Problems with a missing frequency

What you will be able to do

• find a missing frequency when the mean is given;
• find two missing frequencies from the total and the median, or the total and the mean;
• check that an answer is possible.

The plan

Boards often hide one or two frequencies in a table. They give a clue in place: the mean, the median, or the total. The plan is always the same.

1. Call the missing frequency $x$ (and the second one $y$).
2. Write the quantity in the clue as a formula that contains $x$.
3. Put the given number into the formula. This is an equation.
4. Solve it.
5. Check. A frequency is a whole number and is not negative. Put it back and see that the clue is true.

One missing frequency, and the mean

The table has class marks $5, 15, 25, 35, 45$ with frequencies $6, 10, 12, 9$ and $x$. The mean is $30$.

• The total number of values is $N = 6 + 10 + 12 + 9 + x = 37 + x$.
• The sum $\sum fx$ is $30 + 150 + 300 + 315 + 45x = 795 + 45x$.
• The mean is $\dfrac{795 + 45x}{37 + x} = 30$.

Multiply both sides by $37 + x$: $795 + 45x = 1110 + 30x$. Then $15x = 315$ and $x = 21$.

Check. $N = 58$ and $\sum fx = 795 + 945 = 1740$, and $\dfrac{1740}{58} = 30$.

Two missing frequencies, and the median

Seventy values fall in the classes $0$ to $10$, $10$ to $20$, $20$ to $30$, $30$ to $40$, $40$ to $50$ and $50$ to $60$, with frequencies $5, x, 16, 12, y, 7$. The median is $35$.

• The median $35$ is in the class $30$ to $40$. So $l = 30$, $f = 12$, $h = 10$ and $\dfrac{N}{2} = 35$.
• The cumulative frequency before this class is $cf = 5 + x + 16 = 21 + x$.
• The formula gives $35 = 30 + \dfrac{35 - (21 + x)}{12} \times 10$. So $5 = \dfrac{14 - x}{12} \times 10$, and $14 - x = 6$. Then $x = 8$.
• The total is $70$, so $5 + 8 + 16 + 12 + y + 7 = 70$. Then $y = 22$.

Check. The cumulative frequencies are $5, 13, 29, 41, 63, 70$. The $35$th value is in the class $30$ to $40$, and $30 + \dfrac{35 - 29}{12} \times 10 = 35$.

Notice that we decided on the median class first, from the median we were given. The check at the end shows that the choice was right.

Two missing frequencies, and the mean

A table has class marks $10, 20, 30, 40, 50$ with frequencies $4, x, 18, y, 6$. There are $40$ values, and the mean is $29$.

• The total: $4 + x + 18 + y + 6 = 40$ gives $x + y = 12$.
• The sum: $\sum fx = 40 + 20x + 540 + 40y + 300 = 880 + 20x + 40y$, and it equals $29 \times 40 = 1160$. So $20x + 40y = 280$, which is $x + 2y = 14$.

Subtract the first equation from the second: $y = 2$. Then $x = 10$.

Check. $\sum fx = 40 + 200 + 540 + 80 + 300 = 1160$, and $\dfrac{1160}{40} = 29$.

When there is no answer. Keep $N = 40$ but change the mean to $27$. Then $\sum fx = 1080$, so $x + 2y = 10$, and with $x + y = 12$ we get $y = -2$. A frequency cannot be negative. No table of this kind has a mean of $27$. If a question gives numbers like this, the data are inconsistent.

Practise with new numbers

Try it in Python

This program finds the missing frequency of the first example by trying values, and then finds it by solving the equation with fractions. Both should give the same $x$.

Common mistakes

• Forgetting that $N$ also contains $x$. The total is $37 + x$, not $37$.
• Using the wrong class for the median. Find the median class from the given median, and check afterwards.
• Not checking that $x$ is a whole number. A frequency of $4.5$ or $-2$ cannot be right.
• Leaving out the check. Put the answer back into the table and recompute the mean or the median.

Remember these

• Name the missing frequency, write the clue as a formula in it, solve, and check.
• With a missing frequency in the median formula, first decide which class holds the median.
• For two missing frequencies, you need two equations: for example the total and the mean, or the total and the median.
• A missing frequency must be a whole number that is not negative.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A table has frequencies $5$, $x$ and $8$ with class marks $10$, $20$ and $30$. What is the number of values, in terms of $x$?

2. For the table of question 1, write $\sum fx$ in terms of $x$.

3. The two equations are $x + y = 15$ and $x + 2y = 21$. Find $x$ and $y$.

4. A solution gives a missing frequency of $-3$. What does this tell you?

5. Why do we check the median class after finding the missing frequency?

References

• NCERT, Mathematics for Class 10, the chapter Statistics: the exercises on missing frequencies.
• OpenStax, Introductory Statistics, Chapter 2 Descriptive Statistics, for the formulas used.