Calculate Median from Mean and Mode

This problem requires finding the median when the arithmetic mean and mode are provided. We can use the empirical relationship between these three measures of central tendency, which holds for moderately skewed distributions.

Empirical Relationship Formula

The formula connecting mean, median, and mode is:

$ \text{Mean} - \text{Mode} = 3 \times (\text{Mean} - \text{Median}) $

Given Values

• Arithmetic Mean ($\bar{x}$) = 45
• Mode = 48

Calculation Steps

1. Substitute the given Mean and Mode values into the formula:

   $ 45 - 48 = 3 \times (45 - \text{Median}) $

2. Simplify the left side of the equation:

   $ -3 = 3 \times (45 - \text{Median}) $

3. Divide both sides by 3 to isolate the term containing the Median:

   $ \frac{-3}{3} = 45 - \text{Median} $

   $ -1 = 45 - \text{Median} $

4. Rearrange the equation to solve for Median:

   $ \text{Median} = 45 - (-1) $

5. Calculate the final value for the Median:

   $ \text{Median} = 45 + 1 $

   $ \text{Median} = 46 $

Result

The calculated median is 46.