Measures of Variation Cumulative Fequency Box and Whisker Plots Standard Deviation

Transcription

1 UNIT 18 Measures of Variatio: Studet Text Cotets STRAND F: Statistics Uit 18 Measures of Variatio Studet Text Cotets Sectio 18.1 Cumulative Fequecy 18. Box ad Whisker Plots 18.3 Stadard Deviatio deotes that this is ot o the curret CXC/CSEC Mathematics syllabus ad therefore ot examied, but is of relevace to the developmet of the topic. CIMT ad e-learig Jamaica

2 UNIT 18 Measures of Variatio: Studet Text 18 Measures of Variatio 18.1 Cumulative Frequecy Cumulative frequecies are useful if more detailed iformatio is required about a set of data. I particular, they ca be used to fid the media ad iter-quartile rage. The iter-quartile rage cotais the middle 5% of the sample ad describes how spread out the data are. This is illustrated i Worked Example. Worked Example 1 For the data give i the table, draw up a cumulative frequecy table ad the draw a cumulative frequecy graph. Height (cm) Frequecy 9 < h < h < h < h < h < h 15 6 Solutio The table opposite shows how to calculate the cumulative frequecies. Height (cm) Frequecy Cumulative Frequecy 9 < h < h = 7 11 < h = 57 1 < h = < h = < h = 11 1 (15, 11) A graph ca the be plotted usig poits as show opposite. 1 (14, 16) Cumulative Frequecy 8 (13, 88) 6 (1, 57) 4 (11, 7) (9, ) (1, 5) Height (cm) Shadig deotes that the topic is ot o the curret CXC/CSEC Mathematics syllabus ad therefore ot examied, but is of relevace to the cotet of the Uit. CIMT ad e-learig Jamaica 1

3 18.1 UNIT 18 Measures of Variatio: Studet Text Note A more accurate graph is foud by drawig a smooth curve through the poits, rather tha usig straight lie segmets. 1 (15, 11) 1 (14, 16) Cumulative Frequecy 8 (13, 88) 6 (1, 57) 4 (11, 7) (9, ) (1, 5) Worked Example Height (cm) The cumulative frequecy graph below gives the results of 1 studets o a test Cumulative Cumultive Frequecy Test Score CIMT ad e-learig Jamaica

4 18.1 UNIT 18 Measures of Variatio: Studet Text Use the graph to fid: the media score, the iter-quartile rage, (c) the mark which was attaied by the top 1% of the studets, (d) the umber of studets who scored more tha 75 o the test. Solutio Sice 1 of 1 is 6, the media ca be foud by startig at 6 o the vertical scale, movig horizotally to the graph lie ad the movig vertically dow to meet the horizotal scale. I this case the media is 53. Cumulative Frequecy Start at 6 4 Score To fid out the iter-quartile rage, we must cosider the middle 5% of the studets. Media = To fid the lower quartile, start at 1 of 1, which is 3. 4 This gives Lower quartile = 45 To fid the upper quartile, start at 3 of 1, which is 9. 4 This gives Upper Quartile = 64 The iter-quartile rage is the Cumulative Frequecy Lower quartile = Test Score Upper quartile = 67 = 67 Iter - quartile Rage = Upper Quartile Lower Quartile = = 19 CIMT ad e-learig Jamaica 3

5 18.1 UNIT 18 Measures of Variatio: Studet Text (c) 1 18 Cumulative Frequecy Here the mark which was attaied by the top 1% is required. 1%of 1 = 1 so start at 18 o the cumulative frequecy scale. This gives a mark of Test Score 1 13 (d) To fid the umber of studets who scored more tha 75, start at 75 o the horizotal axis. This gives a cumulative frequecy of 15. Cumulative Frequecy So the umber of studets with a score greater tha 75 is 1 15 = Test Score As i Worked Example 1, a more accurate estimate for the media ad iter-quartile rage is obtaied if you draw a smooth curve through the data poits. Worked Example 3 The table below shows the distributio of marks o a test for a group of 7 studets. Mark Frequecy Cumulative Frequecy CIMT ad e-learig Jamaica 4

