gcse statistics coursework

8/18/2019 GCSE Statistics Coursework

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Introduction

The aim of this coursework is to estimate how good the population (HAMstudents) is at estimating string lengths and angles. To do this, I will usesecondary data (rstly ecause it would e !ery time"consuming to collectprimary data from all the population, and secondly ecause I was alreadypro!ided with secondary data). #ecause the data is too e$tensi!e (there aremore than %&& pieces of data), I will analyse it using samples (I ha!e a limitedtime to complete this coursework, and if I analysed e!ery piece of data, Iwouldn't complete it on time). #oth uantitati!e (e.g. I) and ualitati!e data(e.g. hair colour) will e considered when formulating my hypothesis andconclusion. I chose to do this coursework only in the computer ecause thegraphs will e more accurate and not in*uenced y human error.

I was told to use a sample percentage etween +& and -& of the population.I chose to use -&, ecause it will represent the population etter (it's theiggest percentage I could use), ut it's still small enough to handle.

There are se!eral sampling methods I could use (for e$ample, random,systematic, stratied sampling...), ut I chose to use stratied systematicsampling " stratied ecause it chooses the sample proportionally, making thisestimation less iased, and systematic ecause it's easier to select thanrandom sampling (you ust ha!e to choose e!ery nth piece of data on the list).

After my sampling is complete, I'll re!iew it to make sure there are noanormalities or errors, and if I nd any, I'll take those out of my samples andreplace them with another piece of data chosen at random.Afterwards, I will formulate etween three and !e hypothesis (i.e. what I thinkthe outcome of the in!estigation will e / e.g. I think 0ear 11s are etter atestimating than 0ear 2s, or I think 0ear 11s with an I ao!e 1&& are etter atestimating than 0ear 11s with a lower I).3ne of the main reasons why I'm doing this coursework is for my e$aminers tosee if I can test my hypotheses using di4erent methods. To pro!e I can, I willha!e to test my hypotheses using more than one method for each one. I ha!e arange of choices, and here are some e$amples5

6ie charts7graphs (grade 8) 9catter graphs (grade :) A!erages / mean, median, mode (grade :) 8stimate of mean (grade ;) ;umulati!e freuency7o$ plots (grade #)

9pearman's rank (grade #) Histograms (grade A)

<ogically, I will try to choose the higher grade methods so I might achie!e ahigher grade in this coursework. It's not a prolem if my hypotheses turn out toe wrong / ust the fact that I can test them will gi!e me some marks.

=hen looking at all my testing of the hypotheses, I will e ale to write aconclusion saying what I ha!e found out and interpreting the results (this meansthat if, for e$ample, girls' guesses are closer to the actual gures than oys'guesses, girls are etter at estimating than oys).

In spite of all these things, I still ha!e some limitations> for e$ample, I ha!e noassurance that the data pro!ided is correct or uniased (some people may ha!e

+



made up some !alues or een in*uenced y what others wrote). The data alsohas a wide range5 people's guesses are far from each other, and ualitati!edata isn't !ery concentrated (for e$ample, hair colours are !ery !aried).

If I ha!e some time after nishing the coursework, I can add some e$trafeatures as an e$tension / like creating my own sur!ey7uestionnaire, fore$ample. Another thing I can do is to ask my own family and friends> in this wayI could collect some primary data.

Sampling calculations

To do my calculations, I need to choose an appropriate sample of data.

I chose to use -& of the total population in my sample. To do my sampling, I ga!e myself the option to choose etween four methods5

-



?andom sampling 9ystematic sampling 9tratied systematic sampling 9tratied random sampling

I chose stratied systematic sampling, ut I will demonstrate the other threemethods as well.

Random samplingIn random sampling, e!eryone has an eual chance of eing chosen, regardlessof their gender or age. =e don't need to know how many oys and girls weshould choose from each year, we only need to know how many students weshould choose from each year. This means that, sometimes, the sampling is notfair (for e$ample, if there are doule the oys than the girls, and we use randomsampling, it will e !ery likely that there are much more oys than girls who arepicked to e part of the sample.)

To do the calculations, I will need the total population (which I named tp) andthe sample population (sp). sp will always e -& of tp, as I want to use -& of the population in my sample.

