ANSWER TO FRAME 35.
Ranges = ,500 -,400 = ,100. Note that the middle five values could have been
anything between ,400 and ,500 and the range would be the same. The range considers
only two values - the high and the low. All other values in between are not considered.
The second measure of dispersion is the standard deviation. The standard deviation is a
much better measure of dispersion than the range because it considers all observations, rather
than just the two extremes. It measures an "average" dispersion around the mean.
As an example, let's find the standard deviation of the following seven observations: (15, 20,
25, 30, 35, 40, 45).
The first step is to compute the mean. In this case the mean is
u = ΣX = 210 = 30.
One approach to measuring dispersion is to measure the distance between each observation and
the mean. The formula for this relationship is X - u where X is an observation and u is the mean.
Consider the following table:
X - u
15- 30=- 15 Notice that if an observation is
20- 30=- 10 smaller than the mean, X - u is
25- 30=- 5
30- 30= 0
35- 30= 5
40- 30= 10
45- 30= 15
Σ (X-u) = 0
In this example, Σ(X-u) equals zero. This is always true. Consequently, Σ(X-u) is useless in
helping us compute a measure of dispersion. The combination of positive and negative values
always offset each other. To avoid this problem we will square the result of each X-u. (To square
a number means to multiply it by itself.)