6 18.1 UNIT 18 Measures of Variatio: Studet Text Copy ad complete the table to show the cumulative frequecy for the distributio. (i) Usig a scale of 1 cm to represet 5 marks o the horizotal axis ad 1 cm to represet 5 studets o the vertical axis, draw the cumulative frequecy curve for the scores. (ii) What assumptio have you made i drawig your curve through the poit (, )? (c) (d) The pass mark for the test is 47. Use your graph to determie the umber of studets who passed the test. What is the probability that a studet chose at radom had a mark less tha or equal to 3? (CXC) Solutio Mark Frequecy Cumulative Frequecy (i) Graph o followig page. (ii) Assumptio that o studet scored zero. (c) 7 4 = 8 studets passed the test. (d) Probability = =. 35 CIMT ad e-learig Jamaica 5

7 18.1 UNIT 18 Measures of Variatio: Studet Text (i) Cumulative Frequecy Worked Example Marks The heights, i metres, of a radom sample of 4 soldiers from a regimet were measured. The heights are summarised i the followig table. Height i metres (x) Frequecy Cumulative frequecy 175. x < x < x < x < x <. 14. x < x < x < CIMT ad e-learig Jamaica 6

8 18.1 UNIT 18 Measures of Variatio: Studet Text Copy ad complete the cumulative frequecy colum i the table. Costruct a cumulative frequecy curve for the data. (c) Estimate from this cumulative frequecy curve: (i) the media ( Q ) (ii) the upper quartile ( Q 3 ) (iii) the lower quartile Q 1 ( ) ad ( ). (d) (i) Compare your values of Q3 Q Q Q1 (ii) What does this idicate about the shape of the distributio? Solutio The completed cumulative frequecy table is give below. Height i metres (x) Frequecy Cumulative frequecy 175. x < x < x < x < x < x < x < x < ( ) Cumultive Frequecy Height (metres) CIMT ad e-learig Jamaica 7

9 18.1 UNIT 18 Measures of Variatio: Studet Text (c) (i) (ii) 1.99 (iii) (d) (i) Q3 Q=. 35, Q Q1= 4. (ii) Q Q1> Q3 Q, hece (positive) skew to the data; that is, it is ot symmetric. Exercises 1. Make a cumulative frequecy table for each of the three sets of data give below. The, for each set of data, draw a cumulative frequecy graph ad use it to fid the media ad iter-quartile rage. Joh weighed each apple i a large box. His results are give i this table. Weight of apple (g) 6 < w 8 8 < w 1 1 < w 1 1 < w < w 16 Frequecy Paul asked the studets i his class how far they travelled to school each day. His results are give below. Distace (km) < d 1 1 < d < d 3 3 < d 4 4 < d 5 5 < d 6 Frequecy (c) A athletics coach recorded the distaces studets could reach i the log jump evet. His records are summarised i the table below. Legth of jump (m) 1 < d < d 3 3 < d 4 4 < d 5 5 < d 6 Frequecy A farmer grows a type of wheat i two differet fields. He takes a sample of 5 heads of cor from each field at radom ad weighs the grais he obtais. Mass of grai (g) < m 5 5< m 1 1 < m 15 15< m < m 5 5< m 3 Frequecy Field A Frequecy Field B (c) Draw cumulative frequecy graphs for each field. Fid the media ad iter-quartile rage for each field. Commet o your results. CIMT ad e-learig Jamaica 8

10 18.1 UNIT 18 Measures of Variatio: Studet Text 3. A cosumer group tests two types of batteries usig a DVD player. Lifetime (hours) < l 3 3 < l 4 4 < l 5 5 < l 6 6 < l 7 7 < l 8 Frequecy Type A Frequecy Type B 38 6 Use cumulative frequecy graphs to fid the media ad iter-quartile rage for each type of battery. Which type of battery would you recommed ad why? 4. The table below shows how the height of childre of a certai age vary. The data was gathered usig a large-scale survey. Height (cm) 5 < h < h 6 6 < h < h 7 7 < h < h 8 8 < h 85 Frequecy A doctor wishes to be able to classify childre as: Category Percetage of Populatio Very Tall 5% Tall 15% Normal 6% Short 15% Very short 5% Use a cumulative frequecy graph to fid the heights of childre i each category. 5. The maager of a glazig compay employs 3 salesme. Each year he awards bouses to his salesme. Bous Awarded to $5 Best 1% of salesme $5 Middle 7% of salesme $ 5 Bottom % of salesme The sales made durig 5 ad 6 are show i the table below. Value of sales ($1) < V 1 1 < V < V 3 3 < V 4 4 < V 5 Frequecy Frequecy Use cumulative frequecy graphs to fid the values of sales eeded to obtai each bous i the years 5 ad 6. CIMT ad e-learig Jamaica 9

