Transcription

1

2 Prior Kowledge Check 1) State whether each variable is qualitative or quatitative: a) Car colour Qualitative b) Miles travelled by a cyclist c) Favourite pet Qualitative d) Number of sibligs Quatitative Quatitative ) State whether each of these variables is discrete or cotiuous a) Number of pets owed Discrete b) Distace walked by hikers Cotiuous c) Fuel cosumptio of lorries d) Number of peas i a pod Discrete e) Times take by a group of athletes to ru 1500m. Cotiuous Cotiuous 3) Calculate the 3 averages ad rage for the data set below: Peas i a pod Frequecy Mea 5.33 Media 6 Mode 6 Rage 4

3

4 A measure of locatio is a sigle value which is used to represet a set of data. Examples iclude the mea, media ad mode These ca also be kow as measures of cetral tedecy The mode or modal class is the value or class with the highest frequecy The mea, x-bar The sum of x (x represets each umber i the data set) The media is the middle value whe the data is put ito ascedig (or descedig) order The mea is calculated usig the formula: You eed to start usig proper otatio! x ҧ = σ x The umber of bits of data A/B

5 x ҧ = σ x A measure of locatio is a sigle value which is used to represet a set of data. Examples iclude the mea, media ad mode The mea of a sample of 5 observatios is 6.4. The mea of a secod sample of 30 observatios is 7.. Calculate the mea of all 55 observatios Why is this calculatio wrog? Each data set has a differet quatity The mea will be weighted towards the data set with the higher quatity A/B

6 x ҧ = σ x A measure of locatio is a sigle value which is used to represet a set of data. Examples iclude the mea, media ad mode The mea of a sample of 5 observatios is 6.4. The mea of a secod sample of 30 observatios of 7.. Calculate the mea of all 55 observatios. Sample = σ x 5 σ x = 160 Multiply by 5 Sample 7. = σ x 30 σ x = 16 So the total sum of the data is 376 There were 55 bits of data i total Multiply by 30 x ҧ = σ x x ҧ = x ҧ = 6.84 Sub i values Sub i values A/B

7 x ҧ = σ x A measure of locatio is a sigle value which is used to represet a set of data. Examples iclude the mea, media ad mode You will eed to decide which is the best average to use depedig o the situatio Mode Media Mea The mode is used for qualitative data, or where the data has a sigle mode or two modes (bi-modal) This is used for quatitative data, ad is usually used whe the data has some extreme values (sice this will ot affect the media too much) The mea is used for quatitative data ad uses all data i a set. It therefore gives a true measure, but ca be skewed by extreme values A/B

8 x ҧ = σ x x ҧ = σ fx σ f A measure of locatio is a sigle value which is used to represet a set of data. Examples iclude the mea, media ad mode For data give i a frequecy table, you ca calculate the mea by usig the formula below: x ҧ = σ fx σ f The sum of the products of the data values ad their frequecies The sum of the frequecies A/B

9 x ҧ = σ x x ҧ = σ fx σ f A measure of locatio is a sigle value which is used to represet a set of data. Examples iclude the mea, media ad mode Rebecca records the shirt collar size, x, of the male studets i her year. The results are show i the table. For the data, calculate: a) The mode b) The media c) The mea = = 16 d) Explai why a shirt maufacturer might use the mode for settig their productio quota x Collar Size f Number of Studets f = = 48th The media is the 48 th value Add the frequecies up util you get beyod this value So the media must be 16 as it is i that group A/B

10 x ҧ = σ x x ҧ = σ fx σ f A measure of locatio is a sigle value which is used to represet a set of data. Examples iclude the mea, media ad mode Rebecca records the shirt collar size, x, of the male studets i her year. The results are show i the table. For the data, calculate: a) The mode b) The media c) The mea = = 16 = 16. d) Explai why a shirt maufacturer might use the mode for settig their productio quota x Collar Size f Number of Studets x ҧ = σ fx σ f x ҧ = x ҧ = 16. f = 95 Sub i values Calculate fx fx = A/B