Year 7 There are 1&& students in 0ear 2.

sp @ tp $ &.-sp @ 1&& $ &.-sp @ -&

The result tells us we ha!e to choose -& people at random from 0ear 2,regardless of their gender.

Year 8 There are 1+& students in 0ear .

sp @ tp $ &.-sp = 1+& $ &.-sp @ -B

This means we ha!e to choose -B people in 0ear .

Year 9 There are 22 students in 0ear C.

sp @ tp $ &.-sp @ 22 $ &.-sp @ +-.1 D +-

9o we now disco!ered that we ha!e to choose +- people in 0ear C to e part of our sample.

Year 10I'!e counted 2 students in 0ear 1&.

sp @ tp $ &.-

%



sp @ 2 $ &.-sp @ +-.% D +-

=e need to randomly choose +- people in 0ear 1&.

Year 11Einally, there are F students in 0ear 11.

sp = tp $ &.-sp @ F $ &.-sp @ 12.% D 12

=e must choose 12 people from 0ear 11.

Systematic samplingIn this sampling method, e!ery nth piece of data from the list is chosen. To ndthe !alue of n, we must di!ide the total numer of students in the school y thetotal numer of students in each year.8$ample (year 2)5

Total number of students in school @ 1&&G1+&G22G2GF @ %--

Y7 @ %--1&& @ %.-- D %

F

So here are the gures of myrandom samples

Year 7-& students chosen

Year 8-B students chosen

Year 9+- students chosen

Year 10+- students chosen

Year 1112 students chosen



In the 0ear 2 list, we must choose e!ery %th student, regardless of their age orgender.

Year 8 There are 1+& students in 0ear .

Y8 @ %--1+& @ -.B1 D %

=e will choose e!ery %th student in the 0ear list.

Year 9 0ear C has 22 students.

Y9 @ %--22 @ F.B+ D B

=e ha!e to choose e!ery Bth student from the list of 0ear C students.

Year 10In 0ear 1&, there are 2 students.

Y10 @ %--2 @ F.FF D B

=e would also ha!e to choose e!ery Bth student from the list of 0ear 1&students.

Year 11 There are only F students in 0ear 11.

Y11 @ %--F @ 2.%2 D 2

This means we ha!e to choose e!ery 2th student from the list of 0ear 11s.

Stratiedsystematic

sampling This is the method I'll use to otain the samples I will e working with, ecauseit represents the population more fairly than any other sampling method(people are chosen according to their proportions in the population).In this type of sampling, what we need to do is nd the proportion of eachgender to the total numer of students (we do that y di!iding, for e$ample, thenumer of oys y tp). Then, we multiply the result y the sie we want oursample to e, which is sp. As in random sampling, sp will e eual to tp $ -&.

These calculations will gi!e me the numer of oys and girls I ha!e to chooseaccording to their proportion in one particular year.

B

!ere are the gures of my systematic samples

Year 7 Year 10;hoose e!ery %th student ;hoose e!ery Bth student

Year 8 Year 11;hoose e!ery %th student ;hoose e!ery 2th student

Year 9;hoose e!ery Bth student



To see which students I ha!e to choose, I di!ide the total numer of students inone year y the numer of oys (or girls) in that same year.8$ample5

1F oys-& girls%F students

b @ %F1F @ -g @ %F-& @ 1.F D +

This means I should choose e!ery -rd oy and +nd girl in the list (after separatingoys and girls).

Year 7In this case, as we already know, there are 1&& students in 0ear 2. There are %Boys and F% girls.