11 18.1 UNIT 18 Measures of Variatio: Studet Text 6. The histogram shows the cost of buyig a certai toy i a umber of differet shops Frequecy Price ($) Draw a cumulative frequecy graph ad use it to aswer the followig questios. (i) How may shops charged more tha $.65? (ii) What is the media price? (iii) How may shops charged less tha $.3? (iv) How may shops charged betwee $. ad $.6? (v) How may shops charged betwee $. ad $.5? Commet o which of your aswers are exact ad which are estimates. 7. Derrice ad Jeevia played 4 games of golf together. The table below shows Derrice's scores. Scores (x) 7 < x 8 8 < x 9 9 < x 1 1 < x < x 1 Frequecy O a grid similar to the oe below, draw a cumulative frequecy diagram to show Derrice's scores. 4 q y Cumulative Frequecy Score CIMT ad e-learig Jamaica 1

12 18.1 UNIT 18 Measures of Variatio: Studet Text Makig your method clear, use your graph to fid (i) Derrice's media score, (ii) the iter-quartile rage of her scores. (c) Jeevia's media score was 13. The iter-quartile rage of her scores was 6. (i) Who was the more cosistet player? Give a reaso for your choice. (ii) The wier of a game of golf is the oe with the lowest score. Who wo most of these 4 games? Give a reaso for your choice. 8. A sample of 8 electric light bulbs was take. The lifetime of each light bulb was recorded. The results are show below. Lifetime (hours) Frequecy Cumulative Frequecy 4 17 Copy ad complete the table of values for the cumulative frequecy. Draw the cumulative frequecy curve, usig a grid as show below. q y Lifetime (hours) (c) (d) (e) Use your graph to estimate the umber of light bulbs which lasted more tha 13 hours. Use your graph to estimate the iter-quartile rage of the lifetimes of the light bulbs. A secod sample of 8 light bulbs has the same media lifetime as the first sample. Its iter-quartile rage is 9 hours. What does this tell you about the differece betwee the two samples? CIMT ad e-learig Jamaica 11

13 18.1 UNIT 18 Measures of Variatio: Studet Text 9. The umber of joureys made by a group of people usig route taxis i oe moth is summarised i the table. Number of joureys Number of people Copy ad complete the cumulative frequecy table below. Number of joureys Cumulative frequecy (i) Draw the cumulative frequecy graph, usig a grid as below. 4 q y 3 Cumulative Frequecy Number of joureys (ii) (iii) Use your graph to estimate the media umber of joureys. Use your graph to estimate the umber of people who made more tha 44 joureys i the moth. (c) The umber of joureys made usig route taxis i oe moth, by aother group of people, is show i the graph. 4 3 Cumulative Frequecy Number of joureys Make oe compariso betwee the umbers of joureys made by these two groups. CIMT ad e-learig Jamaica 1

14 18.1 UNIT 18 Measures of Variatio: Studet Text 1. The cumulative frequecy graph below gives iformatio o holiday villa retal prices i. The cumulative frequecy is give as a percetage of all retal villas i the Motego Bay area. 1 q y ( ) Cumulative Frequecy Holiday villa retal prices i (J$), per week 4 This grouped frequecy table gives the percetage distributio of villa retal prices (p) i the Motego Bay area i 3. Villa retal prices (p) i J$, 3 Percetage of villas i this class iterval p < p < p < p < p < p < p < 4 Use the data above to complete the cumulative frequecy table below. Villa retal prices (p) Cumulative i J$, 3 Frequecy (%) p < 4 p < 5 p < 68 p < 88 p < 1 p < 16 p < CIMT ad e-learig Jamaica 13