11 x ҧ = σ x x ҧ = σ fx σ f A measure of locatio is a sigle value which is used to represet a set of data. Examples iclude the mea, media ad mode x Collar Size f Number of Studets 15 3 fx 45 Rebecca records the shirt collar size, x, of the male studets i her year. The results are show i the table. For the data, calculate: a) The mode b) The media c) The mea = = 16 = 16. d) Explai why a shirt maufacturer might use the mode for settig their productio quota f = fx = The mode is i this case more useful as it tells the maufacturer what size shirt it eeds to produce the most of A/B

12 x ҧ = σ x x ҧ = σ fx σ f A measure of locatio is a sigle value which is used to represet a set of data. Examples iclude the mea, media ad mode The legth, x mm, to the earest mm, of a radom sample of pie coes is measured. The data is show i the table to the right. a) Write dow the modal class b) Estimate the mea c) Fid the media class Coe legth (mm) x ҧ = σ fx σ f x x ҧ = x ҧ = 34.5 Frequecy f f = 70 Sub i values Calculate fx fx = To calculate the mea from a grouped table you eed to use the midpoit of each group A/B

13 x ҧ = σ x x ҧ = σ fx σ f A measure of locatio is a sigle value which is used to represet a set of data. Examples iclude the mea, media ad mode The legth, x mm, to the earest mm, of a radom sample of pie coes is measured. The data is show i the table to the right Coe legth (mm) Frequecy a) Write dow the modal class b) Estimate the mea c) Fid the media class = 35.5th x f f = 70 The media is the 35.5 th value Add the frequecies up util you get beyod this value So the media class is A/B

14

15 You eed to be able to calculate quartiles ad percetiles of a data set Lowest value Lower Quartile Media Upper Quartile Q 1 Q Q 3 Highest value The media describes the middle of a set of data, splittig the data ito two halves with 50% i each. You ca also calculate quartiles ad percetiles, which are also both measures of locatio The media is also kow as the secod quartile 10% 5% 50% 5% 90% 5% 50% 5% Whe combied with the media, the lower ad upper quartiles split the data ito 4 equal sectios The 10th percetile is the value with 10% of the data lower tha it The 90th percetile is the value with 90% of the data lower tha it So if your test score was i the 90 th percetile, that is a good thig! C

16 You eed to be able to calculate quartiles ad percetiles of a data set The way the quartiles are calculated depeds o whether the data is discrete or cotiuous Discrete Cotiuous Lower Quartile Divide by 4 If whole, the LQ is betwee this value ad the oe above If ot whole, roud up ad take that data poit Divide by 4 ad take that data poit Upper Quartile Divide 3 by 4 If whole, the UQ is betwee this value ad the oe above If ot whole, roud up ad take that data poit Divide 3 by 4 ad take that data poit C

17 You eed to be able to calculate quartiles ad percetiles of a data set From the large data set, the daily maximum gust (kots) durig the first 0 days of Jue 015 is recorded i Hur. The data is show below: Q = +1 Q = th value Q = 10.5th value Q =.5 There are 0 values Calculate Fid this value Note that we treat this as discrete data sice we have all the actual values! Fid the media ad quartiles for this data. Q =. 5 C

18 You eed to be able to calculate quartiles ad percetiles of a data set From the large data set, the daily maximum gust (kots) durig the first 0 days of Jue 015 is recorded i Hur. The data is show below: Fid the media ad quartiles for this data. Q 1 = 18 Q =. 5 Q 3 = 5. 5 The data is discrete For Q = 5 So take the 5.5 th value Q 1 = 18 For Q = 15 So take the 15.5 th value Q 3 = 5.5 C

19 LB + PL GF CW You eed to be able to calculate quartiles ad percetiles of a data set The legth of time (to the earest miute) spet o the iteret each eveig by a group of studets is show i the table below. The data is cotiuous = 5.5th value Fid the group that this is i, ad use liear iterpolatio to estimate the media Time spet o iteret (mis) Frequecy You ca use this formula: a) Fid a estimate for the upper quartile b) Fid a estimate for the 10 th percetile 7 57 LB + Lower boudary of the group PL GF CW Group Frequecy Places ito group Classwidth of the group C

