The table below shows that the age groups have united in order to produce a simple children/adult frequency. So the overall children age group is 11-18 and the adult group is 19+ (19-60). Also note that the group category of the line estimation is both values inclusive. I.e. 7-8 is in more depth like 7.1-8 cm. Also the group category for the line and angle estimations will also be grouped because handling 60 different data with each person’s estimate will not be appropriate and so it will be grouped. The grouping will also help with the calculations that have to be processed before drawing certain graphs.
Children (11-18)
The calculations processed below have been done to work out mainly the Mean but also the Median, Mode and Range.
Frequency Total = 30
Frequency x Mid-Interval Total = 334
Estimated Mean of Line Estimation
- = 11.1cm (1d.p)
30
Modal Group
9-10cm
Range
15cm – 7cm = 8cm
Median
Frequency = 30. Divide this by 2 to get the 15th Term.
The 15th Term lies in the value situated to the group 10-11 as
1+3+7+4 = 15.
The calculations processed below have been done to work out the Mean, Median, Mode and Range.
Frequency Total = 30
Frequency x Mid-Interval Total = 1180
Estimated Mean of Angle Estimation
1180 = 39.3degrees (1d.p)
30
Modal Group
30-40 degrees
Range
60 – 20 = 40 degrees.
Median
Frequency = 30. Divide this by 2 to get the 15th Term.
The 15th Term lies in the value situated to the group 30-40 degrees as 1+16
makes the value of 15 drop in the 30-40 degrees group category.
The proper median can be worked out using the Cumulative Frequency Diagram.
Adults 19-60
The calculations processed below have been done to work out the Mean, Median, Mode and Range.
Frequency Total = 30
Frequency x Mid-Interval Total = 328
Estimated Mean of Line Estimation
328 = 10.9 (1d.p)
30
Modal Group
11-12 cm
Range
15-7 = 8cm
Median
Frequency = 30. Divide this by 2 to get the 15th Term.
The 15th Term lies in the value situated to the group 11-12 cm as 3+5+3+3+8 makes the value of 15 drop in the 11-12 group category.
The proper median can be worked out using the Cumulative Frequency Diagram.
The calculations processed below have been done to work out the Mean, Median, Mode and Range.
Frequency Total = 30
Frequency x Mid-Interval Total = 1210
Estimated Mean of Line Estimation
1210 = 40.3 (1 d.p)
30
Modal Group
30-40
Range
60-20 = 40
Median
Frequency = 30. Divide this by 2 to get the 15th Term.
The 15th Term lies in the value situated to the group 30-40 as 2+13
makes the value of 15 drop in the 30-40 group category.
The proper median can be worked out using the Cumulative Frequency Diagram.
Cumulative Frequency
I will use Cumulative Frequency tables and diagrams,
in order to process my calculations one step further and also make it pertinent to this processing of data by finding the Median and Interquartile Range. Also the work carried out using these statistical terms will help make my work more substantial.
Children (11-18)
The frequency shown below was used in order to obtain the cumulative frequency data. This was done by the addition of values from the frequency column as you go along. This can also be called the running total. When plotting points, it is important to remember to plot the points using the highest value in each group against the corresponding cumulative frequency. This will be an effect to all the cumulative frequency diagrams that will be used in the rest of this coursework,
as it is a common rule for cumulative frequencies.
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
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)
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
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)
Frequency Density = Frequency
Class width
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.
Frequency Density = Frequency
Class width
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)
Frequency Density = Frequency
Class width
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.
Frequency Density = Frequency
Class width
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
Looking at the table on the previous page, you can see that there were many percentage errors and only one correct estimate. However, there were also many close and good estimates that just missed the correct answer. The mean percentage error for the data shown on the previous page is 15.6%.
Looking at the table, you can see that there are many percentage errors with high ranges, many going much over the actual answer.
There is just one estimate that is correct according to the real answer. Although there are many estimates that have no near connection to the actual size of the angle. The mean percentage error for the data shown above is 22.8%.
Adult 19-60
Looking at the above table, you can see that there were many percentage errors and only about 2 correct estimates. However,
as you can see above, there are quite a few estimates which are near to the correct length. The mean percentage error for the data shown above is 17.1%.
Looking at the table above, you can see that there are percentage errors with even higher effects such as a distance between the correct answer and the estimates. However there are about three estimates which have matched the correct answer. This shows there is not much consistency between the estimates done by certain people. The mean percentage error for the data shown above is 23.2%.
Conclusion
Please note that the length of the line is 11 cm and the size of the angle is 35˚.
Looking back at all the data that has been collected and the calculations that had to be processed using the collected data, the thing that can be said regarding my hypothesis is that it was proved wrong. This is because using all the calculations I can see that the adult sector has got more correct estimates than the children sector. The table below how many people were correct according to each sector.
So overall the hypothesis was proved wrong. However I would like to see in more depth how each type of calculations and graph helped me work all this out and how it helped me so that I can pull together all the information.
