Measurement and Description

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Notes

Standardisation

Imagine we have the following scale:

0 1 2 3 4 5 6 7 8 9 10

This scale consists of eleven numbers (n = 11) running from a minimum of 0 to a maximum of 10. The range is \(10 - 0 = 10\).

The mean:

The mean of this scale is: (0+1+2+3+4+5+6+7+8+9+10)/11 = 5

The median:

To calculate the median, or middle value, we order the values in ascending order (it is already listed like that) and we find the value right in the middle of the scale. Because the scale in this case has an odd number of values, we add 1 to the total number of values before we divide by 2: \(11 + 1 / 2 = 6\); so the mmedian is the 6th value in the list when the values are placed in ascending order.

Mean-centring:

\(0:10 - mean(0:10)\) = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5

Standard deviation:

sqrt(sum( (0:10 - mean(0:10) )^2 ) / (length(0:10) - 1)) = 3.3166248

Standard (Z) scoring (standardizing):

\(z = ( 0:10 - mean(0:10) ) / ( sqrt(sum( (0:10 - mean(0:10) )^2 ) / (length(0:10) - 1)) )\) = -1.5075567, -1.2060454, -0.904534, -0.6030227, -0.3015113, 0, 0.3015113, 0.6030227, 0.904534, 1.2060454, 1.5075567