OCR Statistics 1. Working with data. Section 3: Measures of spread
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1 Notes ad Eamples OCR Statistics 1 Workig with data Sectio 3: Measures of spread Just as there are several differet measures of cetral tedec (averages), there are a variet of statistical measures of spread. These otes cotai sub-sectios o: The rage Quartiles ad the iter-quartile rage Bo ad whisker plots Shapes of distributios Idetifig outliers usig quartiles Cumulative frequec tables ad curves Variace ad stadard deviatio The alterative form of the sum of squares Variace ad stadard deviatio usig frequec tables Usig stadard deviatio to idetif outliers Usig codig to calculate stadard deviatio The rage For a set of data, rage = highest item lowest item This is straightforward to calculate, but is highl sesitive to etreme values. For eample, cosider this set of marks for a maths test: {45, 50, 43, 49, 5, 58, 48, 10, 50, 8, 56, 40, 47, 39, 51} The rage of the data is 8 10 = 7 marks, but this does ot give a good measure of the spread, as most of the marks are i the rage Quartiles ad the iter-quartile rage Oe wa of refiig the rage so that it does ot rel completel o the most etreme items of data is to use the iterquartile rage. Iterquartile rage = upper quartile lower quartile. The upper quartile is the media of the upper half of the data, ad the lower quartile is the media of the lower half of the data. MEI, 3/06/09 1/14
2 For a large data set, 5% of the data lie below the lower quartile, ad 75% of the data lie below the upper quartile. The iterquartile rage measures the rage of the middle 50% of the data. For small sets of data, ou use a procedure for placig the lower ad the upper quartile, similar to that used for placig the media. Eample 1 (i) Fid the iterquartile rage of the set of marks below from a test take b 15 studets (ii) Oe studet was abset ad took the test the followig week, scorig 59. Fid the ew iterquartile rage. Solutio (i) First arrage the data i order of size: There are 15 items of data, so the media is the 8 th item, which is 49. Discard this The lower quartile is the media of the lower 7 marks, which is 43. The upper quartile is the media of the upper 7 marks, which is 5. So the iterquartile rage is 5 43 = 9. (ii) The ew set of data has 16 items. For a eve umber of data items, the media falls betwee two items of data, so there is o data item to discard: Media 49.5 The lower quartile is the media of the lower 8 marks, which is 44. The upper quartile is the media of the upper 8 marks, which is 54. The iterquartile rage = = 10 MEI, 07/06/10 /14
3 Bo-ad-whisker plots The media ad quartiles ca be displaed graphicall b meas of a boad-whisker plot, or boplot. This gives a etremel useful summar of the data, ad ca be used to compare sets of data. I this diagram, a bo is draw from the lower to the upper quartile, ad a lie draw i the bo showig the positio of the media. Whiskers eted from the lowest value to the highest: Lowest value Lower quartile Media Upper quartile Highest value Draw to scale Eample Compare the followig sets of data usig their bo ad whisker plots. The represet marks out of 100 for two classes. Class A Class B Solutio The rages of marks are similar, but class A has a lower iter-quartile rage tha class B, which suggests that the majorit of the marks are less spread out for Class A. The media ad quartiles for class A are higher tha those for class B, so o average class A did slightl better o the test. Shapes of distributios The shapes of some histograms for data ca be characterised as follows: Smmetrical Positivel skewed Negativel skewed MEI, 07/06/10 3/14
4 Smmetrical datasets have roughl equal amouts of data either side of a cetral value. Positivel skewed data have greater amouts of data clustered aroud a lower value. Negativel skewed data have greater amouts of data clustered aroud a higher value. Skew ca be see if data is displaed i stem ad leaf diagrams or histograms. Boplots ca also be used to detect skew i the data. The diagram below shows the histogram for a positivel skewed dataset, together with its boplot super-imposed. f.d. You ca see from the boplot that the media is closer to the lower quartile tha the upper quartile, or Upper quartile media > media lower quartile I cotrast, here is a egativel skewed dataset: f.d. MEI, 07/06/10 4/14
