Question

the median of these number cards is 6

what is the missing number?

Explanation:

⚡ Using what you learned: Median

Step 1: Identify the known values and count

The known numbers on the cards are:

$$ 0, 2, 5, 8, 8 $$

Let the missing number be \( x \). There are 6 cards in total.

Step 2: Set up the median condition

For an even number of data points (6 cards), the median is the average of the 3rd and 4th values when arranged in ascending order:

$$ \text{Median} = \frac{\text{3rd value} + \text{4th value}}{2} = 6 $$

This means:

$$ \text{3rd value} + \text{4th value} = 12 $$

Step 3: Determine the missing number

Sort the known values:

$$ 0, 2, 5, 8, 8 $$

• If \( x \le 5 \), the sorted order is \( 0, 2, x, 5, 8, 8 \) (or with \( x \) and \( 2 \) swapped). The 3rd and 4th values would be \( x \) and \( 5 \).

$$ x + 5 = 12 \implies x = 7 $$

But \( 7 \le 5 \) is false.

• If \( x \ge 8 \), the sorted order is \( 0, 2, 5, 8, 8, x \) (or with \( x \) and \( 8 \) swapped). The 3rd and 4th values would be \( 5 \) and \( 8 \).

$$ \frac{5 + 8}{2} = 6.5 eq 6 $$

• If \( 5 < x < 8 \), the sorted order is:

$$ 0, 2, 5, x, 8, 8 $$

The 3rd value is \( 5 \) and the 4th value is \( x \).

$$ 5 + x = 12 \implies x = 7 $$

Since \( 7 \) lies between \( 5 \) and \( 8 \), this is the correct value.

Answer:

7