In 0ear 2, sp @ 1&& $ &.- @ -&.

b @ (%B1&&) $ spb @ &.%B $ -&b @ 1-. D 1%

g @ (F%1&&) $ spg @ &.F% $ -&g @ 1B.+ D 1B"#e ha$e to round the results because %e&re tal'ing about people ( there areno fractions of a person) *f the results are not %hole numbers+ %e ha$e to roundthem),

<et's check if b G g @ sp5b G g @ sp1B G 1% @ -&

The sum of b and g is eual to sp, so our !alues are correct.Jow let's see the systematic numers5

b @ 1&&%B @ +.12 D +g @ 1&&F% @ 1.F D +

This means we ha!e to choose 1% oys and 1B girls from 0ear 2 (e!ery +nd

oyand +nd girl from the list). 0ear There are FC oys and B1 girls (1+& students).

sp @ 1+& $ &.- @ -B

b @ (FC1+&) $ spb @ &.%C $ -Bb @ 12.B% D 1

g @ (B11+&) $ sp

g @ &.F1 $ -Bg @ 1.-B D 1

2



1 G 1 @ -B

b @ 1+&FC @ +.&- D +g @ 1+&B1 @ 1.C2 D +

=e should choose 1 oys and 1 girls from 0ear . Again, we should choosee!ery +nd oy and +nd girl.

Year 9 There are -+ oys and %F girls5 22 students in total.

sp @ 22 $ &.- @ +-.1 D +-

b @ (-+22) $ spb @ &.%+ $ +-b @ C.BB D 1&

g @ (%F22) $ spg @ &.F $ +-g @ 1-.-% D 1-

1& G 1- @ +-

b @ 22-+ @ +.%1 D +g @ 22%F @ 1.21 D +

There will e 1& oys and 1- girls in our 0ear C sample. 8!ery +nd oy and +nd

girl will e chosen.

Year 10%F oys and -- girls are in 0ear 1& (2 students).

sp @ 2 $ &.- @ +-.% D +-

b @ (%F2) $ spb @ &.F $ +-b @ 1-.-% D 1-

g @ (--2) $ sp

g @ &.%+ $ +-g @ C.BB D 1&

1- G 1& @ +- b @ 2%F @ 1.2- D +g @ 2-- @ +.-B D +

=e ha!e to choose 1- oys and 1& girls for our sample (e!ery +nd oy and girl).

Year 11 There are + oys and -& girls in 0ear 11. That makes F students.

sp @ F $ &.- @ 12.% D 12



b @ (+F) $ spb @ &.% $ 12b @ .1B D

g @ (-&F) $ spg @ &.F+ $ 12g @ .% D C

G C @ 12

b @ F+ @ +.&2 D +g @ F-& @ 1.C- D +

=e ha!e to choose oys and C girls for our 0ear 11 sample (e!ery +nd oy andgirl).

Stratied random samplingI will not do the calculations for this sampling method, since I would erepeating myself (the only di4erence etween stratied systematic samplingand stratied random sampling is that stratied random sampling chooses thedata randomly, while stratied systematic sampling chooses the datasystematically). Knowing this, the only thing I will do is ignore the rule of choosing e!ery nth student, ut I will still conser!e the sample sies and theproportion of oys and girls that should e chosen. 9o after doing this, thegures for stratied random sampling would e5

C

These are the gures of my stratied systematic samples

Year 7 Year 9 Year 11-& students chosen +- students chosen 12students chosen1% oys (e!ery +nd) 1& oys (e!ery +nd) oys (e!ery +nd)1B girls (e!ery +nd) 1- girls (e!ery +nd) C girls(e!ery +nd)

Year 8 Year 10-B students chosen +- students chosen1 o s e!er +nd 1- o s e!er +nd

Stratied random sampling gures

Year 7 Year 10-& students chosen +- studentschosen1B oys 1- oys1% girls 1& girls

Year 8 Year 11-B students chosen 12 studentschosen1 oys oys1 girls C girls

Year 9+- students chosen1& o s



And in this way I conclude my sampling calculations. As I said earlier, thegures that I will use are the stratied systematic sampling gures. That takesus to the ne$t part5 sample selection.

Sample selection

9ince I'm using stratied systematic sampling, what I need to do is separateoys and girls from each year. This means I will ha!e 1& tales5

0ear 2 oys 0ear 2 girls 0ear oys 0ear girls 0ear C oys 0ear C girls 0ear 1& oys

0ear 1& girls 0ear 11 oys 0ear 11 girls

The tales will not show all the data, only the selected samples. The rst tale is on the ne$t page.

