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  • Level: GCSE
  • Subject: Maths
  • Word count: 8531

Mayfield data

Extracts from this document...

Introduction

Mathematics Statistics Coursework

image25.pngimage25.png

Mayfield School Mathematics Statistics Courseworkimage00.png

Introduction

I have been assigned to complete a statistical investigation around the fictitious data of Mayfield High School, which has data of a real school. I will be completing this investigate for the subject Mathematics: Statistics. By completing the task that I have been set this will help me achieve my aim which is to gain a General Certificate of Secondary Education in this very particular subject. I will be using various techniques that I have recently studied and learnt and captured to produce a successful & efficient coursework. Alternating Statistical Methods will be used throughout this assignment to prove if my hypothesis is either correct or incorrect.

Task/Situation

I have decided to investigate majorly between the relationship between the height and weight of the pupils and to tell whether or not there is any correlation between them. I will take many actions as possible in achieving pure and efficiently results to meet the needs and requirement of my assignment. To meet my particular aim I will use many statistical interpretations and methods to help me form sufficient conclusions on what I have gained and obtained from the evidence that I will be collecting for this project.

My Hypothesis

A hypothesis is the outline of the idea/ideas which I will be testing and below are the following hypothesis I have decided to investigate for this particular assignment:

  • ‘ Boys at Mayfield School are Taller and Weigh more on average in     comparison to females’
  • ‘Key Stage 4 Students who watch more hours of television on average have a lower IQ Level’
  • ‘ Left Handed Students have higher IQ levels and Key Stage 2 Results in

Comparison right handed student at Mayfield High School’

...read more.

Middle

6

0

1

7

0

8

0

To find the modal weight I will now look at which frequency seems to have appeared the most often and for the Boys Modal Weight it is:

Modal Group for Boys: 40 ≤w< 50         Modal Weight for Girls: 40 kg

Below is Similar Data but Different Calculation methods to find the Mean, Median and also the Mode as this will help me towards proving my hypothesis also this data will help me find the spread and the average of the data which will be helpful throughout this portfolio:image25.pngimage25.png

Boys: Height

Mean  

1.38 + 1.61 + 1.63 + 1.50 + 1.53 + 1.82 + 1.62 + 1.38 + 1.63 + 1.53 + 1.61 + 1.52 + 1.32 + 1.56 + 1.80 + 1.68 + 1.70 + 1.62 + 1.65 + 1.8image20.png

        20

= 31.28 Divided by 20 = 1.6m

Median: 1.65 m (Calculated with the Use of Microsoft Excel)

Range:   1.8 – 1.32 = 0.48 m

Boys Weight

Mean: 48.1 kg

Median: 48 kg

Range:  64-32 =32 kg

Girls: WeightGirls Heightimage21.pngimage21.png

Mean  Mean  

30 + 36 + 36 +37 + 38 + 40 + 40 + 40 + 41                            1.56        m

+ 45 + 46 + 47 + 47 + 52 + 52 + 52 +56

          57 + 59 + 60

image22.png

                             20

= 45.55 which is rounded to 46 kg

Median:  46kg             Median 1.56m

Range:   60 – 30 = 30                                                             Range 1.75 – 1.41 = 0.34m                                        

Histogram and Frequency Polygonsimage25.png

From the data I have collected and formed through my frequency tables and mean averages and many more I will now produce a Frequency Polygon and a Histogram that shows the Boys & Girls Height and Weights From my sample that I have taken a for my assignment. The Frequency Polygon will clearly identify the shape of my variations and both these forms of representing data will help me form a sufficient analysis.

Boys Height

image28.pngimage23.png

image29.png

Histogram for Boys Weightsimage25.png

image30.pngimage24.png

image31.png

image25.pngimage25.png

Girls Height and Weight Frequency Polygons & Histogram

These histograms now give me a clear picture of the data distribution. For the sample there is an even distribution of data. The middle group has the highest frequency which is expected. For the data to be evenly distributed, the other two sides must be fairly symmetrical. It is clear that the histogram do not show this. This shows that the majority of scores were above the median.

Girls Height

image32.pngimage33.pngThese Representations of Data shows admirably that the average height is 150 to 160 which I believe is slightly above average in my honest opinion for my sample and also I have come to find that the trend is rather varied although the frequency are upward to a certain point and downward from the peak onwards.

