= ¼ x 50
= 12.5th piece of data
Lower Quartile Value = 1.40m
Inter-quartile range = Upper Value – Lower Value
= 1.68 – 1.40
= 0.28m
Yr 11 Deviation
Yr 7 Deviation
Conclusion
After carrying out this investigation I have concluded that the year 11’s were more accurate in their estimates. This was because the mean is closer to the actual length and also because the standard deviation is smaller. This means that most people estimated close to the mean which links in with the fact that the inter-quartile range is only 0.28m. My estimate that 50% of year 11 would guess between 1.4 and 1.6m was a bit high as the actual percentage is 42%. I underestimated the year 7’s accuracy as I guessed that only 10% would estimate between 1.4 and 1.6m when in actual fact 24% of them did. The mean of both sets of data is very close with the year 7 mean being 1.52(to 1 d.p) and the year 11 mean being 1.53(to 1 d.p). This shows that both groups had similar guesses. However, the year 11 set of data had one person guessing a length of 0.75m and I think that this threw my data out, as it would have made the mean smaller. I have worked out the percentage errors for both years as shown below.
Year 7
Percentage error = (mean – actual length) x 100%
Actual length
= (1.5156 – 1.58) x 100%
1.58
= -4% (to 1 s.f)
Year 11
Percentage error = (mean – actual length) x 100%
Actual length
= (1.5258 – 1.58) x 100%
1.58
= -3% (to 1 s.f)
The percentage errors show that both the year 11 and year 7 mean was lower than the actual length of 1.58m. As you can see the difference between the means of the two sets of data is not significant however, overall when considering all the figures the year 11 pupils had better estimates.
Year 11 Deviation
Year 7 Deviation
Yr7 bar graph
This graph shows that only 12 people estimated between 1.4 and 1.6m. This shows that 24% of people guessed in the group that includes the actual length of the stick of 1.58m. No one in year 7 underestimated by a large amount however there were a few who overestimated. This has caused the unsymmetrical look of the graph. The range of the data is shown on this graph to be quite large.
Yr11 bar graph
This graph shows that 21 people in year 11 estimated between 1.4 and 1.6m. This illustrates that the year 11 were quite accurate with 42% of them guessing in this group containing the actual length of 1.58m. The graph also shows that there were a handful of people who greatly overestimated and underestimated the length. The graph is quite symmetrical which was expected, as it is equally likely that people will overestimate, as it is that people will underestimate. The graph tells me that the range for the year 11 data is quite large as it is 1.85m.
Polygon
I have drawn a polygon because it is a very clear way of comparing the two sets of data. This graph clearly shows that a much higher percentage of year 11’s than year 7’s estimated in the 1.4 to 1.6m group. This shows that most of year 11 were accurate however the graph also shows that there were a few people in year 11 who greatly overestimated or underestimated the length. The range for year 11 is much larger that that of year 7.
Change to landscape to print
Yr 7 cumulative
I have drawn a graph to show cumulative frequency because this is a more accurate method of showing the range and allows me to look at how the data is spread out. The graph shows that the inter-quartile range is quite small, as it is only 0.40m. The graph shows that the median is 1.56m, which is very close to the actual length of 1.58m.
Yr 11 cumulative
Although the range of the data is quite large the inter-quartile range is shown on the graph to be very small, only 0.28m. Using the inter-quartile range is a more accurate way of measuring how spread out the data is. The graph illustrates that many people estimated close to the median as the inter-quartile range is small. The median is shown on the graph to be 1.54m.
Yr 11 weight
The range of the data is quite large as it is 0.9Kg however the inter-quartile range is only 0.29Kg. Using the inter-quartile range is a more accurate way of measuring how spread out the data is. The graph shows the upper quartile to be 0.58Kg, which is fairly low as it is only 0.13Kg above the actual weight. The lower quartile is 0.29Kg, which is 0.16 Kg away from the actual weight. The median is shown on the graph to be 0.38Kg, which is quite close to the actual weight of 0.45Kg. The graph shows that the year 11 estimates were quite accurate and all estimated similarly because of the small inter-quartile range.
Yr 7 weight
The range of the data is very large as it is 1.48Kg however the inter-quartile range is only 0.26Kg. The graph shows the upper quartile to be 0.69Kg, which is quite high as it is 0.24Kg above the actual weight. The lower quartile is 0.43Kg, which is 0.02Kg away from the actual weight. This is very close to the actual weight, which shows that more people overestimated than underestimated. The median is shown on the graph to be 0.54Kg, which is fairly close to the actual weight of 0.45Kg. This graph shows that the year 7’s were quite accurate.
Year 7 length estimates
Year 11 length estimates
0 1 2 3
Length (m)
Shown above is a box and whisker plot of the estimated length for year 7 and year 11. I have constructed this using a graphical calculator and have obtained a screen print of the graph. This helps me to compare the two sets of data as it shows me how spread out the data is. The plot for the year 11 data shows how small the inter-quartile range is as the box is small. The 25% of data between the upper quartile and the median is more spread out that the 25% of data between the lower quartile and the median. However the year 7 data shows the opposite of this. The 25% of data between the lower quartile and the median is very spread out compared to the 25% of data between the upper quartile and the lower quartile, which is much less spread out.
The box and whisker plot shows the following figures.
