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Cognitive Science

1.2 Psychological Research Methods (PRM)
Source: PRM
See the Notation, and Special Symbols
For more mathematics, see Calculus, Discrete Mathematics, and, if you know Norwegian, The Norwegian Kalkulus pages.

1.2.1 The five number summary

1. Min - The smallest score
2. Q1 - The first quartile
3. Mdn - The median
4. Q3 - The third quartile
5. Max - The largest score

1.2.2 Standard deviation

3 measures of amount of variation in scores (spread):

1. The range, Max - Min
2. The interquartile range, Q3-Q1
3. Standard deviation (s)

We discriminate between population standard deviation and sample standard deviation (that which is based on the whole population, and that which is based on a sample from the population).

How to find the sample standard deviation (s):
(The example is taken from PSY11PYA PRM Assignment 2, autumn 2001, La Trobe University)

1. Calculate the sample mean, called X. This is just the average of all sampled data (the sum of all data divided by the number of data, n)

For example, if your observed lengths (in centimetres) are:
16.2, 10.4, 7.2, 8.4, 18.4, 12.6, then the sample mean
X = (16.2 + 10.4 + 7.2 + 8.4 + 18.4 + 12.6) / 6 = 12.2

2. Then find the Sum of Squares, SS:
SS = (X - X)^2, where ^2 means "squared".
{Alternatively, SS = (X^2) - [( X )^2 / n ] }

Example:
SS =  (16.2 - 12.2)^2 + (10.4-12.2)^2 + (7.2 - 12.2)^2 + (8.4 - 12.2)^2 + (18.4 - 12.2)^2 + (12.6 - 12.2)^2 = 97.28

3. Find s, the sample standard deviation, using the formulae 
s^2 = SS / (n-1)

Example:
s^2 = 97.28 / 5 = 19.456
s = 4.41

The population standard deviation
The population standard deviation s = sqrt( SS / N ), where SS = (X - m)^2 and N is the number of the population. 

The population standard deviation s can be calculated by the formulae
s^2 = ( X^2 - Nm^2 ) / N 

 

1.2.3 Probability distributions

If we were to conduct an experiment in which the result of each trial is either success or failure, then the probability of success
P( success ) = ( No. of success outcomes ) / ( Total no. of trials conducted ) 
where 0 P( success ) 1. For this number to have any meaning, we have to perform the experiment a large number of times. 

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