21.2 Descriptive statistics 597
the heights (in appropriate units) of a sample of the population, the weights of a
sample of manufactured products, or the results of measurements of a physical
quantity such as a rate constant or equilibrium constant.
Table 21.3 50 experimental values
40.6 44.9 47.1 39.5 45.3 38.9 42.9 47.0 45.0 44.2
39.3 40.7 48.4 43.1 48.9 44.9 43.2 37.1 45.3 42.7
47.5 46.5 40.9 40.5 38.9 33.3 49.1 43.7 41.3 41.3
45.3 36.9 49.3 37.3 47.2 44.3 42.9 43.4 43.1 41.1
51.1 43.3 37.9 36.9 53.2 39.3 45.7 42.7 47.1 46.8
For presentation purposes, information of this type is simplified by dividing the
range that contains all the sample values into a suitable number of intervals, called
class intervalsor groups, and allocating the sample values to their appropriate classes.
The number of values in each class is called the class frequency. In our example, the
smallest and largest values are 33.3 and 53.2, respectively, so that all the values lie in
the range 32.0 to 54.0, say, and this is conveniently divided into 11 class intervals,
each of width 2.0. A value that falls on a class boundary is assigned to the upper
class. The resulting frequency distribution is illustrated by a frequency histogram
in Figure 21.2 and a cumulative frequency graphin Figure 21.3. In the latter, each
point on the graph is the frequency of all the values up to and including the value at
that point.
Both the bar chart in Figure 21.1 and the histogram in Figure 21.2 show distributions
of frequencies that are approximately symmetrical about the central value, with
frequency decreasing away from the centre. This is the result expected for many different
types of experiment. Thus, the heights of members of the (human or other) population
are expected to be distributed evenly about some average value, with very small and
very large heights less likely than heights near the average. Similarly, ‘equally good’
1
2
3
4
5
6
7
8
9
10
33 35 37 39 41 43 45 47 49 51 53
Frequency
Value
10
20
30
40
50
323436384042444648505254
.........................................................................................................................................................................
.................
...................
......................................
.....
.....
......
....
.....
......
....
.....
.......
....
.....
.....
.....
.....
.....
....
.....
.....
.....
.....
.....
....
....
....
....
....
.....
....
....
....
....
....
.....
....
....
....
....
...
....
....
....
....
...
....
....
....
...
....
....
....
....
...
....
.....
....
...
.....
...
....
....
....
....
....
...
....
....
....
....
....
....
...
......
....
....
....
....
....
.....
....
....
....
....
....
.....
....
.....
......
......
.....
......
.....
......
......
..........
.................
....................
..................
.............
Cumula tivef requency
Val u e
Figure 21.2 Figure 21.3