1&



Hypotheses

This is the part where I think of some guesses (hypotheses) aout how goodparticular year groups or other groups are at estimation. Here are myhypotheses5

!ypothesis 1K9- girls with an I o!er 1&& estimate lengths more accurately than K9- girlswith an I under or eual to that !alue.

!ypothesis -K9% girls o!er 1.BF m estimate length # more accurately than K9- girls with aheight eual to or smaller than 1.B& m.

!ypothesis .K9% oys estimate angle : more accurately than K9- oys.

!ypothesis /K9- londe girls estimate angle ; more accurately than K9- girls with rownhair.

9o now I ha!e the hypotheses / that means the ne$t step is to start testingthemL

Testing the hypotheses, part 1

Eirst of all, let's look at the rst hypothesis5

11



KS3 girls, IQ > 100

KS3 girls, IQ ≤ 100

KS3 girls, IQ > 100

S. girls %ith an * o$er 100 estimate lengths more accurately than S. girls%ith an * under or e2ual to that $alue)

3stimated mean The rst thing I'm going to do is a tale where I'll group the data. This estimateis for K9- girls with an I o!er 1&&5

9o the estimate mean will e5

=e see the guesses were uite good, ecause the actual !alue of length A is +-cm. The estimate mean of the guesses is only aout 2 mm o4. As a percentage,that de!iation would e5

This is the de!iation in K9- girls' guesses with an I o!er 1&& (related to lengthA).Jow let's look at K9- girls with an I eual to or under 1&&5

An outlier (-) was remo!ed from the original 1 pieces of data.

9urprisingly, the girls with a lower I ha!e guessed closer to the actual !alueL Ididn't need to calculate the de!iation to notice this.

To nd out the gures for length # (FC.F cm)5

1+

687.5

29 =23.71cm

0.71

23 ×100=

71

23=+3.087

397.5

17 =23.38cm




KS3 girls, IQ > 100

The total freuency for this group has gone down from 1 to 1B ecause twooutliers, +& and CF, were remo!ed (I hadn't noticed these when I was doing thesamples).

9o we ha!e a tie5 for length A, girls with an I 1&& ha!e guessed moreaccurately, while for length #, the opposite happened.

To (dis)pro!e my hypothesis, I'll ha!e to use another method.

45 graphs6bo plots "length ,

1-

1747.5

29 =60.26cm

60.26−59.5=0.76cm

0.76×100=

76=+1.28

930

16=58.13cm

58.13−59.5=−1.37 cm

−1.37×100=

−137=−2.3




Eor cumulati!e freuency graphs, the points are always plotted at the upperound of the inter!al, so that's what I'!e done.After plotting the graph (which is in page 1B), I ha!e to nd the positions of 1,+ and -.

Jow I can draw lines on the graph to nd out the !alues of the uartiles. Afterdoing this, I found that51 @ +1.F cm+ @ +%.F cm- @ +2 cm

The median (+) is 1.F cm ao!e the actual gure of +- cm. As a percentage,that is B.F+ ao!e the actual gure. To nd out how consistent the data is, I drew a o$ and whisker plot ased onthe ;E graph (in the ne$t page). Howe!er, the est way to take information outof a o$ plot is to compare it with another o$ plot (which will e related to K9-girls with a lower I), so the ne$t thing I will do is plot another ;E graph andanother o$ plot for K9- girls with an I 1&&.Eirst, we need some tales5

1%

Total frequency = 29

Q2 position =29

2 =14.5

Q1 position =29

4 =7.25



The ;E graph and the o$ plot for this data are in page 12. The information Ican take from these is in page 1.

1F

Q1 position =17

4 =4.25

Q2 position =17

2 =8.5



1 +& ++ +% +B + -& -+ -% -B

KS3 girls, IQ > 100

Estimated length !cm"

1 +& ++ +% +B + -& -+ -% -B&

+

%

B

1&

1+

1%

1B

1

+&

++

+%

+B

+

KS3 girls, IQ > 100


#umulati$e %re&uency

1B



1 +& ++ +% +B + -& -+ -% -B&

+

%

B

1&

1+

1%

1B

1

+&

++

+%

+B

+

KS3 girls, IQ > 100



12



1 1C +& +1 ++ +- +% +F +B +2 + +C -& -1

&

1

+

-

%

F

B

2

C

1&

11

1+

1-

1%

1F

1B

12




1



1 1C +& +1 ++ +- +% +F +B +2 + +C -& -1



1C



<ooking then at the two o$ plots (which now ha!e the same scale so that theycan e compared accurately),