15 18.1 UNIT 18 Measures of Variatio: Studet Text (c) Trace or photocopy the grid for, ad o it costruct a cumulative frequecy graph for your table for 3. I the retal for a holiday villa was J$1. Use both cumulative frequecy graphs to estimate the retal for this house i 3. Make your method clear. 11. The legths of a umber of carpeters' ails were measured to the earest.1 cm ad the followig frequecy distributio was obtaied. Legth of ail Number of ails Cumulative Frequecy (x cm) 98. x < x < x < x < x < x < x < x < Complete the cumulative frequecy colum. Draw a cumulative frequecy diagram o a grid similar to the oe below. 1 8 Cumulative Frequecy Legth of ail (cm) Use your graph to estimate (i) the media legth of the ails (ii) the iter-quartile rage. CIMT ad e-learig Jamaica 14

16 18.1 UNIT 18 Measures of Variatio: Studet Text 1. A weddig was atteded by 1 guests. The distace, d km, that each guest travelled was recorded i the frequecy table below. Distace (d km) < d 1 1 < d < d 3 3 < d 5 5 < d 1 1 < d 14 Number of guests Usig the mid-iterval values, calculate a estimate of the mea distace travelled. (i) Copy ad complete the cumulative frequecy table below. Distace (d km) d 1 d d 3 d 5 d 1 d 14 Number of guests 1 (ii) O a grid similar to that show below, draw a cumulative frequecy curve to represet the iformatio i the table. 1 1 Cumulative Frequecy (c) (i) Use the cumulative frequecy curve to estimate the media distace travelled by the guests. (ii) Give a reaso for the large differece betwee the mea distace ad the media distace. CIMT ad e-learig Jamaica 15 Distace travelled (km)

17 18.1 UNIT 18 Measures of Variatio: Studet Text 13. Mr Isaacs is checkig the time take for documets to be word processed. The first 1 documets which he samples take the followig times, i miutes, to process Calculate (i) the media time, (ii) the iterquartile rage. His completed survey gives him the followig iformatio. Time (miutes) Number of documets Copy ad complete the cumulative frequecy table for this iformatio. Time less tha (miutes) Number of documets 3 8 (c) Use your cumulative frequecy table to draw a cumulative frequecy graph. Use your graph to fid (i) (ii) the media time, the iterquartile rage. You must mark your graph clearly showig how you foud your aswers. (d) Compare the media times ad the iterquartile rages foud i the sample ad the completed survey. Suggest a reaso for ay differeces. Iformatio A quartile is oe of 3 values (lower quartile, media ad upper quartile) which divides data ito 4 equal groups. A percetile is oe of 99 values which divides data ito 1 equal groups. The lower quartile correspods to the 5th percetile. The media correspods to the 5th percetile. The upper quartile correspods to the 75th percetile. CIMT ad e-learig Jamaica 16

18 UNIT 18 Measures of Variatio: Studet Text 18. Box ad Whisker Plots The box ad whisker plot is aother measure of variatio that eables you to illustrate ad compare the measures of both locatio ad variatio through the media ad quartiles. For example, for the data set below, we ca easily fid the media ad quartiles. 4, 5, 1, 1, 11, 1, 15 lower media upper quartile quartile The box is formed by the two quartiles, with the media marked by a lie, whilst the whiskers are fixed by the two extreme values, 4 ad 15. The plot is show below, relative to a scale Box ad whisker plots are particularly useful whe comparig quickly two sets of data. For example, if you wish to compare the data set above with the followig data set, 1, 7, 9, 1, 14, 17, 19 lower media upper quartile quartile the you ca illustrate the two plots together. This is show below you ca immediately see that the data i the secod set are much more spread out tha that i the first set Outliers Outliers are extreme values i data sets ad are ofte igored as they ca distort the data aalysis. We make the cocept precise by defiig a outlier as 'ay value which is either 1.5 times the iterquartile rage (IQR) more tha the upper quartile (UQ) or 1.5 times the IQR less tha the lower quartile (LQ). This is illustrated below. Media LQ UQ 1.5 IQR IQR 1.5 IQR Outliers i this rage CIMT ad e-learig Jamaica 17

