Formation of a Frequency Distribution

When quantitative data is classified by a measurable variable, the result is a frequency distribution — a table showing how many times (the frequency) each value, or each group of values, occurs.

For a small range of distinct values, a discrete frequency distribution simply lists each value against its frequency (tally marks are used while counting):

When the variable can take a very large number of values (marks out of 100, income in rupees), the values are first grouped into class intervals, giving a continuous frequency distribution. Key terms:

• Class limits — the two end values of a class, e.g. in 20–30, the lower limit is 20 and the upper limit is 30.
• Class interval / class width (h) — the difference between the upper and lower boundaries of a class; usually kept equal across all classes for ease of comparison.
• Mid-value (mid-point) of a class $= \dfrac{\text{lower limit} + \text{upper limit}}{2}$ — the single value taken to represent every item that falls in that class.
• Class frequency (f) — the number of observations falling in a class.

Two methods of forming class intervals:

1. Exclusive method — the upper limit of one class is the same as the lower limit of the next (e.g., 0–10, 10–20, 20–30…); by convention, an item exactly equal to the upper limit (e.g., exactly 10) is counted in the next class, not the current one. This method gives continuous, non-overlapping class boundaries and is the one normally used for drawing a histogram.
2. Inclusive method — both limits belong to the same class (e.g., 0–9, 10–19, 20–29…), so there is a visible gap between one class's upper limit and the next class's lower limit. This is easier to read for whole-number data (like marks) but must be converted to the exclusive form before a histogram can be drawn (see the worked example below). …