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We have seen how barplot of proportions show statistical
regularity. Incidentally, there is a name for this: relative
frequency. The actual number of times an outcome occurs is
called the frequency. For instance, if you toss a coin 100
times and get 45 heads and 55 tails, then the frequencies are 45
and 55, while the relative frequencies are 0.45 and 0.55. Often
we like to show the frequencies as a table, like
| Value | Frequency |
| Head | 45 |
| Tail | 55 |
| Total | 100 |
It is called a frequency distribution table. Similarly,
you construct a relative frequency distribution table.
As we have already mentioned, the relative frequency distribution
table (or equivalently, its graphical representation, the
barplot) converges, as the sample size goes to infinity. The
limit is called the probability mass function (pmf).
Here the outcomes were discrete, taking only finitely many
values. Often we encounter continuous outcomes, that can
take any value in an interval, like the height of a person.
The barplot technique cannot be used in these cases.
So we discretise the continuous output into a finite number of
intervals or bins, before applying the barplot technique.
We start with a frequency distribution for the discretised data,
where the bins play the role of values. This called
a frequency distribution for grouped data. The bins are
called classes. They are adjacent to each other, and we
need some convention to decide about which class gets the
boundary point. A typical example could be like
[0,1], (1,2], (2,4], (4,6].
We generally work with data that take value in a bounded
interval $(a,b).$
Then we compute
the relative frequencies. Finally we divide the relative
frequency of a class by the class width to get
the relative frequency density for that class. A barplot
for this is called a histogram, which is powerful
graphical device to harness statistical regularity.
We say that $X_1,...,X_n$ constitute a random sample from a
distribution if they are the outcomes of repeated independent
trials of the same random experiment, and their barplot or
histogram converges to that distribution. We also say
that $X_1,...,X_n$ are independently and identically
distributed (IID) from that distribution.