19 18. UNIT 18 Measures of Variatio: Studet Text Ay outlier should be marked o the box ad whisker diagram but the whisker should exted oly to the lowest ad highest values which are ot outliers. Worked Example 1 The umber of goals scored by the 11 members of a football team i 7 were as follows: (c) (d) (e) Fid the media. Fid the upper ad lower quartiles. Fid the iterquartile rage. Explai why, for this set of data, the iterquartile rage is a more appropriate measure of spread tha the rage. The goals scored by the 11 members of the hockey team i 8 are summarised i the box ad whisker plot below Goals scored (i) (ii) O a copy of the diagram, summarise the results for 7 i the same way. Do you thik the team scored more goals i 8? Explai your reasoig. Solutio We first put the umber of goals i icreasig order, i.e: There are 11 data poits, so the media is the + th value, i.e. the 6th value, which is The lower quartile is the + th value, i.e. the 3rd value, which is 1; the 4 upper quartile is the 9th value, i.e. 9. CIMT ad e-learig Jamaica 18

20 18. UNIT 18 Measures of Variatio: Studet Text (c) Iterquartile rage = 9 1= 8. (d) (e) The iterquartile rage is a better measure to represet the 'average' spread, rather tha the rage, as it excludes the outlyig values Goals scored The team scored more goals i 8; the media is much lower. Worked Example The cumulative frequecy curve represets the times take to ru 15 metres by each of the 4 members of the athletics club, Westo Harriers. y y 4 Cumulative frequecy x Time (miutes) From the graph, fid: (i) the media time; (ii) the upper quartile ad the lower quartile. CIMT ad e-learig Jamaica 19

21 18. UNIT 18 Measures of Variatio: Studet Text (c) Draw a box ad whisker plot to illustrate the data. Use your box ad whisker plot to make oe commet about the shape of a histogram for these data. A rival athletics club, Eastham Ruers, also has 4 members. The time take by each member to ru 15 metres is recorded ad these data are show i the followig box ad whisker plot Time (miutes) (d) Use this diagram to make oe commet about the data for Eastham Ruers compared with the data for Westo Harriers. Solutio (i) From the dashed lie (1 o the vertical axis), the media is 5 1 miutes 4 or 5 miutes 15 secods. (ii) Similarly: the upper quartile is miutes or 5 miutes 45 secods, the lower quartile is 4 mi 48 sec (width of each small square is 3 secods) Miutes (c) (d) The data are almost symmetric about the media. The data for Eastham Ruers are skewed to the left, with a lower media time. Hece Eastham Ruers' data are sigificatly better tha Wester Harriers' data, idicatig relatively more athletes with faster times. Worked Example 3 The ages (i years) of a group of people visitig a club are give below Idetify ay outlier. Illustrate the data usig a box ad whisker plot. CIMT ad e-learig Jamaica

22 18. UNIT 18 Measures of Variatio: Studet Text Solutio First idetify the media ad upper ad lower quartiles of the 7 data values. Puttig the data i icreasig order gives: The media is the + th value, i.e. the 14th value 3 The lower quartile is the th value, i.e. the 7th value LQ = 7 The upper quartile is the ( ) th value, i.e. the 1st value UQ = 37 4 The iterquartile rage is 37 7 = 1. Now check for outliers, rememberig that the outliers must be less tha LQ 1.5 IQR = = 7 15 = 1 or more tha UQ IQR = = = 5 The oly outlier is 76. This box ad whisker plot illustrates the data. outlier Exercises 1. A maufacturig compay eeds to place a regular order for compoets. The maager ivestigates compoets produced by three differet firms ad measures the diameters of a sample of 5 compoets from each firm. The results of the measuremets for the samples of compoets from Firm B ad Firm C are illustrated i the two box plots show below. CIMT ad e-learig Jamaica 1

