Understanding Mean, Median, and Mode

In statistics, the mean, median, and mode are measures of central tendency used to describe a dataset.

• Mean: The average of all values, calculated by summing all values and dividing by the number of values.
• Median: The middle value in a dataset when arranged in ascending or descending order.
• Mode: The value that appears most frequently in a dataset.

The Empirical Relationship

For moderately skewed distributions, there is an approximate empirical relationship between the mean, median, and mode. This relationship is often stated as:

Mode = 3 × Median - 2 × Mean

Alternatively, it can be expressed as:

Mean - Mode = 3 × (Mean - Median)

This formula helps us estimate one measure if we know the other two.

Applying the Formula to the Problem

We are given:

• Median = 4
• Mode = 6

We need to find the Mean.

Let's use the first version of the empirical relationship formula:

Mode = 3 × Median - 2 × Mean

Calculation Steps

Substitute the known values into the formula:

$6$ = 3 × $4$ - 2 × Mean

Now, let's solve for the Mean:

1. Calculate 3 × 4:
   $6$ = 12 - 2 × Mean
2. Rearrange the equation to isolate the term with Mean:
   2 × Mean = 12 - $6$
3. Perform the subtraction:
   2 × Mean = $6$
4. Divide by 2 to find the Mean:
   Mean = $6$ / 2
5. Final calculation:
   Mean = $3$

Conclusion

Using the empirical relationship for skewed distributions, the calculated Mean value is 3.