20 LB + PL GF CW You eed to be able to calculate quartiles ad percetiles of a data set The legth of time (to the earest miute) spet o the iteret each eveig by a group of studets is show i the table below. LB + PL GF CW The value is 5.5 places ito the group (it is the 5.5 th value, ad we have already had 7 before the group started) Remember for cotiuous data, you will eed to use 33.5 ad 36.5 as the class boudaries Calculate Time spet i iteret (mis) Frequecy = a) Fid a estimate for the upper quartile = 5.5th value b) Fid a estimate for the 10 th percetile 7 57 C

21 LB + PL GF CW You eed to be able to calculate quartiles ad percetiles of a data set The legth of time (to the earest miute) spet o the iteret each eveig by a group of studets is show i the table below. Time spet i iteret (mis) Frequecy The data is cotiuous The 10 th percetile is calculated as follows = 7th value Fid the group that this is i, ad use liear iterpolatio to estimate it a) Fid a estimate for the upper quartile = b) Fid a estimate for the 10 th percetile C

22 LB + PL GF CW You eed to be able to calculate quartiles ad percetiles of a data set The legth of time (to the earest miute) spet o the iteret each eveig by a group of studets is show i the table below. LB + PL GF CW The value is 5 places ito the group (it is the 7 th value, ad we have already had before the group started) Remember for cotiuous data, you will eed to use 31.5 ad 33.5 as the class boudaries Calculate Time spet i iteret (mis) Frequecy P 10 = a) Fid a estimate for the upper quartile b) Fid a estimate for the 10 th percetile 7 = This otatio is usually used for the 10 th percetile C

23

24 LB + PL GF CW A measure of spread is a value which idicated how spread out the data set is. Examples iclude the rage ad iterquartile rage. The rage is the differece betwee the largest ad smallest values, ad measures the spread of all the data The iterquartile rage is the differece betwee the upper ad lower quartiles, ad measures the spread of the middle 50% of the data The iterpercetile rage is the differece betwee give percetiles D

25 LB + PL GF CW A measure of spread is a value which idicated how spread out the data set is. Examples iclude the rage ad iterquartile rage. The table to the right shows the masses (toes) of 10 Africa elephats. The rage is the biggest possible value subtract the smallest possible value =.5 Mass, m (t) Frequecy 4.0 m < m < m < m < m Fid estimates for: a) The rage. 5 b) The iterquartile rage c) The 10 th to 90 th percetile rage D

26 LB + PL GF CW A measure of spread is a value which idicated how spread out the data set is. Examples iclude the rage ad iterquartile rage. The table to the right shows the masses (toes) of 10 Africa elephats. Fid estimates for: a) The rage b) The iterquartile rage c) The 10 th to 90 th percetile rage Q 1 = For Q 1 4 = 10 4 LB + = 30th value Now use liear iterpolatio PL GF CW = Mass, m (t) Frequecy 4.0 m < m < m < m < m Sub i values Calculate = 4.87 D

27 LB + PL GF CW A measure of spread is a value which idicated how spread out the data set is. Examples iclude the rage ad iterquartile rage. The table to the right shows the masses (toes) of 10 Africa elephats. Fid estimates for: a) The rage b) The iterquartile rage c) The 10 th to 90 th percetile rage Q 1 = Q 3 = = For Q = LB + = 90th value Now use liear iterpolatio = = 5.84 PL GF CW Mass, m (t) Frequecy 4.0 m < m < m < m < m Sub i values Calculate D

28 LB + PL GF CW A measure of spread is a value which idicated how spread out the data set is. Examples iclude the rage ad iterquartile rage. The table to the right shows the masses (toes) of 10 Africa elephats. Fid estimates for: a) The rage b) The iterquartile rage c) The 10 th to 90 th percetile rage P 10 = For P = LB + = 1th value Now use liear iterpolatio = = 4.46 PL GF CW Mass, m (t) Frequecy 4.0 m < m < m < m < m Sub i values Calculate D

29 LB + PL GF CW A measure of spread is a value which idicated how spread out the data set is. Examples iclude the rage ad iterquartile rage. The table to the right shows the masses (toes) of 10 Africa elephats. Fid estimates for: a) The rage b) The iterquartile rage c) The 10 th to 90 th percetile rage P 10 = P 90 = = For P = LB + = = 108th value Now use liear iterpolatio = 6.18 PL GF CW Mass, m (t) Frequecy 4.0 m < m < m < m < m Sub i values Calculate D

