# Formulation Of A Theory.

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Introduction

Formulation Of A Theory

The certain factors that influence the accuracy of estimation is mainly: Age, the time of day it is processed, gender and the style of the survey.

The reason age has an affect on the accuracy of estimation, is because if the person is too young or too old the survey may not be reliable enough in order for it to be processed into a data collection sheet the reason being that they may not be suited to the imperial or metric measurement used in modern days. The young may not have learnt the imperial measures and the elderly may not be used to the correct terms.

The effect, time has to getting good reliability is that the survey should be taken out at a convenient time of the day where food has been consumed properly and the people’s minds are focused. The times which may be irrelevant to practice a survey is early in the morning or late at night where people may be tired.

The gender may have a certain effect on the outcome of results because the males may differ in technical skills to females or it is possible that perhaps females are more capable of handling academical situations rather than males.

The style of a survey may affect the result because if badly constructed questions are used, people will not answer the survey properly and will try to avoid them. An example of a badly constructed question is ‘What is your age?’-This is personal. Many people will not want to answer.

An example of a well constructed question is ‘Which age group are you in?

0-20 21-30 31-50 etc.

A questionnaire is usually put together to test a hypothesis. The hypothesis that will be used in this questionnaire is:

Middle

3

4

(9.5,4)

9-10

7

11

(10.5,11)

10-11

4

15

(11.5,15)

11-12

4

19

(12.5,19)

12-13

5

24

(13.5,24)

13-14

3

27

(14.5,27)

14-15

3

30

(15.5,30)

See next page for the cumulative frequency diagram which represents

the data shown in the table above.

Looking at the cumulative frequency diagram on the previous page,

you can see that it shows the Median and Interquartlie Range being worked out. The interquartile range is not the same as the fundamental

range itself but it offers a similar view of finding it and a more advanced version of it too. The Box Plot offers an easier process of working out the Interquartile Range as it is done in a way which will make it easier to extract information and data from the diagram. The diagram shows that there is a Positive Distribution to the data according to the structure of the contents in the box plot itself.

Median = 11.5 cm

Upper quartile = 13.2 cm

Lower quartile = 9.9 cm

Interquartile Range = 13.2 cm – 9.9 cm

= 3.3 cm

Angle (degrees) | Frequency | Cumulative Frequency | Plotting of Values |

20-30 | 1 | 1 | (30.5,1) |

30-40 | 16 | 17 | (40.5,17) |

40-50 | 12 | 29 | (50.5,29) |

50-60 | 1 | 30 | (60.5,30) |

See next page for the cumulative frequency diagram which represents

the data shown in the table above.

Looking at the cumulative frequency diagram on the previous page,

you can see that it shows the Median and Interquartile Range being worked out. The interquartile range is not the same as the fundamental

range itself but it offers a similar view of finding it and a more advanced version of it too. The Box Plot offers an easier process of working out the Interquartile Range as it is done in a way which will make it easier to extract information and data from the diagram. The diagram shows that there is a Symmetric Distribution to the data according to the structure of the contents in the box plot itself.

Median = 39 ˚

Upper quartile = 44.5 ˚

Lower quartile = 33.6 ˚

Interquartile Range = 10.9 ˚

Adults (19-60)

Line (cm) | Frequency | Cumulative Frequency | Plotting of Values |

7-8 | 3 | 3 | (8.5,3) |

8-9 | 5 | 8 | (9.5,8) |

9-10 | 3 | 11 | (10.5,11) |

10-11 | 3 | 14 | (11.5,14) |

11-12 | 8 | 22 | (12.5,22) |

12-13 | 2 | 24 | (13.5,24) |

13-14 | 1 | 25 | (14.5,25) |

14-15 | 5 | 30 | (15.5,30) |

See next page for the cumulative frequency diagram which represents the data shown in the table above.

Looking at the cumulative frequency diagram on the previous page,

you can see that it shows the Median and Interquartile Range being worked out. The interquartile range is not the same as the fundamental

range itself but it offers a similar view of finding it and a more advanced version of it too. The Box Plot offers an easier process of working out the Interquartile Range as it is done in a way which will make it easier to extract information and data from the diagram. The diagram shows that there is a Negative Distribution to the data according to the structure of the contents in the box plot itself.

Median = 11.6 cm

Upper quartile = 12.8 cm

Lower quartile = 9.4 cm

Interquartile Range = 3.4 cm

Angle (degrees) | Frequency | Cumulative Frequency | Plotting of Values |

20-30 | 2 | 2 | (30.5,2) |

30-40 | 13 | 15 | (40.5,15) |

40-50 | 12 | 27 | (50.5,27) |

50-60 | 3 | 30 | (60.5,30) |

See next page for the cumulative frequency diagram which represents the data shown in the table above.

