To solve this, we will calculate the missing frequency value using the mean formula.

Step 1: Recall the mean formula

$\text{Mean} = \frac{\text{Sum of all items}}{\text{Total frequency}}$

The mean number of items bought is given as $3$. Using this formula: $3 = \frac{\text{Sum of all items}}{\text{Total frequency}}$

Step 2: Express the total frequency

The table provides frequencies except for the missing one (corresponding to $4$ items bought). Let the missing frequency be $x$. The total frequency is: $3 + 10 + 1 + x + 5 = 19 + x$

Step 3: Calculate the sum of all items

The sum of all items bought is the product of each number of items and its frequency: $(1 \times 3) + (2 \times 10) + (3 \times 1) + (4 \times x) + (5 \times 5)$ This simplifies to: $3 + 20 + 3 + 4x + 25 = 51 + 4x$

Step 4: Set up the equation

Substitute into the mean formula: $3 = \frac{51 + 4x}{19 + x}$

Multiply through by $19 + x$: $3(19 + x) = 51 + 4x$ $57 + 3x = 51 + 4x$

Step 5: Solve for $x$

Rearrange terms: $57 - 51 = 4x - 3x$ $6 = x$

Final Answer:

The missing frequency is $\mathbf{6}$.

Would you like further details or additional examples?

Related Questions:

1. What is the total number of items bought after finding the missing value?
2. How does changing the mean affect the missing frequency?
3. Can the frequency ever be a non-integer value in such problems?
4. How would you verify the solution is correct?
5. What happens to the sum if the frequency of $5$ items is doubled?

Tip: Always double-check your work by substituting your result back into the original formula!