Investigating the Relationship Between the Length and Width of a Leaf
Aim:
To investigate if there is a relationship between the length and width of the leaves. I will also investigate summary measures to represent my data.
Data Collection:
I choose to collect my leaves from a tree in my back garden, as it was easy to access. I picked the leaves from different places and different heights around the tree so that my sample of leaves would be random. I choose a sample size of 35 leaves because it is large enough to reflect the trend amongst the width and length of the leaves on the tree, but still keeping it to a manageable size. When I collected 35 leaves I measured the length of each leaf
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I added all the widths and called them y.
x=3522/35
x=100.63mm
y=1709/35
y=48.83mm
The formula for standard Deviation is:
Measurement
Total
Mean
Standard Deviation
LENGTH
3522
100.63
19.99
WIDTH
1709
48.83
11.63
To check that these values are correct I shall group my data and estimate the mean and standard deviation. I will lose detail by grouping the data and I need to make the assumption that the mid class value is representative of data within each interval. Grouped frequency tables are used for continuous data or for discrete data when a lot of data has to be recorded. I am going to organise the data into classes. I am going to draw two tables. One for the length and one for the width.
Grouped frequency table for length.
Length mm
Tally
Frequency (f)
Mid-point (x)
f*x
fx²
50-59
/
1
54.5
54.5
2970.25
60-69
/
1
64.5
64.5
4160.25
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The medians are 100mm for length and 47mm for width. In both cases the box plots indicate a positive skew.
To investigate the relationship of the length and width of the leaves as there are two [IMAGE] variables I shall draw a scatter graph.
The scatter graph indicates a strong positive correlation, as the points are close to a straight line. As the length increase the width also increases.
I shall work out the Product Moment Correlation Coefficient (PMCC)
As a check I will use the function key on the computer to find the PMCC. From the computer the PMCC=0.9232572 this matches my calculator value. This is a very strong correlation between length and width of leaves.
I shall now test whether it is statistically correct.
H p=0 there is no correlation
H p>0 there is some positive correlation that implies one tailed test.
Testing at 5% level of significance
From tables for n=35 the critical value=0.2826
As 0.9232>0.2826 the result is significant therefore we reject H .
There is evidence of a positive correlation between length and width of leaves.
As there is evidence of correlation I can calculate the
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