Girls Weightimage25.pngimage25.png

image34.pngimage35.png

This data shows with great intent that the highest frequency I 40 – 49 kg which shows that both boys and girls from the stratified sample that I have taken for this area of my assignment and this hypothesis in particular that they have a lot In common in terms of frequent data sources. In addition to this I have also come to find that the girls do not have many students above 69 kg whereas for boys there are 3 as times as many students above this height.

Cumulative Frequency Diagrams & Tablesimage25.pngimage25.png

I will now produce cumulative frequency diagram for both girls & boys height and weight and this will help me gain sufficient evidence towards forming my conclusion and I will also find percentiles and will produce box and whisker plots as this will help me view my data and trend efficiently.

1) BOYS HEIGHTS:

BOYS

Height (cm)

Cumulative

Frequency

Less than 140

2

Less than 150

2

Less than 160

8

Less than 170

16

Less than 180

17

Less than 190

20

Less than 200

20

Using a Statistical Program that I have downloaded I entered my data into the appropriate field and below Is a diagram of for the Cumulative Frequency which will help me identify percentiles and trends in my data. The graph will also include a line which helps me identify the trend clearly

image36.png

image25.pngimage25.png

2) BOYS WEIGHT

BOYS

Weight(kg)

Cumulative

Frequency

Less than 40

4

Less than 50

11

Less than 60

17

Less than 70

20

Less than 80

20

image37.png

From this particular Cumulative Frequency Diagram I have been able to find that the average weight from the sample that I have taken is 60kg which I my opinion is generally quite high and the Interquartile Range that I will be finding at a later stage will help me view the spread of data and the margin of error. I believe that all the graphs that I will produce will help me complete my objective and conclude efficiently and successfully.

Boys Height Box and Whisker Plotimage25.pngimage25.png

I then used the same statistical software I created a box and whisker which is a vital representation of data and it will

image38.png

Boys Weight Box and Whisker Plot

image39.png

Comparison of Box and Whisker Plots

Form this Box and Whisker Plot I will be able to find the median which will show the middle frequency of my data and also will be able to view the maximum and minimum values for both height and the weight and find the percentiles and the quartiles. For the Height the Median is 170cm and the interquartile range is 30 cm and the maximum value is 200 cm and minimum value is 140cm. whereas for the weight I have found the Median as 60kg which is respectively what I had predictable and is suitably accurate although the range of data for the weight is less and the data is negatively skewed in comparison to the height for the boys.

I will no be producing Cumulative Frequency Diagram and Tables for the Girls Height and Weight since this will help me form a sufficient and reliable comparison in height between Girls and Boys and form a successful and accurate conclusion to my hypothesis.image25.pngimage25.png

GIRLS HEIGHT

GIRLS

Height (cm)

Cumulative

Frequency

Less than 140

0

Less than 150

5

Less than 160

16

Less than 170

17

Less than 180

20

Less than 190

20

Cumulative Frequency Diagram

image40.pngimage02.png

GIRLS

Weight (kg)

Cumulative

Frequency

Less than 40

5

Less than 50

13

Less than 60

19

Less than 70

20

Less than 80

20

GIRLS WEIGHTimage25.pngimage25.png

Cumulative Frequency Diagram

My variable will be the Height

image41.png

Median 60 kg

Lower Quartile 50 kg

Interquartile Range 22g

Upper Quartile 72 kg

From this I have found the spread on data efficient and on the following page I will compare my results from the cumulative frequency against both boys and girls height and weight and make a suitable conclusion from this representation.

Comparison of Height and Weight of Boys and Girls from C.Frequency and Box Plotsimage25.pngimage25.png

From the Cumulative Frequency Diagram I have come to find that the Median height for boys in the sample that I have taken is larger in comparison to the girl’s median height. Boys Median Height is 170 cmwhereas Girls Median Height is 162 cmand shows that there is an 8cm difference in the middle figure from both sets of data I have collected although this may on some occasion be accurate due to the fact my sample may not be efficient and also the range of data varies where the IQ range for Boys is 21 cm and Girls in 33 cm which shows that the spread of data that I have from my sample for boys is narrower whereas girls have a wider spread and make the results reliable as a whole. As far as the Weights are concerned the range is similar and the range is rather symmetrical also and this may be since my graphs may be irregular.