Finding the Estimated Mean
Looking back at this section I can find that when the estimated mean was worked out for the children section’s line estimates, it shows that it got 11.1 cm as the mean when the actual answer is 11 cm. This does prove that overall working out as the group mean, it was close to the actual answer. However, on comparison to the children the adult section’s estimated mean for the line estimates is 10.9 cm with the actual still being 11 cm. This does not show a difference to which one is closest or not as both sections have the same amount of difference from the actual answer.
Also with the estimated mean for the size of the angle on the children section is 39.3˚ when the actual answer is 35˚.
This shows that as a group mean, it is not very close to the actual answer. However, on comparison to the children, the adult section’s estimated mean for the angle estimates as a group is 40.3 with the actual answer being 35˚. This shows that in this case, the children’s estimation of the size of the angle was better as a group mean than the adult’s group mean.
Deviation
Looking back at this section, I can see that the actual purpose of this use of method is in order to find how wrong people were from the actual answer and the percentage error also helps us in getting the percentage of how wrong they were, by adding the plus for if they estimated over the actual value and minus for if they estimated under the actual answer. The mean percentage error was also worked out in order to make things quite uncomplicated for comparison between the children and adult sectors.
The mean percentage error for the line estimates in the children sector is 15.6%. This value shows that overall as a group; the children sector was not very far off when estimating the length of the line. Although, similarly the adult sector’s estimation for the length of the line is 17.1%. This also shows that as a group the adult sector is quite close to the actual answer.
The mean percentage error for the estimation of the size of the angle in the children’s sector is 22.8%.This value shows that as a group the children’s sector was not close as it was with the mean for the line estimates. However, quite similarly, the adult sector’s mean percentage error for the angle size estimation is 23.2%.
This shows that both sectors were not very close to the actual answer as a group. I think that in this case for the adult estimation of the size of the angle, the mean percentage error may have been distorted to extreme values as three people did estimate the correct size of the angle.
Cumulative Frequency
Looking back at this section of the coursework, I can see that a couple of matters were worked out using this method, like the median and the interquartile range. The interquartile range was worked out because it is a more advanced version of the simple range and a more accurate method too. The interquartile range was more easily interpreted through a range of box plots, all being drawn under the cumulative frequency diagram.
The working out of the median in this case was accurate because of the many calculations the data had to undergo through the cumulative frequency calculations.
The median for the children sectors’ line estimation is 11.5 cm. This value shows that through all data collected through in the children’s sector, the median of all data was very close to the actual length of the line- 11 cm.
Conversely, the median for the adult sector’s line estimates is
11.6 cm. This shows that the children sector was better through the method of the median than the adult sector when it came to the estimation of the line.
The median for the children sector’s angle size estimation is 39˚.
This shows that as a group, the children sector is quite close to the actual answer through the method of the median.
Although quite similarly, the adult sector’s median for the estimation of the size of the angle is 40.5˚. This shows that through the median the children sector have done better as a group than the adult sector.
Box Plots
Looking at this part of the coursework, I can see that linking in with the cumulative frequency, the use of box plots makes it more easier to interpret the interquartile range than the ordinary cumulative frequency diagram because the upper and lower quartiles have been worked into a box where it is more easier to read off the results and it also shows certain distributions such as positive, negative and symmetric for example.
The interquartile range for the children sector for the line estimation is 3.3 cm. This shows that there is not much of a big range between certain data. Also in the adult sector, the interquartile range for the line estimation is 3.4 cm which also shows not much difference although it works out that the children overall have been quite close of estimating the length of the actual line rather than the adult sector.
The interquartile range for the children sector for the angle size estimation is 10.9˚.This shows that there is quite a bit of difference between the data that was submitted through as each estimate.
Also in the adult sector, the interquartile range for the angle size estimation is 11.6˚ which also shows quite a bit of difference although it works out again, the children sector overall have been closer to the actual angle size than the adult sector.
Histograms
Looking at this section of calculations the data had to be processed through; I can see that in order to get my data through an advanced section I had to use histograms. The use of histograms determined the frequency density and this equals the frequency the group get divided by how big the group is which is also knows as class width.
The frequency density for the children sector in the line estimation group 9-12 is 3.75. This shows that there was a large frequency to this group and this proves that a lot of people estimated around the actual answer 11 cm. The group 9-12 determines that if the bigger the frequency to this group the closer each sector was to achieving the correct estimate. However the frequency density for the adult sector in the line estimation group 9-12 is 4. This shows there was a bigger frequency to this group then there was to the children sector.
The frequency density for the children sector in the angle size estimation group 20-40 is 0.76. This shows that there was a large frequency, however, only to a group with a big class width. It does show though, that there was a large amount of children estimating
around the actual answer which is 35˚. Nevertheless, the frequency density for the adult sector in the angle size estimation group 20-40 is 0.71. This shows there was a smaller frequency to this group rather than the children sector.
Standard Deviation
Looking back at this final section of calculations my collection of data had to undergo, I have noticed the understanding of certain formulae can have the effect of accurate calculations. As said earlier, the standard deviation was used only in the means of comparing the children and adult sectors.
The standard deviation was found to be 20.6 cm for the estimation of length of line in the children’s sector compared to a value of 22.4 cm for the estimation of the length of the line in the adult sector. This shows that the children’s sector were quite consistent
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.
