GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.

Transcription

1 GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all the umbers ad divide by how may there are. b) Media - Put all the umbers i order ad fid the middle oe. If there are two, the fid the mea of them. c) Mode - The umber that appears most ofte (there ca be more tha oe). 4) How to calculate the rage: The differece betwee the biggest umber ad the smallest umber. 5) How to use a frequecy table to fid averages: NUMBER TALLY FREQUENCY NUMBER x FREQUENCY Total - 8 Total - 13 MEAN - 13 divided by 8 gives 1.65 MEDIAN - workig dow the table gives ad, the mea of which is. MODE - it is clear that is the mode. RANGE = 1 6) How to use multiple bar charts ad compoet bar charts. 7) How to use grouped data. 8) How to draw a histogram, ad a frequecy polygo (lies to joi the middle of each bar together). 9) How to calculate averages for grouped data:

2 GROUP MID-POINT FREQUENCY MID-POINT x FREQUENCY > 0 <= > 50 <= > 100 <= ,375 Total - 5 Total - 15 MEAN - estimated at 15 / 5 = 85 MODAL GROUP - > 100 <= 150 MEDIAN - the umber will be i the > 50 <= 100 group. The media is the 13 th umber, ad there are the 7 th up to the 14 th umbers i this group. There are 8 umbers, so the 13 th umber will be 7/8 ito the group. It covers 50, so 7/8 of 50 = This is the estimated media. 10) How to write out questioaires. 11) That a sample ca be used i a survey whe you caot ask everyoe. It should be selected at radom, for if groups of people are ot icluded the it will be biased. The bigger a sample is, the better. If it is ot possible to use radom samplig, the systematic samplig should be used. This is whe the ames are i a list, ad you choose, for istace, every 5 th perso o the list. 1) That the expected frequecy is the amout of times you would expect somethig to happe, e.g. if you tossed a coi 100 times, you would expect 50 heads ad 50 tails. 13) That the relative frequecy is the relative umber of times you would expect somethig to happe, e.g. you would expect a coi to show heads ½ of the time ad tails ½ of the time whe it is tossed. This is the probability of a evet happeig, ad the sample space is the list of evets that could happe (i.e. heads or tails). 14) That the absolute error of a measuremet is the maximum error that ca be made, e.g. a measuremet to the earest millimetre has a absolute error of ½ a millimetre. The relative error is the absolute error divided by the measuremet take, ad the percetage error is the relative error times ) That data ca be either quatitative (whe it is umbers) or qualitative (whe it is somethig else, like a descriptio). 16) Quatitative data ca be either discrete (whe it is oly certai umbers ad ca t be i betwee them) or cotiuous (whe it ca be absolutely ay umber ad to ay degree of accuracy, like a measuremet for istace). 17) That you must use the class boudaries of data to fid the mid-poit of each group. 18) That to estimate the mode with grouped data, you eed to fid the modal group o the histogram, ad draw a lie from the top left had corer of the bar to the top left

3 had corer of the bar o its right, ad do the same for the top right had corer ad the bar o the left. You ca the draw a lie straight dow to the x-axis, ad take the readig as a estimate of the mode. 19) How to draw a cumulative frequecy curve. 0) That to fid the media o a cumulative frequecy curve, you fid the middle value o the y-axis, draw a horizotal lie to the curve, the a vertical lie straight dow to the x-axis to take a readig. 1) That the lower quartile has a y value of the total umber of values divided by 4, ad the upper quartile y value is the total umber of values mius the lower quartile y value. You ca draw these o the graph i the same way as the media. The iterquartile rage is the upper quartile mius the lower quartile, ad it shows how spread out the data is. ) That the semi-iterquartile rage is half the iterquartile rage. Sometimes data is split ito deciles (10 equal parts) or percetiles (100 equal parts) istead. 3) How to draw a scatter graph ad lie of best fit. 4) That the lie of best fit ca show the correlatio of the data. If it is slopig upwards away from the origi, the it is positively correlated. If it is slopig the other way, the it is egatively correlated. If you caot draw a lie of best fit, the the data is ucorrelated. 5) That if you wat to fid a value beyod the lie of best fit, the it is called extrapolatig. Fidig a value betwee the kow values is called iterpolatig. 6) That a stratified radom sample is whe you split people for a survey ito groups, ad choose people at radom from each group. The umber chose though, depeds o the size of the group. 7) That quota samplig is whe you choose certai types of people to ask. It is ot radom though, so it ca be biased. 8) That a evet is aythig that happes whe you are calculatig probabilities. Several evets make up a outcome. 9) That evets are idepedet whe they do t affect each other. Evets are depedat if they affect each other. 30) How to draw a tree diagram.

