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Telepathy Investigation

Introduction
The aim of this investigation is to statistically analyse the results of telepathy tests run on a selection of people to determine whether none/one/some/all of them poses telepathic powers. These results could be used as a representative sample of other similar people.
The selection of people will be children, all 16 or 17 years old, from a further mathematics class. These children are similar in age and assumed to be similarly intelligent. A group of people similar in these characteristics has been chosen, so this group could be used as a representative sample of other similar people.
The sample size will be 10 people large, as this should be large enough to show if any of them have telepathic powers compared to the rest. This size should also be large enough to show if further investigation into this group of society about their telepathic powers should be considered, should some of them exhibit telepathic powers. This size should also be large enough to determine whether the "transmitter" (see below), possesses telepathic powers.
A person will take card from a pack of cards and will look at it, while not letting anyone but him/her to view the card. Another person will then state what suit they think the card is, heart, diamond, club or spade. Whether they got the suit right or wrong will be recorded.
This investigation will try and determine whether any people in the group can receive telepathic messages, and so the person viewing the card - the "sender", will be the same person throughout the investigation. This means that the telepathic "sending" ability of the person viewing the card is also tested. When looking at the results from the group as a whole, the telepathic powers of the sender can be analysed, while looking at individual results of people trying to received messages will give an idea as to their own telepathic powers.
Each person will try and receive the suit of 20 cards, as this size appears to be a large enough set of tests to determine whether the subject has any telepathic abilities. The card will be picked from a shuffled pack of cards, in case any tampering has been involved.
The results of the test will be compared to a binomial model that assumes that the suit that the receiver picks is picked at random (that the receiver or transmitter has no telepathic powers). This distribution model requires the events to be independent of each other. Certain precautions must be taken to ensure this. The subject must not be told whether or not they got a suit right or wrong, as this may affect their next answer. Also, the subject must not be able to see the transmitter, as this may also affect their answer, as the transmitter's expression may give an idea to whether they stated the correct suit.
The mean and standard deviation will be calculated
from the real data, and from the binomial model, so
they can be compared. They can then be compared to see
if there are any significant differences that could
mean telepathic powers are present.


Results
Person No of correct Answers
1 6
2 8
3 8
4 2
5 3
6 8
7 6
8 6
9 9
10 9

Number of correct answers 2 3 4 5 6 7 8 9
Frequency 1 1 0 0 3 0 3 2

Calculations Using Data
Let x be number of correct answers
Mean

Standard Deviation

Binomial Model
The binomial probability model can be calculated from
the following formula, where given n trials of an
event, the probability of r occurrences of an outcome
that has a probability p occurring at each trial.

To calculate the probability distribution for the set
of telepathy tests, assuming that the chance of each
subject stating the correct card is ¼, and there are
20 tests run on each subject, the formula is
therefore:

Using this formula, the probability distribution for
correctly stating the suit of card is as follows,
where X is the number of suits correctly chosen.
x p(x)
0 3.1712 x 10-3
1 2.1141 x 10-2
2 6.6695 x 10-2
3 1.1339 x 10-1
4 1.8969 x 10-1
5 2.0233 x 10-1
6 1.6861 x 10-1
7 1.1241 x 10-1
8 6.0887 x 10-2
9 2.7061 x 10-2
10 9.9223 x 10-3
11 3.0068 x 10-3
12 7.5169 x 10-4
13 1.5419 x 10-4
14 2.5699 x 10-5
15 3.1712 x 10-6
16 3.5693 x 10-7
17 2.7994 x 10-8
18 1.5552 x 10-9
19 5.4570 x 10-11
20 9.0949 x 10-13

Mean
The mean of this probability distribution can be
calculated thus:

Standard Deviation

Analysis
The mean of the real data is 30% higher than the mean
of the probability distribution. There could be
several reasons for this. It could mean that the
transmitter has some telepathic powers. It could mean
that many of the people in the group to receive
messages are telepathic. It could be a combination of
both.
However the standard deviation for the test data is
approximately 18% higher than for the probability
model. This means that there is a higher spread, that
there are more results further from the mean than
there would be if the subjects were simply guessing
the suit. This does suggest that some of the subjects
that could have possessed telepathic powers were not
getting the correct suit right on purpose. To put it
simply, they were "fighting" the messages they were
sent. It could also mean that the transmitter could
have been sending them incorrect messages.
I think that data doesn't show that anyone in the test
had any extremely significant telepathic powers. The
highest number of correct cards in the test was 9.
This happened twice. This is quite an improbable
event; the probability of such an event happening
once, is 0.027061. However, as there are so many
things that could happen - from getting 0 right to
getting all 20 right, they are all improbable, just
some are more probable than others.
Here I have found I have contradicted myself. The mean
and standard deviation show that some of the people
have telepathic powers, but looking at the data they
don't. I think this is because this test was severely
limited. The results of the test could have been
random, and the mean and standard deviation simple
"happen" to be higher. I think that the test was far
to short and the test group too small to be certain of
any telepathic powers. A future test should test each
person more and ask him or her to identify more suits
of cards. The sender would have to be tested more
also, and try and transmit more cards to more people.
To make sure that this group of people doesn't have
telepathic powers within it, they should all be
subject to longer tests, as this will confirm whether
they have telepathic powers or not.
A large limitation of this test was deciding whether
the results show if it is the sender, or the receivers
that were telepathic. This could be done by making the
receivers try and receive messages from a selection of
people, and analysing the results to find which people
were the best transmitters.

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