Scatter Diagram

I will now produce a Scatter Graph which shows the Height vs Weight for all the Boys and girls data that I have collected and I will be able to find whether there is a pure correlation and I will then compare my results and I will see whether my two sets of data are related from my sample:

I will conclude after both graphs

Boys Height vs Weight

image42.png

Girls Height vs. Weightimage25.pngimage25.png

image43.png

Conclusion of Scatter Graphs

Boys & Girls Height vs. Weight = From the Scatter Graph that I have produced I have come to find that there is a Correlation or a Trend between both variables which are Height and Weight and there is a Fairly Strong Positive Correlation and this shows me that the taller the person the higher the weight although there are always some irregularities which is unique as a symmetrical trend is rather impossible as every human being has various growth period. In addition to this will be the major form of representation in my and I believe this shows the trend clearly and efficiently.

Body Mass Index

Calculation= WEIGHT / (HEIGHT) ²

I will be Doing this calculation for 5 people randomly from the sample that I have taken:

7

Austin

Steven

1.54

43

7

Lloyd

Mark

1.61

56

9

Bagnall

Veronica

1.49

37

10

Bhatti

Hannah

1.72

56

10

Durst

Freda

1.75

60

Calculation= WEIGHT / (HEIGHT) ²image25.pngimage25.png

  1. 43/(1.54) ² = 43/2.3716 = 18.13
  1. 56/(1.61) ² = 56/2.5921 = 20.64

3) 37/(1.49) ²= 37/2.2201 = 16.67

4) 56/(1.72) ²= 56/2.9584 = 18.93

      5) 60/(1.75) ²= 60/3.0625 = 19.6

From this calculations I can conclude that the Higher the Height in Equivalent to the Weight the Higher the Value of the Body Mass Index which shows me that both variable have a certain trend and also shows that on some occasions the height is parallel to the Weight.

Spearman Rank Correlation Co-Efficient

I will now using the same data sample of five random students to find the spearman rank correlation.The Spearman's Rank Correlation Coefficient is used to discover the strength of a link between two sets of data:

image44.png

X Value

Y Value

X Rank

Y Rank

D

1.54

43

4

4

0

0

1.61

57

3

2

1

1

1.49

37

5

5

0

0

1.72

56

2

3

-1

1

1.75

60

1

1

0

0

R = 1 – 6(2)        = 1-   12                        = 1 – 12/120 = 1 -1/10 = 0.9image03.png

image04.png

5(25-1)               24 Times 5= 120

Evaluation of Hypothesis 1image25.pngimage25.png

In my honest opinion I feel that I successfully completed and analyzed my hypothesis and I have gained a sufficient evidence to back up my theories. I would like to remind you that my main objective for this hypothesis was to find out whether I was correct or incorrect in my thinking that Boys at Mayfield School are taller and weigh more on average than the Girls at the same school. Within this aim I was also aiming to find whether there is a certain trend or relationship between the height and weight of the students that I have chosen to analyse and as I explained earlier due to the large number of students I was not possible to analyse all students so I gained a sufficient sample which I made as unbiased as possible. MY HYPOTHESIS WAS CORRECT

Conclusion of Hypothesis

  • The Histograms, frequency polygons proved that the results were more accurate and made more sense than that from the random sampling.
  • There is a positive correlation between height and weight. In general tall people will weigh more than smaller people.
  • In general boys tend to weigh more and be taller then girls.
  • By doing stratified sampling, there were a fewer exceptional values caused by different year groups and therefore ages. I was bound to find irregularities within my data
  • The cumulative frequency curves confirm that boys have a more spread out range in weight, with more girls having smaller weights. In height, boys tend to be taller.
  • The spearman rank correlation coefficient shows that the correlation between height and weight is strong.
  • My Body Mass Index showed that there is a strong trend between height and weight
  • In general the taller a person is, the more they will weigh.
  • There is a positive correlation between height and weight. In general tall people will weigh more than smaller people.
  • There therefore is a positive correlation between height and weight across the school as a whole. This correlation seems to be stronger when separate genders are considered
  • If I had taken larger samples my hypothesis may become more accurate.