30

31 The variace ad stadard deviatio ca also be used to aalyse a set of data The variace ad stadard deviatio are both measures of spread, ad ivolve the fact that each data poit deviates from the mea by the amout: x xҧ Where x is a data poit ad xҧ is the mea of the data as a whole E

32 Variace = σ x xҧ Variace = σ x σ x Variace = S xx The variace ad stadard deviatio ca also be used to aalyse a set of data The variace is defied as: The average of the squared distaces from the mea So the distaces of each data poit from the mea are all squared, ad divided by how may there are. Variace = σ x xҧ Mea of the squares subtract the square of the mea = σ x σ x = S xx This formula is equivalet This formula is equivalet The otatio S xx is short for σ x xҧ or σ x σ x This gives the formula to the right: E

33 Variace = σ x xҧ Variace = σ x σ x S xx σ = σ x xҧ Variace = σ = σ x σ x σ = S xx σ x ҧ x σ x σ x S xx The variace ad stadard deviatio ca also be used to aalyse a set of data The stadard deviatio is the square root of the variace. The symbol σ (lower case sigma) is used to represet stadard deviatio Therefore σ is usually used to represet the variace The square root of ay of these gives the Stadard Deviatio This is what it looks like i the formula booklet! E

34 σ = σ x xҧ σ = σ x σ x σ = S xx σ x ҧ x σ x σ x S xx The variace ad stadard deviatio ca also be used to aalyse a set of data The Stadard Deviatio tells you the rage from the mea which cotais aroud 68% of the data (if data is ormally distributed you will lear about this at a later date) For example, if 100 studets have a mea height of 150cm ad a stadard deviatio of 10cm of the studets are withi oe Stadard Deviatio 95 of the studets are withi two Stadard Deviatios E

35 σ = σ x xҧ σ = σ x σ x σ = S xx σ x ҧ x σ x σ x S xx The variace ad stadard deviatio ca also be used to aalyse a set of data The marks gaied i a test by seve radomly selected studets are: x x Fid the variace ad stadard deviatio of the marks of the seve studets. σ x = 36 σ x = 18 σ = σ x σ x σ = σ = Square root Sub i values we eed σ x ad σ x Calculate The middle of the 3 formulae above is most commoly used whe you have the raw data E

36 σ = σ x xҧ σ = σ x σ x σ = S xx σ x ҧ x σ x σ x S xx The variace ad stadard deviatio ca also be used to aalyse a set of data The marks gaied i a test by seve radomly selected studets are: Mea = 5.14 Fid the variace ad stadard deviatio of the marks of the seve studets..17 So this meas that 68% of the data is withi.17 of the mea. Note that 68% is ot always a possible percetage though! So out of the origial 7 values, 4 are withi 1 stadard deviatio of the mea This is oly 57% rather tha 68%, because the data size is very small! E

37 σ = σ x xҧ σ = σ x σ x The variace ad stadard deviatio ca also be used to aalyse a set of data Shamsa records the time spet out of school durig the luch hour to the earest miute, x, of the female studets i her year. The results are show i the table. σ = S xx σ x ҧ x σ x σ x Time (mis) x f fx fx Frequecy σ f = σ fx = 308 σ fx = S xx Calculate the stadard deviatio of the time spet out of school. For tabled data, we eed to use a modified formula σ x σ x σ fx σ f σ fx σ f σ fx σ f σ fx σ f Sub i values we eed to use the table to calculate these! Calculate E

38 σ = σ x xҧ σ = σ x σ x σ = S xx σ fx σ x ҧ σ f σ fx σ f x σ x σ x S xx The variace ad stadard deviatio ca also be used to aalyse a set of data For a grouped table we eed to use the midpoits (as i the previous sectio) Ady recorded the legth, i miutes, of each telephoe call he made for a moth. The data is summarized i the x table below. f fx fx Legth of call (mis) Frequecy 0 < l < l < l < l < l < l σ f = 7 σ fx = 85 σ fx = Calculate a estimate of the stadard deviatio of the legth of the phoecalls E