23 18. Diameter of compoet (mm) UNIT 18 Measures of Variatio: Studet Text Firm A Firm B Firm C * R (i) Fid the rage of the sample of measuremet for Firm B. (ii) Fid the iterquartile rage of the sample of measuremet for Firm C. (iii) Explai why the result labelled R is show as a outlier o the box plot for Firm C. (c) The results of the measuremets for the sample from Firm A are summarised as follows. Media = 5. mm, lower quartile = 3.4 mm, upper quartile = 6.5 mm, lowest value =.5 mm, highest value = 7.3 mm. Draw, o a copy of the grid above, a box plot to illustrate the sample results for Firm A. The maager studies the three box plots to decide which firm's compoets he should use. The compoets he requires should have a diameter of 5 mm, but some variatio above ad below this measuremet is boud to happe ad is acceptable. Ay compoets with diameters below 4 mm or above 6 mm will have to be throw away. State which firm's compoets you thik the maager should choose. Explai carefully why you thik he should choose this firm rather tha the other two.. A radom sample of 51 people were asked to record the umber of km they travelled by car i a give week. The distaces, to the earest km, are show below CIMT ad e-learig Jamaica

24 18. UNIT 18 Measures of Variatio: Studet Text (c) Fid the media ad the quartiles of this distributio. Draw a box plot to represet these data. Give oe advatage of usig the box plot to illustrate data such as that give above. 3. The weights, to the earest kilogram, of 19 pigs were: Fid the iter-quartile rage of the weights. Fid ay weights that are outliers. The media of the data is 37 kg. (c) Draw a box plot for the data Weight (kg) (d) Name a distributio that could be used to model the weight of these pigs. Give a reaso for your choice. The weight of a full-grow pig is about 45 kg. (e) What does this suggest about the 19 pigs? 4. The legth of reig of each of the last 19 Eglish moarchs is give i the table. George VI 16 years George IV 1 years James II 3 years Edward VIII years George III 6 years Charles II 5 years George V 6 years George II 33 years Charles I 4 years Edward VII 9 years George I 13 years James I years Victoria 64 years Ae 1 years Elizabeth I 45 years William IV 7 years William III 14 years Mary 5 years Edward VI 6 years Fid the media ad quartiles of the legth of reig of these 19 moarchs. Write dow the ame of ay moarch whose legth of reig is a outlier. You must show calculatios to support your aswer. CIMT ad e-learig Jamaica 3

25 18. UNIT 18 Measures of Variatio: Studet Text (c) The box ad whisker plot shows the legth of reig of the last 19 Popes Number of years Draw a box ad whisker plot for the legth of reig of the last 19 moarchs o a copy of the diagram. (d) Compare the legth of reig of moarchs ad Popes. 5. The cumulative frequecy polygo below shows the times take to travel to a city cetre school by a group of studets. Cumulative frequecy Frequecy Time (mi) Estimate from the graph: (i) the media; (ii) the iterquartile rage; (iii) the percetage of studets takig more tha 35 miutes to reach school. CIMT ad e-learig Jamaica 4

26 18. UNIT 18 Measures of Variatio: Studet Text A school of equivalet size i a rural area showed the followig distributio of times take to travel to school. Time take (mi) No. of studets Cumulative frequecy table Time No. of pupils ad uder 5 8 <5 8 5 ad uder 1 44 <1 1 ad uder <15 15 ad uder 5 9 <5 5 ad uder 4 7 <4 4 ad uder <55 (i) (ii) (iii) Complete a copy of the cumulative frequecy table for the data. Draw o the same axes the cumulative frequecy polygo for this school, labellig the polygo clearly. Estimate from this polygo the media ad the iterquartile rage. (c) Costruct box ad whisker plots for each set of data ad commet o the mai differeces that are apparet betwee the two distributios. 6. I a village a record was kept of the ages of those people who died i 8. The data are show o the stem ad leaf diagram Key: 3 deotes 3 years How may people died i this village i 8? At the start of the year there were 75 people i the village. Calculate the death rate for this village. (c) Use the stem ad leaf diagram to obtai values for: (i) the media, (ii) the lower ad upper quartiles. CIMT ad e-learig Jamaica 5