1 +& ++ +% +B + -& -+ -% -B

KS3 girls, IQ > 100


1 +& ++ +% +B + -& -+ -% -B



+&



B&BF

2&2F

&F

C&CF

1&&1&F

11&11F

1+&

-F

%&

%F

F&

FF

B&

BF

2&

2F

&

F

KS3 girls

IQ

Estimated length ' !cm"

Aftercollecting all the data (which I did in a separate sheet), I otained this graph5

++



KS+ girls > 1() m

1

+

-

%

F

KS3 girls ≤ 1()0 m !guesses in cm"

The same thing was done for K9% girls N 1.BF m5

+%



1

+

-

KS+ girls > 1() m !guesses in cm"

The correct !alue of length # is FC.F cm. If we look at the inter!al with theiggest freuency (the one with the largest slice), it contains the correct guessin oth situations, so we can say that oth groups ha!e mostly guessedaccurately. #ut for me to know which of the groups has guessed the mostaccurately, I'll test this hypothesis again with a di4erent method.

rdered stem and leaf diagrams6a$erages

+F



KS3 girls ≤ 1()0 m

To test this hypothesis, I need to nd the asic a!erages for oth groups /mean, median and mode. #ut this time I want to do more than ust nd thea!erages> instead, I'll nd them through an ordered stem and leaf diagram (onediagram for each group) showing the guesses of e!ery girl in the sample (on thene$t page)5

-edian To nd the median's position, we count the numer of !alues (which is +B), andthen5

In this case, we will ha!e to calculate the mean of the 1- th and 1%th !alues(which we nd y counting the lea!es from right to left), and this will gi!e us themedian5

9o we ust found that the median for these K9- girls is FC cm / this numer is!ery close to the correct !alue of FC.F cm, showing the a!erage guess is uiteaccurate.

-ean=e ha!e to add up all the !alues (the result is 1F2F) and di!ide it y thenumer of !alues (the freuency)5

The mean is slightly higher than the median (ecause it considers all the piecesof data), ut B&.+ cm is still uite an accurate guess.

-ode.modal group

+B

26 1

2= 13.5t! "alue

59 59

2= 59 c

1575

26= 60.58 c



KS+ girls > 1() m

To nd the mode, we look at the numer which appears the most in the stemand leaf diagram. This is uite easy to work out / the mode for this group is,again, FC cm (which is !ery accurate).

The modal group is the group which contains most of the data (in this case, thestem which contains most of the data). The modal group for this set of data isclearly F+"FC. This group does not contain the actual !alue, ut it is !ery close(only &.F cm o4).

/ange The range is otained y sutracting the lowest !alue from the highest !alue. If we do this, we will get 1 / -2 @ %% cm, which is a high !alue (ut we need toconsider the sample was also ig> therefore, we should compare this to therange of the other group efore taking any conclusions).

Qntil now, this group has guessed !ery accurately in general, ut let's take alook at the other group and then we can make some comparisons.

-edian

The median for this group is further o4 from the actual !alue. According to thehypothesis, this was the group supposed to guess more accurately, so thehypothesis is looking dispro!en at the moment. <et's look at the other a!erages

efore making conclusions5

-ean

The mean guess for this group is also further o4 from the correct !alue than themean guess of the other group. At this point, it looks like K9- girls are guessingmore accurately.

-ode.modal group

There are two modal groups5 F%"F and 2&"2F. =hen this happens, we say a setof data is imodal.

+2

∑ f+1

2=

12+1

2= 6.5t! "alue

6t! 7t!

2=

61+65

2= 63 cm

∑ f#

∑ f=

767

12= 63.92 cm



KS3 girls ≤ 1()0 m

The mode for this set of data is also more than one !alue> F%, F and 2+ allappear twice.

This doesn't allow us to reach much of a conclusion with this a!erage.

