For which of the following lists of 5 numbers is the average (arithmetic mean) greatest than the median?

Options

A. 3, 3, 5, 8, 8
B. 2, 3, 5, 6, 7
C. 3, 3, 5, 7, 7
D. 3, 4, 5, 6, 7

Answer

To determine which list of 5 numbers has an average (arithmetic mean) greater than the median, we will follow these steps:

### Step 1: Identify the Lists
We have four lists of numbers:
- Option A: 3, 3, 5, 8, 8
- Option B: 2, 3, 5, 6, 7
- Option C: 3, 3, 5, 7, 7
- Option D: 3, 4, 5, 6, 7

### Step 2: Calculate the Median
For a list of 5 numbers, the median is the third number when the numbers are arranged in ascending order.

- **Option A**: The numbers are 3, 3, 5, 8, 8. The median is 5 (the third number).
- **Option B**: The numbers are 2, 3, 5, 6, 7. The median is 5 (the third number).
- **Option C**: The numbers are 3, 3, 5, 7, 7. The median is 5 (the third number).
- **Option D**: The numbers are 3, 4, 5, 6, 7. The median is 5 (the third number).

### Step 3: Calculate the Average (Mean)
The average is calculated by summing all the numbers and dividing by the total count (which is 5 in this case).

- **Option A**:
\[ \text{Mean} = \frac{3 + 3 + 5 + 8 + 8}{5} = \frac{27}{5} = 5.4 \]

- **Option B**:
\[ \text{Mean} = \frac{2 + 3 + 5 + 6 + 7}{5} = \frac{23}{5} = 4.6 \]

- **Option C**:
\[ \text{Mean} = \frac{3 + 3 + 5 + 7 + 7}{5} = \frac{25}{5} = 5 \]

- **Option D**:
\[ \text{Mean} = \frac{3 + 4 + 5 + 6 + 7}{5} = \frac{25}{5} = 5 \]

### Step 4: Compare Mean and Median
Now we compare the mean and median for each option:

- **Option A**: Mean = 5.4, Median = 5 (Mean > Median)
- **Option B**: Mean = 4.6, Median = 5 (Mean < Median)
- **Option C**: Mean = 5, Median = 5 (Mean = Median)
- **Option D**: Mean = 5, Median = 5 (Mean = Median)

### Conclusion
The only option where the average (mean) is greater than the median is **Option A: 3, 3, 5, 8, 8**.

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