39 σ = σ x xҧ σ = σ x σ x σ = S xx σ fx σ x ҧ σ f σ fx σ f x σ x σ x S xx The variace ad stadard deviatio ca also be used to aalyse a set of data Ady recorded the legth, i miutes, of each telephoe call he made for a moth. The data is summarized i the x table below. f Legth of call (mis) Frequecy 0 < l < l < l < l < l < l σ fx σ f σ fx σ f Sub i values we eed to use the table to calculate these! Calculate Calculate a estimate of the stadard deviatio of the legth of the σ f = 7 phoecalls σ fx = σ fx = 85 E

40

41 σ fx σ f σ fx σ f σ x σ x Codig ca be used to make a set of values simpler to work with x ҧ = σ x Imagie we have the followig data o people s heights (cm) S xx If umbers i a data set are particularly large, they ca all be altered i the same way to make them smaller This will however affect the measures of locatio ad dispersio that we have bee calculatig, ad you eed to be aware of these effects Mea = Rage = 7 If all the values were multiplied by, what would happe to the measures above? Mea ad rage would double If all the values above had 0 added to them, what would happe to the measures above? Mea would icrease by 0, but the rage would stay the same If you chage a set of data by addig or subtractig a amout, this will ot affect the rage, or ay other measures of spread, such as the IQR or stadard deviatio F

42 σ fx σ f σ fx σ f σ x σ x x ҧ = σ x S xx Codig ca be used to make a set of values simpler to work with x 33 y 3. y 10.4 A scietist measures the temperature, x at five differet poits i a uclear reactor. Her results are give below: , 355, 306, 317, 340 σ y = 15 σ y = a) Use the codig y = x 300 this data 10 to code b) Calculate the mea ad stadard deviatio of the coded data c) Use your aswer to b) the calculate the mea ad stadard deviatio of the origial data. തy = σ y തy = 3 To code the data. take each startig value, subtract 300 from it, ad divide the aswer by 10 Now we ca calculate the mea ad stadard deviatio of this ew iformatio Sub i values തy = 15 5 Work out σ y = σ y = σ y = 1.7 σ y σ y 15 5 Sub i values Work out F

43 σ fx σ f σ fx σ f σ x σ x x ҧ = σ x S xx Codig ca be used to make a set of values simpler to work with A scietist measures the temperature, x at five differet poits i a uclear reactor. Her results are give below: 33, 355, 306, 317, 340 a) Use the codig y = x 300 to code 10 this data ഥy = 3 b) Calculate the mea ad stadard deviatio of the coded data c) Use your aswer to b) the calculate the mea ad stadard deviatio of the origial data. σ y = 1. 7 Origial mea The origial data had 300 subtracted, ad the was divided by 10 We eed to reverse this, so multiply by 10, ad the add = 330 Origial stadard deviatio The origial data had 300 subtracted, ad the was divided by 10 The subtractig 300 will ot have affected the stadard deviatio, so we oly eed to multiply by = 17. F

44 ҧ σ fx σ f σ fx σ f σ x σ x x ҧ = σ x S xx Codig ca be used to make a set of values simpler to work with From the large data set, date o the maximum gust, g kots, is recorded i Leuchars durig May ad Jue 015. The data was coded usig h = g 5 10 ad the followig statistics foud: S hh = തh = = 61 The mea has had 5 subtracted ad the bee multiplied by 10 തh = You ca write this as a formula: gҧ 5 10 = gҧ = gҧ 5 5 = gҧ We kow the mea of h Multiply by 10 Add 5 This is a better way to show your workigs! Calculate the mea ad stadard deviatio g = 5 of the maximum gust i kots. F

45 ҧ σ fx σ f σ fx σ f σ x σ x x ҧ = σ x S xx Codig ca be used to make a set of values simpler to work with From the large data set, date o the maximum gust, g kots, is recorded i Leuchars durig May ad Jue 015. The data was coded usig h = g 5 10 ad the followig statistics foud: S hh = തh = = 61 Calculate the mea ad stadard deviatio g = 5 of the maximum σ g = gust 8.45 i kots. We eed to fid the stadard deviatio of h first! σ h = S hh σ h = σ h = Use the formula above As the subtractio has ot affected the stadard deviatio, we oly eed to udo the divisio by 10 Like with the previous example, we ca write this as a formula σ h = σ g = σ g = σ g Sub i values Calculate Sub i values Multiply by 10 F