/ange The range for this group is 2F / F% @ +1 cm. This is less than half of the othergroup's range, meaning this set of data is much more consistent, and therefore,etter. Qnfortunately for the hypothesis, this is the only statistical !aluepointing in fa!our of this group.

This hypothesis seems dispro!en at the moment, ecause K9- girls' 1.B& mha!e guessed length # more accurately than K9% girls N 1.BF m. Howe!er, I stillthink K9%'s sample was slightly too small, and that might ha!e in*uenced someof the results. I think I will ha!e to try another method to e sure.

Standard de$iation This method considers all of the data, and it gi!es us an idea of how spread out

or consistent the data is in relation to an a!erage (the mean). #oth groups ha!ea similar mean, ut the group with more consistent data and an a!erage closerto the correct !alue will e the group who has guessed more accurately.

KS3 girls ≤ 1()0 m The formula for the calculation of standard de!iation is the following5

The lower case letter sigma symolises standard de!iation, and it is ust one of the symols. The $ ar is the symol for the mean (which I ha!e calculatedearlier / B&.F cm). The calculations reuire a tale, which is elow5

+

σ =√∑( x−´ x)2

n



KS+ girls > 1() m

The numer emphasised in lue is the main gure we need for the calculation,which we can now make5

=e now disco!ered that the standard de!iation for the guesses of K9- girls 1.B& m is C.2 cm. This means that, in general, aout B of the totalpopulation would e within C.2 cm ao!e or elow the mean (B&.F cm). Inother words, aout B of the population has guessed etween F&. cm and2&.-B cm.<et's make the calculations for K9% girls N 1.BF m and compare them (the meanfor this group is B-.C+ cm)5

This means K9% girls' guesses are less spread out from the a!erage / meaning

K9%'s data is more consistent and closer to the a!erage of B-.C+ cm.;onsidering this, I conclude...

!ypothesis - conclusion The hypothesis was pro!en / K9% girls can estimate length # more accuratelythan K9- girls, ecause their data was more consistent and closer to the correct!alue.

+C

σ =√∑( x−´ x)2

n❑⇔

σ =√2486.3464

26 ❑

⇔

σ =√ 95.63=9.78cm

σ =√∑( x−´ x)2

n❑⇔

σ =√664.9168

12 ❑

⇔

σ =√ 55.41=7.44cm



KS3 oys

Testing the hypotheses, part 3

The third hypothesis was5

S/ boys estimate angle > more accurately than S. boys)

<ercentagesI'm planning to count e!ery correct estimate in K9- and K9% and di!ide thatnumer for the total of indi!iduals in the sample. The result multiplied y 1&&will gi!e me the percentage of correct guesses, and the group with the largestgure will e the group that can guess more accurately.

KS3 oys The total numer of indi!iduals in the K9- sample is %+. 3ut of these, % oys

guessed angle : would e 1%FP, which is true. In a percentage, that is5

9o the percentage of correct guesses in K9- is C.F+.

KS+ oys There are +1 pieces of data in total in the K9% sample. 3ut of all theseindi!iduals, + ha!e guessed the e$act !alue of 1%FP.#ut wait a minuteL

The proportion % out of %+ has the same !alue than + out of +1L That meansK9- oys and K9% oys ha!e the e$act same proportion of correct guesses, sowe ha!e no means of knowing who guesses more accurately y ust using thismethod.

45 graphs6bo plots6erhaps we might e ale to reach a conclusion y nding out each group'smedian and determining which one has the most consistent data.

KS3 oys This group has %+ pieces of data, so the median's position will e half of this(+1). 1's position will e half way through the median (1&.F). - will e R of

the way through the total freuency (+1 G 1&.F @ -1.F). The points are alwaysplotted at the upper ound of the inter!al.

-&

4

42×100=9.52



1&& 1&F 11& 11F 1+& 1+F 1-& 1-F 1%& 1%F 1F& 1FF 1B&

KS3 oys

Estimated angle !2"

After plotting the ;E graph, I found that5

<owest !alue @ 1&&P1 @ 11.FP

-+



KS+ oys

+ @ 1-FP- @ 1%1.FPHighest !alue @ 1B&P

;onsidering this data, I drew a o$ plot which is shown elow the ;E graph. The central F& of the population (represented y the o$) looks uite centred,meaning that the data is not mainly made up of high or low !alues. It looks likethe -rd uarter of the data is the section which contains most of the !alues (+Fof the data is etween 1-FP " 1%1.FP).