4 31) That the probability of somethig happeig is the umber of your outcomes over the total umber of outcomes. 3) That evets are mutually exclusive if they have o poits i commo, ad evets that have poits i commo are o-mutually exclusive. 33) That you ca use special symbols to represet umbers i statistics: x - represets all the data collected. - stads for the umber of items collected. - (sigma) meas to add all the umbers up. x - (x bar) represets the mea. e.g. x x = 34) That the deviatio from the mea is the umber mius the mea (x - x). The variace of a set of umbers shows how spread out they are: variace = ( x x) 35) That the stadard deviatio ca be used to measure the spread of data. It is useful because it takes all the umbers ito accout: stadard deviatio = ( x x) 36) That the mea deviatio ca also be used, but it does t take ito accout the sig, ad does t do ay squarig. It uses the modulus ( ) to show that it does t matter what the sig is: mea deviatio = x x 37) That to compare two sets of data which are t from the same experimet, you eed to scale oe or both of the sets of data. You first eed to stadardise the data, the you scale it: x x stadardise - s = σ scale - x + ( sσ )

5 38) That the frequecy distributio shows all the possible results, ad the frequecy of each result. 39) That the shape of the frequecy distributio ca be described: SYMMETRICAL - has the mode at the cetre. BIMODAL - has two peaks. SKEW - ot symmetrical. POSITIVELY SKEWED - skewed towards the y-axis. NEGATIVELY SKEWED - skewed away from the y-axis. NORMAL DISTRIBUTION - a special curve with few high ad low values, but with the highest frequecies aroud the middle. 40) How to draw a box plot, ad a stem ad leaf diagram. 41) That to fid the equatio of a lie of best fit, you eed the gradiet of the lie, ad the y-itercept of the lie. The gradiet ca be foud by drawig a lie horizotally across from the y-axis, the goig vertically up or dow agai, to reach the x-axis. The legth of the vertical lie divided by the legth of the horizotal lie gives the gradiet. The y-itercept is the y-coordiate of the poit the lie crosses the y-axis (it may eed to be exteded). This will give the followig equatio: y = (gradiet)x + (y-itercept) OR y = mx + c 4) That the gradiet is called the regressio coefficiet. The higher the regressio coefficiet, the steeper the lie. 43) How to use Spearma s coefficiet of Rak Correlatio (d is the differece betwee raks, ad is the umber of thigs placed i order): 6d ( 1) r = s 1 44) That diagrams ca be misleadig by chagig scales, ad by icreasig the width of bars / pictures, as well as the height. 45) That 5! meas 5 factorial, or 5 x 4 x 3 x x 1. The same applies for ay umber (e.g. 3! = 3 x x 1). 46) That to calculate the umber of ways of choosig several items out of a group of items (e.g. 5 out of 4), you use the followig formula ( is the umber to choose from, ad r is the umber to choose):

6 C r =! r! ( r)! 47) To fid the probabilities of several evets (all the same) happeig oe after aother i a experimet, you use the followig formula: r Probability (r successes) = C p ( 1 p) 48) That a time series is whe you collect data over a series of time. If a graph goes up ad dow quite regularly, it is called variatio. There are three types of variatio: SEASONAL - a regular chage that ca happe over ay period of time (weeks, moths, years etc.). CYCLICAL - a chage that keeps o happeig but is ot regular. RANDOM - a chage that is t seasoal or cyclical. 49) That movig averages are used to show the geeral tred by eveig out seasoal variatio, ad how to use movig averages. 50) That you ca exted the tred lie o a time series graph to predict what could happe i the future. After extedig the tred lie, you eed to add the seasoal variatio. To do this you look at the previous times whe seasoal variatio takes place, ad calculate how far above or below the tred lie you should be. You the fid the mea, ad add it to (or subtract it from) the x value of the tred lie at that poit. 51) That the birth rate is the umber of births for every 1000 i the populatio i oe year. It ca be calculated usig the followig formula: r r total umber of births i the year x 1000 total populatio at the middle of the year 5) How to stadardise the birth rate ad death rate for differet populatios. 53) That to calculate the weighted mea you use the followig formula: weighted mea = w1 score1 + w score + w3 score w + w + w ) That a idex umber is a percetage used to compare prices. A base price is used to compare other prices with, ad is give the idex umber of 100. The formula for calculatig other idex umbers is as follows: idex umber = price x 100

7 base price 55) That a retail price idex is a idex umber for lots of differet items. It ca be used to compare the cost of livig from oe year to aother. The formula to calculate a retail price idex is as follows (weightig = w, ad idex umber = d): retail price idex = ( w d) w