used to make predictios usig the lie of best fit e.g if a idividual has a average residual of 5 the to predict for this particular perso usig the lie, 5 should be added to the value predicted usig the equatio. q RELIABILITY OF PREDICTIONS - Iterpolatio predictig usig a x-value withi the rage of x-values used to calculate the a ad b cosidered to be a reliable predictio - Extrapolatio predictig usig a x-value outside of the rage of x-values used to calculate a ad b UNRELIABLE estimate - because you are assumig that the liear tred cotiues idefiitely use your commo sese to explai why this may be icorrect - watch out for NEGATIVE (or urealistic) y values which may result for the x values suggested agai use your commo sese to explai why this is urealistic
q SCALING either the x or the y values will chage the equatio e.g y = 0.5 1.2x If the x values are doubled the the equatio becomes y = 0.5 1.2(2x) y = 0.5 2.4x CORRELATION If 5 is added o to each of the y values the the equatio becomes y + 5 = 0.5 1.2x y = - 4.5 1.2 x (always rearrage to get y = a + b x) The Product Momet Correlatio Coefficiet : is a umerical measures of the stregth ad type of correlatio deoted by r ad will lie i the rage -1 r 1 q idicates how well the data, whe plotted i a scatter graph, fits a straight lie patter q NOT APPROPRIATE if the data does ot follow a liear patter whe plotted (straight lie) so scatter graph is eeded to check this q If you are give a table of values use your calculator to fid r (If you have time it s a good idea to check that you have etered your values correctly) q If you are calculatig usig summary values -make sure you use the formula book showig how you substituted the values ito the formula INTERPRETING r make sure you do this i the cotext of the questio (ot just positive correlatio) e.g. There appears to be a fairly strog relatioship betwee temperature ad ice cream sales, higher temperatures appear to correspod to higher values of ice-cream sales ad vice versa. q Scalig data a liear trasformatio or scalig of oe or both of the variables will ot affect the correlatio coefficiet all of the poits will stay i the same positio RELATIVE to each other TAKE CARE q Not all correlatio will be liear For this data, the correlatio coefficiet is close to 0 This does ot mea that there is o correlatio but simply mea that there is o liear correlatio (patter appears to be quadratic) q Spurious Correlatio A strog correlatio betwee 2 variables does ot mea that oe thig causes the other high marks i a maths exam do ot ecessarily cause high marks i a Statistics exam, they are likely to both be depedet o a commo third variable : the studets mathematical ability q Outliers Oe or two outliers ca have a dramatic effect o a correlatio coefficiet