Box Plots

Note that for all cumulative frequency diagrams you will see a box plot underneath each one. This is because another way of displaying data for comparison is by means of a box plot. These data are always placed against a scale so that their values are accurately plotted.

The diagrams also show how the cumulative frequency curves and the box plots are connected for three common types of distribution.

These include: Symmetric, Negative and Positive Distribution.

Looking at the cumulative frequency diagram on the previous page,

you can see that it shows the Median and Interquartile Range being worked out. The interquartile range is not the same as the fundamental

range itself but it offers a similar view of finding it and a more advanced version of it too. The Box Plot offers an easier process of working out the Interquartile Range as it is done in a way which will make it easier to extract information and data from the diagram. The diagram shows that there is a Positive Distribution to the data according to the structure of the contents in the box plot itself.

Median = 40.5 ˚

Upper quartile = 46.5 ˚

Lower quartile = 34.9 ˚

Interquartile Range = 11.6 ˚

Histograms

For all past calculations, diagrams and tabulation the class groups were the the values shown inclusive. However in the histograms, the values will be exclusive so for example 7-8 will mean in simple terms 7- 8.9,

and so on.

Sometimes the data in a frequency distribution are grouped into classes, whose intervals are different. This means the histogram has bars of unequal width. It is the area of a bar in a histogram that represents the frequency. I will use Histograms to represent my data because I feel that using a histogram to represent my data more in depth than most other graphs. Certain calculations have to be done in order to achieve a proper and accurate histogram.

Children (11-18)

Line (cm) | 7-8 | 9-12 | 13-15 |

Frequency | 4 | 15 | 11 |

Class interval | Lower bound. | Upper bound. | Interval width |

7-8 | 6.5 | 8.5 | 2 |

9-12 | 8.5 | 12.5 | 4 |

13-15 | 12.5 | 15.5 | 3 |

Frequency Density = Frequency

Class width

Class interval width | Frequency | Frequency Density |

2 | 4 | 2 |

4 | 15 | 3.75 |

3 | 11 | 3.6 (1d.p) |

See histogram on next page which represents data from table above.

The histogram shown in the previous page shows from the frequency density, that there was a large frequency to the 9-12 cm group. As a whole, it shows that the frequency can be worked out simply by multiplying the difference between the two values in which the bars extend to by the frequency density it shows. For example, if the group

7-9 was chosen we would know that the difference between them is 2.

The frequency density is also the value 2. If both values were multiplies you would get the frequency. That is 2 x 2 = 4. So 4 is the frequency.

This is the information that can easily be interpreted from the histogram.

Angle (degrees) | 20-40 | 41-50 | 51-60 |

Frequency | 16 | 13 | 1 |

Class interval | Lower bound. | Upper bound. | Interval width |

20-40 | 19.5 | 40.5 | 21 |

41-50 | 40.5 | 50.5 | 10 |

51-60 | 50.5 | 60.5 | 10 |

Frequency Density = Frequency

Class width

Class interval width | Frequency | Frequency Density |

21 | 16 | 0.76 (1d.p) |

10 | 13 | 1.3 |

10 | 1 | 0.1 |

See histogram on next page which represents data from table above.

The histogram shown in the previous page shows from the frequency density, that there was a large frequency to the 20-40 degrees group. As a whole, it shows that the frequency can be worked out simply by multiplying the difference between the two values in which the bars extend to by the frequency density it shows. For example, if the group

40-50 was chosen we would know that the difference between them is 10. The frequency density is the value 1.3. If both values were multiplies you would get the frequency. That is 1.3 x 10 =13. So 13 is the frequency. This is the information that can easily be interpreted from the histogram.

Adults (19-60)

Line (cm) | 7-8 | 9-12 | 13-15 |

Frequency | 8 | 16 | 6 |

Class interval | Lower bound. | Upper bound. | Interval width |

7-8 | 6.5 | 8.5 | 2 |

9-12 | 8.5 | 12.5 | 4 |

13-15 | 12.5 | 15.5 | 3 |

Frequency Density = Frequency

Class width

Class interval width | Frequency | Frequency Density |

2 | 8 | 4 |

4 | 16 | 4 |

3 | 6 | 2 |

See histogram on next page which represents data from table above.

The histogram shown in the previous page shows from the frequency density, that there were two large frequency groups of 7-8 and 9-12 cm. As a whole, it shows that the frequency can be worked out simply by multiplying the difference between the two values in which the bars extend to by the frequency density it shows. For example, if the group

9-12 was chosen we would know that the difference between them is 4 because using their lower and upper boundaries as shown on the previous page, the value of 4 is extracted.