5 Here, the boplot shows that the media is closer to the upper quartile tha the lower quartile, so Upper quartile media < media lower quartile. Idetifig outliers usig quartiles Oe defiitio of a outlier uses the quartiles ad iterquartile rage. A outlier ca be idetified as follows (IQR stads for iterquartile rage): a data which are 1.5 IQR below the lower quartile; a data which are 1.5 IQR above the upper quartile. For eample, here is the dataset from Eample 1(ii) Lower quartile 44 Media 49.5 Upper quartile 54 The iterquartile rage is = IQR = = IQR below the lower quartile = = 9, so 10 is a possible outlier. 1.5 IQR above the upper quartile = = 69, so 8 is a possible outlier. The Geogebra resource Boplots ad outliers ca be used to eplore the media ad quartiles, ad ivestigate outliers usig the media ad iterquartile rage. Cumulative frequec tables ad curves Cumulative frequec curves are useful for estimatig the quartiles ad the iter-quartile rage of a large data set. The et eample was also used i sectio to fid the media. Here the iterquartile rage is foud as well. MEI, 07/06/10 5/14
6 Eample 3 Estimate the media ad iterquartile rage of the followig dataset, which gives the mass of 100 eggs: Solutio Mass, m (g) Frequec 40 m < m < m < m < m < m < m < m < 80 0 Mass, m (g) Frequec Mass Cumulative frequec m < m < 45 4 m < m < m < m < m < m < 60 m < m < m < m < m < m < m < The cumulative frequec curve is draw below: c.f mass (kg) 5 of the eggs lie below the lower quartile, show b the ellow lie. 50 of the eggs lie below the media, show b the red lie. 75 of the eggs lie below the upper quartile, show b the blue lie. MEI, 07/06/10 6/14
7 Media = 58 Lower quartile = 53 Upper quartile = 66 Iterquartile rage = = 13. Variace ad stadard deviatio Cosider a small set of data: {0, 1, 1, 3, 5} The mea of this data is give b The deviatio of a item of data from the mea is the differece betwee the data item ad the mea, i.e.. The set of deviatios for this set of data is: {, 1, 1, 1, 3} These deviatios give a measure of spread. However, there is o poit i just addig them up, because their sum is alwas zero! Istead, square each deviatio ad add them up. The sum of their squares is deoted S : For the set of data above: S ( ) ( 1) ( 1) I geeral: ( i ) or i1 S S ( ) Dividig this quatit b, the umber of data, gives the variace. The square root of this quatit is called the stadard deviatio. For the set of data above: 16 Variace S 3. 5 S Stadard deviatio I geeral: ( ) Variace MEI, 07/06/10 7/14
8 Stadard deviatio ( ) Eample 4 Calculate the variace ad stadard deviatio of the data {0,, 3, 6, 9} Solutio S (0 4) ( 4) (3 4) (6 4) (9 4) Variace 10 5 Stadard deviatio The alterative form of the sum of squares Whe the mea does ot work out eatl, the deviatios will also be difficult to work with. I this case, it is easier to work with a alterative formula for S : S ( ) For the first dataset {0, 1, 1, 3, 5}: S as before. The variace ad stadard deviatio ca ow be writte i the alterative forms: Variace Stadard deviatio Eample 5 Calculate the variace ad stadard deviatio of the data set {1, 1,, 3, 3, 3, 4}. Sice the mea is ot a roud Solutio umber, it is easier to use the secod forms of the formulae Variace Alwas do the whole calculatio at oce. Do ot use a rouded versio of the mea! MEI, 07/06/10 8/14
9 Stadard deviatio For large sets of data, ou are sometimes give a summar of the data: the values of, ad. Eample 6 A set of sample data is summarised as: = Fid (i) (ii) the mea the stadard deviatio 15. Solutio 140 (i) (ii) stadard deviatio Variace ad stadard deviatio usig frequec tables I sectio, ou saw how the formula for the mea ca be adapted for use with data give i a frequec table: f f I the same wa, the formulae for the measures of spread ca be adapted for data give i a frequec table. Be careful: f² meas square, the multipl b f. S f S Variace f Stadard deviatio S f It is ofte coveiet to set out the calculatio i colums, as show i the followig eample: MEI, 07/06/10 9/14
