Chance of that
Problem
This is an open investigation. It can be taken to various levels of complexity and note that the construction of non-trivial examples is, in itself, difficult.
I was looking at two seemingly random lists of 12 whole numbers
chosen from 1 to 5. The sample correlation was exactly zero.
Convince yourself that such a list of numbers is possible.
Can you get a feel for the properties of such lists?
Explore the properties and frequencies of such lists of numbers,
perhaps varying the two numbers involved.
For more investigations see
our Stage 5 pages.
Getting Started
You might like to note that the sample correlation $r_{xy}$ of a pair of lists of $N$ numbers $x_i$, $y_i$ is given by
$$r_{xy} = \frac{\sum^{N}_{i=1} \left((x_i-\bar{x})(y_i-\bar{y})\right)}{\sqrt{\sum^N_{i=1}(x-\bar{x})^2\sum^{N}_{i=1}(y_i-\bar{y})^2}}$$
Student Solutions
Yaseen from LAE Tottenham in the UK visualised the lists of numbers and used computer programming to generate pairs of lists whose correlation was exactly zero. This is Yaseen's work:
I tried manually creating two seemingly random lists adhering to the parameters given, however, I had no luck. The value of the Pearson product-moment correlation coefficient (PPMCC) never hit exactly 0. Therefore, I used my Python skills to create a program that would generate the two lists for me with zero correlation.
Click here to see Yaseen's code
Click here to run Yaseen's code
The way it works:
- Generate two random lists
- 12 whole numbers
- Each number between 1 and 5 inclusive
- Calculate the correlation between the two lists.
- This is done by importing a function that calculates r.
- Check if r = 0
- If it does, exit the loop.
- If it does not, go back to step 1.
- Output the two lists
Click below to see examples of lists generated by Yaseen's program.
|
List x |
List y |
Graph |
|
3, 2, 1, 1, 4, 3, 4, 2, 4, 4, 3, 5 |
2, 5, 1, 1, 4, 2, 2, 5, 5, 1, 4, 1 |
Figure 1 |
|
4, 2, 1, 4, 4, 2, 1, 1, 1, 2, 3, 3 |
3, 5, 3, 2, 2, 1, 1, 2, 5, 1, 4, 4 |
Figure 2 |
|
2, 1, 4, 3, 3, 3, 4, 2, 3, 2, 4, 1 |
3, 1, 3, 2, 1, 4, 2, 2, 3, 3, 4, 5 |
Figure 3 |
|
3, 4, 4, 3, 3, 1, 2, 2, 4, 2, 5, 3 |
3, 1, 5, 2, 4, 1, 2, 2, 1, 5, 2, 5 |
Figure 4 |
|
5, 2, 3, 4, 3, 3, 3, 2, 1, 5, 3, 2 |
5, 5, 5, 4, 4, 4, 3, 3, 5, 4, 2, 4 |
Figure 5 |
Analysis
When I saw the graphs I was confused as there were [fewer] than twelve data points on each. I quickly realised that this is due to there being more than one data point at [the same] point on the graph. This meant that the graph was slightly misleading as acquiring a proper understanding of the graph requires knowledge of the raw data.
Yaseen also thought about patterns which might be present in the underlying numbers. Can you use these ideas to manually generate lists of numbers whose correlation coefficient is equal to zero?
Pearson's r has the following equation:
$r=\frac{\Sigma(x-\bar{x})(y-\bar y)}{\sqrt{(x-\bar x)^2(y-\bar y)^2}}$
A shorter representation:
$r=\dfrac{S_{xy}}{S_xS_y}$
When there is no correlation:
$r=0 \Rightarrow S_{xy}=0$
$\therefore\Sigma(x-\bar x)(y-\bar y)=0$
While I am not sure how to fully interpret this, I conjecture that there is some sort of cancelling occurring within the set of $(x-\bar x)(y-\bar y)$ values. It could consist of identical values with different signs (e.g. $-3,$ $-2,$ $-1,$ $+1,$ $+2,$ $+3$) or varying values that altogether sum to $0$ (e.g. $-10$, $-8$, $-4$, $-3$, $-1$, $+3$, $+5$, $+6$, $+12$). I feel the key to this lies within the product of the deviation scores.
Yaseen also imposed some constraints to his program to see whether it could still find lists with correlation coefficient equal to zero. Can you find a pair of lists manually which satisfies Yaseen's constraints? Two minutes might not have been long enough for the program to find lists which work.
I edited my Python code to generate two lists with zero correlation without the number $1$ present in either of the lists. I left the program running for over $2$ minutes yet no results were outputted. After this, I ran another experiment setting the condition that the number $1$ cannot be present in list $x$ and the number $2$ cannot be present in list $y.$ This also produced no results. This
implies that two lists, following the given parameters, cannot exist with at least one number from $1$ to $5$ inclusive not being present in both lists or one number missing from list $x$ and another missing from list $y.$
With real data, you would not expect to get a value of $r$ of exactly $0.$ Even the slightest shift of a data point towards the line of best fit would change the value of $r.$