KS+ oysEirst, we need a tale5

The total population of this sample is +1, so +'s position will e half of this(1&.F). 1's position is half of + (F.+F). -'s position is the triple of 1 (1F.2F).

=hen analysing the ;E graph (on the ne$t page), I disco!ered that5

<owest !alue @ 1+FP1 @ 1+P+ @ 1-%.FP- @ 1%-.FPHighest !alue @ 1B&P

I then used this data to draw the o$ plot (shown elow the ;E graph). The comparison of the two o$ plots is shown in page -&.

--



1+F 1-& 1-F 1%& 1%F 1F& 1FF 1B&&

1

+

-

%

F

B

2

C

1&

11

1+

1-

1%

1F

1B

12

1

1C

+&

+1

KS+ oys

Estimated angle !2"


-%



1+F 1-& 1-F 1%& 1%F 1F& 1FF 1B&

KS+ oys

Estimated angle !2"

-F



1&& 1&F 11& 11F 1+& 1+F 1-& 1-F 1%& 1%F 1F& 1FF 1B&

KS3 oys

#omparing o plots

1&& 1&F 11& 11F 1+& 1+F 1-& 1-F 1%& 1%F 1F& 1FF 1B&

KS+ oys

Estimated angle !2"

K9- oys ha!e a larger spread (this is not a4ected y the sie of the sample,ecause K9% could ha!e data as spread out of K9-, regardless of the fact

-B



KS3 runette and londe girls, respecti$ely

that the sample is only half of K9-'s sample). This means K9%'s data is moreconsistent and, possily, more accurate.

K9%'s interuartile range is smaller than K9-'s (1%-.F " 1+ @ 1F.FP against1%1.F / 11.F @ +-P).

Jeither group's interuartile range contains the correct !alue of 1%FP.

K9-'s median is larger y &.FP (1-FP against 1-%.FP). 9ince K9%'s data is more consistent and closer to the correct !alue, we cansay...

!ypothesis . conclusion This hypothesis was pro!en / K9% oys can estimate angle : more accuratelythan K9- oys.

Testing the hypotheses, part +

The fourth hypothesis stated5

S. blonde girls estimate angle 4 more accurately than S. girls %ith bro%nhair)

rdered double stem and leaf diagrams

The real !alue of angle ; is BFP. The stem and leaf diagram on the left showsthe guesses of K9- runette girls and the right hand side shows the guesses of K9- londe girls (as the title indicates). This way of showing a pair of stem andleaf diagrams is enecial, ecause you can compare them more easily and at aglance. 9ome information we can immediately draw from this method5

#oth samples' modal group is B&P " BCP (the longest leaf), which contains thecorrect guess.

The sample of runettes is multimodal (trimodal). The - modes are %FP, F%Pand BFP (all of these appear - times), with BFP eing the correct !alue of theangle.

The sample has a range of +P " %FP @ -2P.

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The total freuency of the sample is +&. The position of the median is the1&.Fth !alue, which is etween B+P and B+P " this means the median is B+P.

The sample of londes is unimodal / the mode is BFP (the correct !alue),which appears three times in the sample.

The range of this sample is 2FP " -1P @ %%P. This !alue is 2P larger than therange of the sample of K9- runette girls. Although the range is higher, it is only so ecause of the least - !alues " -1P,

-+P and -+P. The rest of the !alues are concentrated etween F&P and 2FP,which would e practically a range of only +FP.

The total freuency of this sample is ++. The position of the median istherefore the 11.Fth !alue / this corresponds to the !alue of (B- G BF) 5 + @B%P.

The median of the guesses of londe girls is +P larger than the median of theguesses of runette girls, ut closer to the actual !alue of the angle.

All these statistics point K9- londe girls as the group who can estimateangle ; more accurately, ut I should test the hypothesis with more thanone method to remo!e any douts.