27 18. UNIT 18 Measures of Variatio: Studet Text (d) O a copy of the diagram below, draw a box ad whisker diagram to illustrate the data Age 18.3 Stadard Deviatio The two frequecy polygos draw o the graph below show samples which have the same mea, but the data i oe are much more spread out tha i the other. 5 Frequecy Legth The rage (highest value lowest value) gives a simple measure of how much the data are spread out. Stadard deviatio (s.d.) is a much more useful measure ad is give by the formula: s.d. = ( x i x) i= 1 ( ) where x i represets each datapoit x 1, x,..., x x is the mea, is the umber of values. CIMT ad e-learig Jamaica 6

28 18.3 UNIT 18 Measures of Variatio: Studet Text The ( x x) gives the square of the differece betwee each value ad the mea (squarig exaggerates the effect of data poits far from the mea ad gets rid of egative values), ad i sums up all these squared differeces. i= 1 ( x x) i The expressio 1 xi x ( ) i= 1 gives a average value to these differeces. If all the data were the same, the each x i would equal x ad the expressio would be zero. Fially we take the square root of the expressio so that the dimesios of the stadard deviatio are the same as those of the data. So stadard deviatio is a measure of the spread of the data. The greater its value, the more spread out the data are. This is illustrated by the two frequecy polygos show above. Although both sets of data have the same mea, the data represeted by the 'dotted' frequecy polygo will have a greater stadard deviatio tha the other. Worked Example 1 Fid the mea ad stadard deviatio of the umbers, 6, 7, 8, 5, 9 Solutio The mea, x, is give by, x = Now the stadard deviatio ca be calculated. = 35 5 = 7 s.d. = = = ( 6 7) + ( 7 7) + ( 8 7) + ( 5 7) + ( 9 7) = = (to 3 decimal places) CIMT ad e-learig Jamaica 7

29 18.3 UNIT 18 Measures of Variatio: Studet Text A alterative formula for stadard deviatio is s.d. = x i i= 1 x This expressio is much more coveiet for calculatios doe without a calculator. The proof of the equivalece of this formula is give below although it is beyod the scope of the GCSE syllabus. Proof You ca see the proof of the equivalece of the two formulae by otig that ( xi x) = ( xi xi x + x ) i= 1 i= 1 = ( ) + x x x x i i i= 1 i= 1 i= 1 i i i= 1 i= 1 i= 1 = x x x + x 1 (sice the expressios x ad x are commo for each term i the summatio). But 1 =, sice you are summig =, ad x i= 1 defiitio, thus terms = 1 i= 1 x, by 1 1 xi x xi x xi x i= 1 i= 1 i= 1 ( ) = + (substitutig 1 i= 1 = ) i = xi i x + x (dividig by ) xi = 1 = 1 x i = i= 1 x + x (substitutig x for i= 1 x i ) ad the result follows. x i = i= x 1 CIMT ad e-learig Jamaica 8

30 18.3 UNIT 18 Measures of Variatio: Studet Text Worked Example Fid the mea ad stadard deviatio of each of the followig sets of umbers. 1, 11, 1, 13, 14 5, 6, 1, 18, 19 Solutio The mea, x, is give by x = = 6 5 = 1 The stadard deviatio ca ow be calculated usig the alterative formula. s.d. = = = = (to 3 decimal places) The mea, x, is give by x = 5 = 1 (as i part ) The stadard deviatio is give by s.d. = = = (to 3 decimal places). Note that both sets of umbers have the same mea value, but that set has a much larger stadard deviatio. This is expected, as the spread i set is clearly far more tha i set. CIMT ad e-learig Jamaica 9

31 18.3 UNIT 18 Measures of Variatio: Studet Text Worked Example 3 The table below gives the umber of road traffic accidets per day i a small tow. Accidets per day Frequecy Fid the mea ad stadard deviatio of this data. Solutio The ecessary calculatios for each datapoit, x i, are set out below. Accidets per day Frequecy ( x i ) f i ( ) x i xi fi x f TOTALS i i From the totals, = 5, The mea, x, is ow give by x i fi = 43, x = i 17 i= 1 x = i= 1 = 43 5 i= 1 x f = 17. i i The stadard deviatio is ow give by s.d. = x i f i i= 1 x 17 = = Most scietific calculators have statistical fuctios which will calculate the mea ad stadard deviatio of a set of data. CIMT ad e-learig Jamaica 3

