Mathematics
YEAR 10 – LEAVES PROJECT
Hypotheses:
- As the length increases so will the width.
- The length and width will be greater in 2002 than 2001.
- The spread of the length and width will be great in 2002 than in 2001.
- The length and width of the leaves will follow a normal to almost normal distribution.
- The length of leaf for which 10% of the leaves are longer will be greater in 2002.
From the data gathered on leaves, attached overleaf, I performed statistical techniques to attempt to prove my hypotheses.
Factors Effecting Results:
The above therefore leaves my results questionable.
Before we can analyse the data gathered on leaves, I have to identify if there are any anomalies. I have highlighted these in yellow:
I think that these numbers are anomalies because they are either too small or too big to fit in with the consistency of the data. I am going to discount these numbers from the rest of my work as including them would make my results unreliable or faulty.
Statistical Technique: Averages, Quartiles and Largest and Smallest
Numbers.
From this table we can see that the mean length increases over the year, as does the mean width. The reasoning behind the averages being different in the two years could be due to the factors I listed early. I ...
This is a preview of the whole essay
I think that these numbers are anomalies because they are either too small or too big to fit in with the consistency of the data. I am going to discount these numbers from the rest of my work as including them would make my results unreliable or faulty.
Statistical Technique: Averages, Quartiles and Largest and Smallest
Numbers.
From this table we can see that the mean length increases over the year, as does the mean width. The reasoning behind the averages being different in the two years could be due to the factors I listed early. I produced two scatter graphs to compare the lengths and widths in 2001 and the same again but in 2002. These can be located over the next few pages.
The first scatter graph shows us that there is a weak, positive correlation between the length and width in 2001. This means that as the length increases, so does it’s width (to a certain extent). The formula of the graph shows us that it crosses the y-axis at +22.06mm meaning that when a leaf is 0mm long it is +22.06mm in width, this is impossible but as we don’t have data to continue the trendline then it is impossible to tell if this is actually accurate. The gradient tells us that as the length increases by 1mm the width increases by 0.2901mm.
The second scatter graph shows a stronger, positive correlation but there are still points being placed about the trendline. Once again the formula for the trendline shows us that the trendline crosses the y-axis at +33.474mm showing that when the length is 0mm the width is +33.474mm, once again this is impossible. The gradient of this scatter graph tells us that as the length increases by 1mm the width increases by 0.1278mm.
My first hypothesis was:
As the length increases so will the width.
According to the scatter graphs (shown over the page) this hypothesis has been proved as on both of the scatter graphs, the gradient shows us that as the length increases so does the width.
Statistical Technique: Box Plots.
Another way of showing this table is in the form of a “box and whisker” diagram, located overleaf.
My next hypothesis was:
The length and width will be greater in 2002 than 2001.
In order to attempt to prove this hypothesis I am going to look at the mean on the table above and also represent the data above in the form of box plots (shown overleaf). The mean length and width in 2001 are 62.57mm and 35.14mm, and the mean length and width in 2002 are 76.71mm and 43.28.
From the table above we can see that the length and width is greater in 2002 than in 2001.
The box plots show us the range from the largest to the smallest. In 2001 the “whiskers” are fairly long ranging from 27mm – 98mm, shows us that there is a wider distribution of data or that there are anomalies at the ends of the two “whiskers” causing the whiskers to be elongated. In 2002 the “whiskers” are shorter, ranging from 45mm – 102mm. But, the longest length in 2001 is 98mm whereas in 2002, 102mm. The same applies with the width. In 2001 ranging from 11mm to 56mm and in 2002 ranging from 29mm to 56mm.
In conclusion the mean length and width are greater in 2002 than in 2001, and the box plots show that the longest length is in 2002, but the largest width is equal in 2001 and 2002 (56mm).
My third hypothesis was:
The spread of the length and width will be great in 2002 than in 2001.
In order to prove or disprove this hypothesis we look at the box plots (shown overleaf) and look at the range and IQR.
The range for lengths in 2001 is 71mm and widths are 45mm. The range for lengths in 2002 is 57mm and widths are 27mm. If we look at the stems, the ones in 2001 are a lot longer than in 2002 - which suggests a wider spread of data. This possibly could mean that there was just one oddity for a minimum, so we have to look at the “box”. The boxes for 2002 are smaller than the boxes for 2001, which disproves my hypothesis that the spread of leaves is greater in 2002 than 2001.
The box plots show us that the two “whiskers” are almost equal in all of the box plots; this shows us that they are of an even distribution. This supports my forth hypothesis:
The length and width of the leaves will follow a normal to almost normal distribution.
Length 2001:
Key: 2 | 7 = 27
Width 2001:
Length 2002:
Width 2002:
N.B. Graphs are not included for width as we are only looking at lengths in 2001 and 2002.
My final hypothesis was:
The length of leaf for which 10% of the leaves are longer will be greater in 2002.
In order to prove this, I have to find out 10% of the totals on both cumulative frequency graphs.
For lengths in 2001, 4.9 and in 2002, 5.
From the two graphs I can tell that the 10th percentile in 2001 is equal to 40 and in 2002 the 10th percentile is equal to 63.
From the above I can say that my hypothesis has been proven.
In conclusion I have proven 4 out of 5 of my hypotheses and disproved the remaining one.