image25.pngimage25.png

Hypothesis 2 ‘Key Stage 4 Students who watch more hours of television on average have a lower IQ Level’

Planning

For this particular hypothesis I will be only be using the data of the School Year of 10 and 11 due to the fact I have specifically chosen to investigate Key Stage 4 and this is the Stage studied by these 2 school years. I have chosen to only base my investigation on this key stage since I believe if I used the complete Mayfield High School data for this hypothesis the range and spread and range of data would be too large to make a sufficient analysis of the results that I will gain. I will be a Sampling Method is which I can sufficiently break down the number of student and meanwhile keep the investigation fair as possible.

Sampling

The sampling method that I have chosen to use for this very particular hypothesis is RANDOM SAMPLING. This is where every item in the population will have an equal chance of being selected. Below is the method that I chose to do the successfully and conveniently. I believe this method will help my results and outcome stay unbiased.

  1. I Printed a Copy of the Students and there personal qualities and detail.
  1. I then cut out the student First Name and Surnames.
  1. As you may remember from my two ways table earlier in this portfolio the total number of student that I have in this particular investigation is 370.
  1. In my sampling the student Gender is not an issue of comparison so I will be using all the names in one sample
  1. I then used a Hat which I had and put the all the names into this Hat
  1. I then shook the hat so that I cannot tell or neither can anyone else see the order of the names, which are in the hat.
  1. I then instantly decided to withdraw 30 students from the hat. An unbiased person who was besides me while I was completing this sample adjudged this procedure.

image25.pngimage25.png

After I had sufficiently completed the sample below are the results that I had obtained. I then typed the results into a Microsoft Excel Spreadsheet. Below is the data that I have gained from my sample. I have decided to only include the Name, Year, Number of Hours of TV, and Favourite TV Show and IQ Level since these is the field, which are important to me whereas other fields are invaluable.

Random Sample Results:

Year Group

Surname

Forename

Favourite TV programme

Average number of hours TV watched per week

IQ

10

Air

Jason

Match Of The Day

2

116

10

Black

Mia

Ali G

14

103

11

Compass

Sharon

Big Brother

40

106

11

Dixon

Graham

The Simpsons

30

102

10

Doens

John

The News

16

101

10

Ewards

Michael

Match Of The Day

27

104

11

Flawn

Elise

Neighbours

10

101

10

Grimshaw

Katie

Blind Date

17

104

11

Jackson

Debi

Eastenders

22

90

11

McCreadie

Jenny

The Simpsons

25

104

10

McDonald

Harold

The Simpsons

21

100

11

McDonald

James

The Simpsons

21

122

10

Edd

Michael

0

81

11

Fillstin

Rowena

Holly Oaks

14

104

10

Jones

Nathan

Bad Girls

8

92

11

Zarrent

Donna

Buffy

36

103

11

Thompson

Kamara

Big Brother

6

89

11

Thomson

Jade

Brookside

18

96

11

Solomons

Ian

The Simpsons

16

100

10 ...read more.

Conclusion

If I had taken large sample my hypothesis may become more accurate and able to form a successful conclusion. The IQ also depends on the persons surrounding, ability and knowledge and stimulation and motivation which can all play a factor in the results.Overall I have found that my Hypothesis was incorrect and the statistical evidence that I had gained did not back up my theory. Another reason behind my misfortune is the range of data from 7-11 is too wide and I should have narrowed the frame down but now helps me in the future.

Conclusion/Evaluation of Hypothesis 3

‘Left Handed Students have higher IQ levels and Key Stage 2 Results in

Comparison Right handed student at Mayfield High School’

In my honest opinion I feel that I have proved my hypothesis in a short span and I feel that the statistical evidence that I have duly gained is more then sufficient to form a hypothesis and if I used any other forms of representing data it would have meant I was generally repeating. Also I feel that I have used the correct statistical evidence to prove my theory and used the right evidence. In addition to that I feel that Left Handed students are generally have higher IQ Levels according to my sampling frame and investigate in this is what I have based my theory upon.

        -  -Jimit Vyas

...read more.

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