Relati$e fre2uency I am thinking of otaining the relati!e freuency of the correct guess for each of the groups. The way to otain this statistics is to di!ide the freuency of the!alue we want to measure (in this case, BFP) y the total freuency of thesample. After doing this for each of the groups, the sample with the highestrelati!e freuency of correct guesses will e the sample who can guess more

accurately.

KS3 runette girls Total freuency @ +&Ereuency of correct guesses @ -

;alculation of relati!e freuency of correct guesses5

KS3 londe girls

Total freuency @ ++Ereuency of correct guesses @ -

;alculation of relati!e freuency of correct guesses5

The relati!e freuency is pointing K9- runette girls as the ones who guessmore accurately (although ust y a short margin). This means the two methodsI ha!e used are against each other, ecause each one identies a di4erent

sample as the one who can estimate more accurately.

-

$orrect %uesses

Total %uesses =

3

20=0.15

$orrect %uesses

Total %uesses =

3

22≈0.14



Howe!er, relati!e freuency is more accurate when a !ery large numer of trials ha!e een made (normally at least -&&), and that suggests that theseresults do not portray the data !ery well.

Qnfortunately, I am short of time to de!elop this coursework further, so I willha!e to say...

!ypothesis / conclusion#ased on my ordered doule stem and leaf diagram and the analysis I took fromit, this hypothesis is pro!en / K9- londe girls ha!e more consistent data andthe median is closer to the actual !alue. The londe girls' mode is also moreaccurate (the sample is unimodal against the trimodal result of the runettegirls. #ecause of all these facts, I elie!e K9- londe girls are the ones who canestimate angle ; more accurately, ust like my original hypothesis stated.

#onclusion

Here are the conclusions to each of my hypotheses5

!ypothesis 1S. girls %ith an * o$er 100 estimate lengths more accurately than S. girls%ith an * under or e2ual to that $alue)

This hypothesis was partly pro!ed / K9- girls with I N 1&& can estimate length# more accurately, ut not length A.

!ypothesis -S/ girls o$er 1):; m estimate length more accurately than S. girls %ith aheight e2ual to or smaller than 1):0 m)

The hypothesis was pro!en / K9% girls can estimate length # more accuratelythan K9- girls, ecause their data was more consistent and closer to the correct!alue.

!ypothesis .

S/ boys estimate angle > more accurately than S. boys) This hypothesis was pro!en / K9%'s data was more consistent and closer to theactual !alue.

!ypothesis /S. blonde girls estimate angle 4 more accurately than S. girls %ith bro%nhair)Although this hypothesis was a it rushed, it was pro!en / ased on mystatistics, K9- londe girls can estimate angle ; more accurately andconsistently.

There were some hindrances to my work, and a particular e$ample was the lack

of time. This was mainly notorious in the testing of hypothesis %, where mytesting was limited to only one method. There were also some occasions where

-C



the samples might ha!e een a it small / in hypothesis +, for e$ample. K9%'ssample had only 1+ people.#esides my teacher, I had other sources of aid for the completion of thiscoursework, like mymaths.co.uk, where I got an idea of what methods might emore suitale and how to apply them properly. In some occasions I searched onSoogle for help (this was mainly for me to learn how to draw graphs in 8$cel,particularly o$ plots). I do not resent the decision of writing this courseworkcompletely in computer, ecause it gi!es a neater presentation and moreaccuracy in the graphs (e.g. when drawing lines of est t in scatter graphs ordrawing the angles for a pie chart).If I had the chance of doing this again, I would like to ha!e had some more timeso I could de!elop my testing further. I elie!e this would ha!e oosted mygrade further. 3ther ways of collecting data would also ha!e een helpful(instead of using secondary data, perhaps I could ha!e created my ownuestionnaire to get only the information I want / as you will ha!e noticed, Ididn't e!en touch some of the data that was pro!ided to me, for e$ample K9+statistics or the distance pupils ha!e to co!er e!ery day to come to school.)

:espite these ostacles and errors (and some others you might nd), I still ha!ehope that I might achie!e a grade in this coursework that will satisfy me, ut asyou know, that is not my decision, ut yoursL

gcse statistics coursework

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