Year 7
Minimum = 1.0m
Maximum = 2.50m
Lower Quartile = 1.25m
Upper Quartile = 1.67m
Inter quartile range = 0.42m
Median = 1.525m
Year 11
Minimum = 0.75m
Maximum = 2.60m
Lower Quartile = 1.40m
Upper Quartile = 1.65m
Inter-quartile range = 0.25m
Median = 1.50m
These deviate from the figures gained from the cumulative frequency graph because using the calculator is more accurate. The cumulative frequency graph used grouped data which made the curve less accurate.
Year 7 weight estimates
Year 11 weight estimates
0 1
Weight (Kg)
Shown above is a box and whisker plot of the estimated weight for both year 7 and year 11. I have constructed this using a graphical calculator and have obtained a screen print of the graph. This helps me to compare the two sets of data, as it is an accurate method of showing how spread out the data is. Both sets of data have quite a large inter-quartile range as shown by the width of the boxes. However the inter-quartile range of the year 7 data is slightly smaller that the inter-quartile range of the year 11 data.
The box and whisker plot shows the following figures.
Year 7
Minimum = 0.100Kg
Maximum = 1.000Kg
Lower Quartile = 0.400Kg
Upper Quartile = 0.659Kg
Inter-quartile range = 0.259Kg
Median = 0.500Kg
Year 11
Minimum = 0.020Kg
Maximum = 1.500Kg
Lower Quartile = 0.274Kg
Upper Quartile = 0.571Kg
Inter-quartile range = 0.297Kg
Median = 0.3775Kg
These figures again deviate from the figures obtained from the cumulative frequency graph because the cumulative frequency graph used grouped data which made the curve less accurate.
Data comparing length estimates of male and females
There was not a significant difference between the estimates of the year 11 pupils and the year 7 pupils. I have decided to find out whether males are better than females or vice versa. I have sorted the data already collected into 2 lists one for males and one for females as shown on this page.
Female Male
I have worked out the figures shown below using a graphical calculator and I have found that there is a more significant difference between the two means. Overall I think that the males were more accurate as the mean was much closer to the actual length. The males has a mean of 1.5549m while the females was only 1.4904m, this is a difference of 0.0645m. Also the males only had a percentage error of –2% whereas the females had a percentage error of 6%.
Extension
As an extension to my coursework I have also investigated the estimated weight. I have done this because there are not a large difference between the estimates in length between the year 7 and year 11.
Introduction
I have collected data from year 7 and year 11 pupils and have recorded their estimates of how heavy they think the bamboo stick is. A bamboo stick of weight 0.45Kg was held up in front of 173 year 7 pupils and 178 year 11 pupils. The pupils were then asked to estimate the weight of the bamboo stick in kilograms. This data was collected together and placed in a table with the lengths giving me a data sheet with both sets of data on it. Once I had collected the random sample of data from this sheet using the method stated earlier I had sets of data for the length and the weight. This has allowed me to compare the estimates of length and weight made by each person as I have a set of data for each pupil. The tables of data are shown on the following pages.
Hypothesis
I think that the year 11 pupils will have much more accurate estimates than the year 7 because they have been using measurements for longer than the year 7. I believe that the year 7 pupils will not be as accurate and the mean of their estimates will be further away from the actual weight of the stick. I expect the year 11 data to deviate from the mean only slightly and therefore have a smaller standard deviation. I believe that the year 7 data will have a larger standard deviation that the year 11 because I believe that each set of data will deviate from the mean by a large amount. I expect the scatter graph of the year 11 data showing the length against the weight to be less widely spread as I expect the year 11 pupils to have similar estimates. However I expect the scatter graph of the year 7 data to be more spread out, as their estimates will be less similar. I expect both sets of data to some positive correlation because if they overestimate the length they are likely to overestimate the weight.
Data comparing year 7 and year 11 weight estimates
Year 7
Year 11
Year 7 Cumulative Frequency
Median = ½ x total cumulative frequency
= ½ x 50
= 25th piece of data
Median = 0.54Kg
Upper Quartile = ¾ x total cumulative frequency
= ¾ x 50
=37.5th piece of data
Upper Quartile Value = 0.69Kg
Lower Quartile = ¼ x total cumulative frequency
= ¼ x 50
= 12.5th piece of data
Lower Quartile Value = 0.43Kg
Inter-quartile range = Upper Value – Lower Value
= 0.69 – 0.43
= 0.26Kg
Year 11 Cumulative Frequency
Median = ½ x total cumulative frequency
= ½ x 50
= 25th piece of data
Median = 0.38Kg
Upper Quartile = ¾ x total cumulative frequency
= ¾ x 50
=37.5th piece of data
Upper Quartile Value = 0.58Kg
Lower Quartile = ¼ x total cumulative frequency
= ¼ x 50
= 12.5th piece of data
Lower Quartile Value = 0.29Kg
Inter-quartile range = Upper Value – Lower Value
=0.58 – 0.29
= 0.29 Kg
Conclusion
After carrying out this extension to the investigation I have concluded that the year 11’s were more accurate in their estimates. The mean of the year 11 data is much closer to the actual weight than the mean of the year 7 data. This is because the year 11 data has a percentage error of only –4% whereas the year 7 data has a percentage error of 19%. However the year 11’s had a higher standard deviation. This shows that the year 7 greatly overestimated the weight while the year 11 slightly underestimated. The two scatter graphs comparing length and weight have very different patterns. The graph for the year 11 data shows a slight positive correlation while the graph for the year 7 data shows no correlation. This makes the year 7 estimates seem more random whereas the year 11 estimates are more logical. However both graphs are quite scattered with the year 11 graph being slightly more clustered than the graph of the year 7 data. When considering all the data shown in the table below I have found that the year 11s had more accurate estimates.