The frequency density is also the value 4. If both values were multiplies you would get the frequency. That is 4 x 4 = 16. So 16 is the frequency.

This is the information that can easily be interpreted from the histogram.

Angle (degrees) | 20-40 | 41-50 | 51-60 |

Frequency | 15 | 12 | 3 |

Class interval | Lower bound. | Upper bound. | Interval width |

20-40 | 19.5 | 40.5 | 21 |

41-50 | 40.5 | 50.5 | 10 |

51-60 | 50.5 | 60.5 | 10 |

Frequency Density = Frequency

Class width

Class interval width | Frequency | Frequency Density |

21 | 15 | 0.71 (1d.p) |

10 | 12 | 1.2 |

10 | 3 | 0.3 |

See histogram on next page which represents data from table above.

The histogram shown in the previous page shows from the frequency density, that there was a large frequency to the 20-40 degrees group. As a whole, it shows that the frequency can be worked out simply by multiplying the difference between the two values in which the bars extend to by the frequency density it shows. For example, if the group

50-60 was chosen we would know that the difference between them is 10. The frequency density is the value 1.2. If both values were multiplies you would get the frequency. That is 1.2 x 10 =12. So 12 is the frequency. This is the information that can easily be interpreted from the histogram.

The Deviation

This is usually used as a comparison because it represents what is known as the percentage error. Usually when we have a fixed value,

the deviation measures how much wrong a person is if they for example estimated wrongly and the percentage of it. So the data that is represented below will be to show observation to the difference between the age groups more clearly and easier to represent.

Children 11-18

Actual length of line | Estimation by people | Error Value | Percentage Error | ||

11 cm | 10 cm | -1 | -9% | ||

11 cm | 10 cm | -1 | -9% | ||

11 cm | 10 cm | -1 | -9% | ||

11 cm | 9.5 cm | -1.5 | -14% | ||

11 cm | 14 cm | +3 | +27% | ||

11 cm | 14 cm | +3 | +27% | ||

11 cm | 10.5 cm | -0.5 | -5% | ||

11 cm | 10 cm | -1 | -9% | ||

11 cm | 12.5 cm | +1.5 | +14% | ||

11 cm | 12.5 cm | +1.5 | +14% | ||

11 cm | 15 cm | +4 | +36% | ||

11 cm | 12.5 cm | +1.5 | +14% | ||

11 cm | 8 cm | -3 | -27% | ||

11 cm | 10 cm | -1 | -9% | ||

11 cm | 11 cm | 0 | 0% | ||

11 cm | 15 cm | +4 | +36% | ||

11 cm | 13 cm | +2 | +18% | ||

11 cm | 9 cm | -2 | -18% | ||

11 cm | 14 cm | +3 | +27% | ||

11 cm | 10 cm | -1 | -9% | ||

11 cm | 12 cm | +1 | +9% | ||

11 cm | 13 cm | +2 | +18% | ||

11 cm | 15 cm | +4 | +36% | ||

11 cm | 12 cm | +1 | +9% | ||

11 cm | 10.5 cm | -0.5 | -5% | ||

11 cm | 11.5 cm | +0.5 | +5% | ||

11 cm | 9 cm | -2 | -18% | ||

11 cm | 12 cm | +1 | +9% | ||

11 cm | 10.5 cm | -0.5 | -5% | ||

11 cm | 8.5 cm | -2.5 | -23% |

Conclusion

and better in estimating the length of the line while in the adult sector the estimations seemed more spread out and not as consistent.

The standard deviation was found to be 136.9˚ for the estimation of the size of the angle in the children’s sector compared to the value of 126.4˚ for the estimation of the size of the angle done by the adult sector.

This shows that the adult sector were far more consistent in their estimation of the size of the angle than the children’s sector because a difference like this between the both sectors show that overall, with the estimation of both length of line and size of angle, the adult sector are better at estimating than children and so this supports the reason for why my hypothesis was proved wrong.

So as said before, looking back at all this that needed to be concluded, my hypothesis was proved wrong and the adult sector were better at estimating than the children sector.

Evaluation

Looking back through this piece of course work, I think the things that can be done to make this coursework even better and to top standard would be the use of more graphs form other kinds such as pie charts, frequency polygons and scatter graphs etc. The reason that the use of these graphs would have made this coursework look better is because it would have tested my data on different graphs and made the coursework look more substantial as well. However, looking back at the graphs that were used like Histograms and Cumulative Frequency Diagrams, it has made my coursework look quite advanced and also with the knowledge of standard deviation being tested out on my data has made my working out of different terms like all kinds of the averages easier and simple, for interpreting the results from.

This student written piece of work is one of many that can be found in our GCSE Waves section.

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