10 Eample 7 The table below shows the umber of occupats of each house i a small village. Number of occupats Frequec Total 194 Fid the mea ad stadard deviatio of the umber of occupats. Solutio Mea f f ² f² f 194 f 690 f 938 f f f ( 194) Stadard deviatio I practice, of course, calculatios like these ca be carried out much more easil usig a spreadsheet, or b eterig the data ito a calculator (most calculators allow ou to eter either raw data or frequecies, ad the will calculate the various statistical measures for ou). You ca also look at the PowerPoit presetatio Variace ad stadard deviatio, which shows fidig the variace ad stadard deviatio of raw data ad data preseted i a frequec table. MEI, 07/06/10 10/14
11 For practice i fidig stadard deviatio, tr the iteractive questios Mea ad stadard deviatio. If the data is grouped, the ou must use mid-iterval values, just as ou did i estimatig the mea. Remember that the results for measures of spread will also be estimates usig this method. Eample 8 Estimate the mea ad stadard deviatio of the data with the followig frequec distributio: Solutio Weight, w, (grams) Frequec, f 0 w < w < w < w < w < 50 4 w Mid-iterval f f ² f² value, 0 w < w < w < w < w < f 30 f 760 f Mea f ( 30 ) Stadard deviatio Usig stadard deviatio to idetif outliers Stadard deviatio ca be used to idetif outliers, usig the followig rule: All data which are over stadard deviatios awa from the mea are idetified as outliers. Eample 9 Use the stadard deviatio to idetif a outliers i the followig set of data: MEI, 07/06/10 11/14
12 Solutio = S S 7714 Stadard deviatio = stadard deviatios below the mea is stadard deviatios above the mea is So a outliers are below 4.0 or above The ol value outside this rage is 99; so this is the ol outlier. The Geogebra resource Histograms, mea ad stadard deviatio ca be used to eplore the shapes of histograms, ad ivestigate outliers usig the mea ad stadard deviatio. Usig codig to calculate stadard deviatio It is sometimes possible to simplif the calculatios of variace ad stadard deviatio b codig the data, i the same wa as for the mea. You ca trasform the data usig a liear codig: a b a You ca udo this codig: b Sice each data item has bee trasformed usig this codig, the mea of the data udergoes the same trasformatio. So the mea of the coded data,, is related to the mea of the origial data,, b the equatio a b. Sice stadard deviatio is a measure of spread, the addig a to all the items of data does ot affect the stadard deviatio. However, multiplig all the data items b b makes the data b times more spread out tha previousl. So the stadard deviatio of the coded data, s, is related to the stadard deviatio of the origial data, s, b the equatio s bs. For eample, the data set {30, 50, 0, 70, 40, 0, 30, 60} could be simplified b dividig all the data b 10. This meas usig the codig. 10 which gives the ew data set {3, 5,, 7, 4,, 3, 6}. MEI, 07/06/10 1/14
13 You ca fid the mea, ad the stadard deviatio, s, of this ew data set. The, sice = 10, ou ca fid the mea of the origial data usig the equatio 10 ad the stadard deviatio of the origial data usig the equatio s 10s. Alterativel, the umbers could be made smaller b subtractig 0 before 0 dividig b 10. This is the codig 10 which gives the ew data set {1, 3, 0, 5,, 0, 1, 4} You ca fid the mea,, ad the stadard deviatio, s, of this ew data set. The, sice = , ou ca fid the mea of the origial data usig the equatio 10 0 ad the stadard deviatio of the origial data usig the equatio s 10s. Codig is especiall useful whe dealig with grouped data, sice i these cases ou are dealig with mid-iterval values which follow a fied patter. For eample, if ou were dealig with heights grouped as , etc., ou would be workig with mid-iterval values of 104.5, 114.5, 14.5 etc B usig the codig, ou would be workig with values of 0, 1, 10, etc. Eample 10 Use liear codig to calculate the mea ad stadard deviatio of the followig data: Weight, w, (grams) Frequec, f 0 w < w < w < w < w < 50 4 Solutio The mid-iterval values (deoted b ) are 5, 15, 5, etc. A coveiet codig is 5 10 The correspodig values become 0, 1,, f f ² f² f 30 f 61 f 169 MEI, 07/06/10 13/14
14 s f s 10s For practice i usig liear codig, tr the iteractive questios Liear codig. MEI, 07/06/10 14/14
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