Understanding the Relationship Between Mean, Median, and Mode

In statistics, the mean, median, and mode are measures of central tendency that describe the center point of a data set. While they measure central location, they do so in different ways. The arithmetic mean is the average value, the median is the middle value, and the mode is the most frequent value.

For moderately skewed distributions, there exists an empirical relationship between these three measures. This relationship is often expressed by the formula:

\(\text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean}\)

This empirical formula allows us to estimate one measure if the other two are known.

Calculating the Arithmetic Mean using the Empirical Formula

We are given the following values:

• Mode = 25.5
• Median = 16.5

We need to find the value of the arithmetic mean using the empirical relationship formula:

\(\text{Mode} = 3 \times \text{Median} - 2 \times \text{Arithmetic Mean}\)

Substitute the given values into the formula:

\(25.5 = 3 \times 16.5 - 2 \times \text{Arithmetic Mean}\)

First, calculate the product of 3 and the median:

\(3 \times 16.5 = 49.5\)

Now, the equation becomes:

\(25.5 = 49.5 - 2 \times \text{Arithmetic Mean}\)

To isolate the term with the arithmetic mean, subtract 49.5 from both sides of the equation:

\(25.5 - 49.5 = -2 \times \text{Arithmetic Mean}\)

\(-24 = -2 \times \text{Arithmetic Mean}\)

Finally, divide both sides by -2 to find the value of the arithmetic mean:

\(\text{Arithmetic Mean} = \frac{-24}{-2}\)

\(\text{Arithmetic Mean} = 12\)

Determining the Value of the Arithmetic Mean

Following the calculation using the empirical formula relating the mode, median, and arithmetic mean, we found the value.

The calculation shows that the arithmetic mean is 12.