Grouped data and class marks

Before computing any statistics, we must understand the language of frequency tables. A frequency table lists the classes (intervals of values) and the frequencies (number of observations in each class).

For example:

This says: $4$ students scored between $0$ and $10$, $6$ scored between $10$ and $20$, etc. The total number of students is $4 + 6 + 10 + 5 + 3 = 28$.

Class marks

When computing the mean of grouped data, every observation in a class is represented by the midpoint of that class, called the class mark:

$\text{Class mark} = \frac{\text{lower limit} + \text{upper limit}}{2}.$

For the table above, the class marks are $5, 15, 25, 35, 45$.

The use of class marks is a simplification: we assume the observations in each class are evenly spread and use the midpoint as a representative value. This is the source of a tiny error in the computed mean, but for board-exam purposes it is the standard assumption.

Continuous and discontinuous classes

The classes in the example are continuous: the upper limit of one class equals the lower limit of the next. The boundary value (say $10$) is conventionally assigned to the next class (i.e., $10-20$, not $0-10$).

Sometimes classes are written discontinuously, with gaps: $0-9, 10-19, 20-29, \ldots$. To make computations consistent, you must first make the classes continuous by adjusting the boundaries. Subtract $0.5$ from each lower limit and add $0.5$ to each upper limit:

$0-9 \to (-0.5)-9.5, \quad 10-19 \to 9.5-19.5, \quad 20-29 \to 19.5-29.5.$

(In general, the adjustment is half the difference between the upper limit of one class and the lower limit of the next.)

This adjustment does not change the class marks (still $4.5, 14.5, 24.5, \ldots$), but it does affect the class size (which becomes $10$ in both cases, but the boundaries are different).

Class size

The class size (or class width) is the difference between the upper limit and the lower limit of one class. It is denoted $h$ and is typically the same across all classes in a table.

For $0-10$: $h = 10$. For $25-50$: $h = 25$. For $9.5-19.5$: $h = 10$.

Cumulative frequency

The cumulative frequency of a class is the total of the frequencies of that class and all preceding classes. It is denoted $cf$.

For the table above:

Cumulative frequency is essential for computing the median and for drawing ogives.

The cumulative frequency of the last class equals the total frequency $n$.

More-than and less-than tables

Sometimes data is given as less than cumulative frequencies. For instance, "less than $10$: $4$ students; less than $20$: $10$ students; ..." From a less than table you can reconstruct the original frequencies by subtracting consecutive cumulative values.

A more than table reverses this: "more than $0$: $28$ students; more than $10$: $24$ students; ...". The cumulative frequency in this case counts from the top down.

Worked examples

Example 1. Find the class marks for the classes $0-5, 5-10, 10-15, 15-20$.

Class marks: $2.5, 7.5, 12.5, 17.5$. Class size $h = 5$.

Example 2. A frequency table has classes $0-10, 10-20, 20-30$ with frequencies $5, 8, 7$. Find the cumulative frequencies.

$cf$: $5, 13, 20$. Total: $20$.

Example 3. Convert the discontinuous classes $1-5, 6-10, 11-15$ to continuous form.

Subtract $0.5$ from each lower limit and add $0.5$ to each upper limit: $0.5-5.5, 5.5-10.5, 10.5-15.5$. Class size $5$.

Example 4. Given a less than table: $< 10: 5, < 20: 15, < 30: 35$. Find the class frequencies.

Classes $0-10, 10-20, 20-30$ have frequencies $5, 10, 20$.

Example 5. Construct a frequency table for the marks $4, 7, 9, 12, 15, 18, 22, 25, 27$ using class size $5$, starting at $0$.

Classes $0-5, 5-10, 10-15, 15-20, 20-25, 25-30$. Frequencies $1, 2, 1, 2, 1, 2$.

Try it yourself

1. Find the class mark of the class $25-35$.
2. Convert $0-9, 10-19, 20-29$ to continuous classes.
3. A frequency table has classes $10-20, 20-30, 30-40$ with frequencies $5, 12, 7$. Find the cumulative frequencies.
4. The total frequency is $40$ and the cf of the third class is $30$. Find the frequency of the third class given the cf of the second is $18$.
5. Find the class size for $5-15$.
6. A less-than table reads $<5: 3, <10: 8, <15: 18, <20: 24$. Find class frequencies.
7. Construct a frequency table for $7, 12, 14, 18, 22, 25, 25, 28, 33$ with class size $5$ starting at $5$.
8. Find the class marks for classes of size $20$ starting at $0$: $0-20, 20-40, \ldots$
9. A more-than table reads $> 0: 50, > 10: 42, > 20: 30, > 30: 12, > 40: 0$. Find the class frequencies for $0-10, 10-20, 20-30, 30-40$.
10. If the class marks are $5, 15, 25, 35$, what are the classes (assuming class size $10$)?
11. A class has lower limit $25$ and upper limit $35$. Find the class mark and class size.
12. Total frequency $50$. The cf of the class $20-30$ is $30$. The cf of the class $10-20$ is $15$. Find the frequency of the class $20-30$.

Pitfalls / Insight

(1) Discontinuous classes must be made continuous before applying mean/median/mode formulas with class size.

(2) The class mark, not the upper or lower limit, is what represents observations in calculations.

(3) Cumulative frequency is cumulative from the top, not the bottom , i.e., add as you go down the table.

(4) The total frequency $n$ is the cf of the last class. Always verify this with the column sum.