Statistical Analysis for Education and Psychology Researchers

year the value reduces to 90 per cent of the value it had at the beginning of the second
year and in the third year the value reduces still further to 80 per cent of the value it had
at the beginning of the third year. The average rate of decrease in value over the three-
year period that would result in the same value of the house at the end of the three years
is given by the geometric mean of the three rates.
This can be evaluated as rate 1 ×rate 2 ×rate 3 =95×90×80=684000=rate^3 , so
rate=cube root of 684000=88.1 per cent.

A general notation is the nth root of the product (multiplication) of the values. The n
refers to the number of values, for example, two values would be the square root of the
product of the two values. A simplified way of calculating the geometric mean is to take
the antilogarithm of the mean of the natural logarithm of the rates.
Logarithms to the base e, denoted as logexi (where xi is any positive real number) is
called a natural logarithm. For example, loge2 is=0.693. The geometric mean of the three
rates, 95 per cent, 90 per cent and 80 per cent is=(loge95+
loge90+loge80)/3=13.436/3=4.479. The antilogarithm of this value is=88.1.

Measures of Shape

The shape of a distribution is often compared to what is called a normal distribution.
This is actually a mathematically defined theoretical distribution for a population which
when drawn is characterized by a number of properties:

Figure 3.16: Theoretical normal

distribution curve

• The curve is unimodal, it is smooth, has one highest point which is in the centre of the
  distribution.


• The mode, median and mean all have the same value and indicate the centre of the
  distribution.

Statistical analysis for education and psychology researchers 66