32 18.3 UNIT 18 Measures of Variatio: Studet Text Exercises 1. Fid the mea ad stadard deviatio of each set of data give below. A B C Describe the relatioship betwee each set of umbers ad also the relatioship betwee their meas ad stadard deviatios.. Two machies, A ad B, fill empty packets with soap powder. A sample of packets was take from each machie ad the weight of powder (i kg) was recorded. A B Fid the mea ad stadard deviatio for each machie. Which machie is most cosistet? 3. Two groups of studets were tryig to fid the acceleratio due to gravity. Each group coducted 5 experimets. Group A Group B Fid the mea ad stadard deviatio for each group, ad commet o their results. 4. The umber of matches per box was couted for 1 boxes of matches. The results are give i the table below. Number of Matches Frequecy Fid the mea ad stadard deviatio of this data. CIMT ad e-learig Jamaica 31

33 18.3 UNIT 18 Measures of Variatio: Studet Text 5. Whe two dice were throw 5 times the total scores show below were obtaied. Score Frequecy Fid the mea ad stadard deviatio of these scores. 6. The legth of telephoe calls from a office was recorded. The results are give i the table below. Legth of call (mis) < t < t < t.. < t 5. Frequecy Estimate the mea ad stadard deviatio usig this table. 7. The charges (to the earest $) made by a jeweller for repair work o jewellery i oe week are give i the table below. Charge ($) Frequecy Use this table to estimate the mea ad stadard deviatio. 8. Thirty families were selected at radom i two differet coutries. They were asked how may childre there were i each family. Coutry A Coutry B Fid the mea ad stadard deviatio for each coutry ad commet o the results. 9. Calculate the stadard deviatio of the umbers 3, 4, 5, 6, 7. Show that the stadard deviatio of every set of five cosecutive itegers is the same as the aswer to part. CIMT ad e-learig Jamaica 3

34 18.3 UNIT 18 Measures of Variatio: Studet Text 1. Te studets sat a test i Mathematics, marked out of 5. The results are show below for each studet. 5, 7, 35, 4, 49, 1, 1, 45, 45, 48 Calculate the mea ad stadard deviatio of the data. The same studets also sat a Eglish test, marked out of 5. The mea ad stadard deviatio are give by mea = 3, stadard deviatio = 3.6. Commet o ad cotrast the results i Mathematics ad Eglish. 11. Te boys sat a test which was marked out of 5. Their marks were 8, 4, 35, 17, 49, 1, 48, 38, 4 ad 7 Calculate (i) (ii) the mea of the marks, the stadard deviatio of the marks. Te girls sat the same test. Their marks had a mea of 3 ad a stadard deviatio of 6.5. Compare the performaces of the boys ad girls. 1. There are twety studets i class A ad twety studets i class B. All the studets i class A were give a I.Q. test. Their scores o the test are give below. 1, 14, 16, 17, 19, 11, 113, 114, 116, 117, 118, 119, 119, 11, 14, 15, 17, 17, 13, 134. The mea of their scores is 117. Calculate the stadard deviatio. Class B takes the same I.Q. test. They obtai a mea of 11 ad a stadard deviatio of 1. Compare the data for class A ad class B. (c) Class C has oly 5 studets. Whe they take the I.Q. test they all score 15. What is the value of the stadard deviatio for class C? 13. The followig are the scores i a test for a set of 15 studets (i) Calculate the mea score. (ii) Calculate the stadard deviatio of the scores. CIMT ad e-learig Jamaica 33

35 18.3 UNIT 18 Measures of Variatio: Studet Text A set of 1 differet studets took the same test. Their scores are listed below After makig ay ecessary calculatios for the secod set, compare the two sets of scores. Your aswer should be uderstadable to someoe who does ot study Statistics. 14. I a survey o examiatio qualificatios, 5 people were asked, "How may subjects are listed o your GCSE certificate?" The frequecy distributio of their resposes is recorded i the table below. Number of subjects Number of people Calculate the mea ad stadard deviatio of the distributio. A Normal Distributio has approximately 68% of its data values withi oe stadard deviatio of the mea. Use your aswers to part to check if the give distributio satisfies this property of a Normal Distributio. Show your workig clearly. CIMT ad e-learig Jamaica 34