Understanding Measures of Central Tendency: Mode, Mean, and Median

The question asks us to find the median of a dataset given its mode and mean. In statistics, the mode, mean, and median are all measures of central tendency, representing the typical value in a dataset.

For a moderately skewed distribution, there is an empirical relationship between the mode, median, and mean. This relationship is commonly expressed by the formula:

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

We are given:

• Mode = 18
• Mean = 21

We need to find the Median. Let's substitute the given values into the empirical formula:

\(18 = 3 \times \text{Median} - 2 \times 21\)

Now, let's solve this equation step-by-step to find the value of the Median.

1. First, calculate the value of \(2 \times 21\):
   \(2 \times 21 = 42\)
2. Substitute this value back into the equation:
   \(18 = 3 \times \text{Median} - 42\)
3. Next, isolate the term involving the Median. To do this, add 42 to both sides of the equation:
   \(18 + 42 = 3 \times \text{Median}\)
   \(60 = 3 \times \text{Median}\)
4. Finally, solve for the Median by dividing both sides of the equation by 3:
   \(\text{Median} = \frac{60}{3}\)
   \(\text{Median} = 20\)

Therefore, using the empirical relationship for moderately skewed distributions, the median of the data is 20.

This empirical formula is a useful approximation when the distribution is not perfectly symmetrical. In a perfectly symmetrical distribution (like a normal distribution), the mean, median, and mode are all equal.

Revision Table: Measures of Central Tendency Formulas

Additional Information: Skewness and Central Tendency

The relationship between the mean, median, and mode depends on the skewness of the distribution:

• Symmetrical Distribution: In a symmetrical distribution (like the normal distribution), the mean, median, and mode are all located at the same point. Mean = Median = Mode.
• Positively Skewed Distribution: In a positively skewed distribution, the tail is longer on the right side. The mean is typically greater than the median, which is typically greater than the mode. Mode < Median < Mean. The empirical formula can be used, but the approximation might be less accurate for heavily skewed data.
• Negatively Skewed Distribution: In a negatively skewed distribution, the tail is longer on the left side. The mean is typically less than the median, which is typically less than the mode. Mean < Median < Mode. Again, the empirical formula is an approximation.

The empirical formula used in this problem is based on the observation of many real-world datasets that are moderately skewed. It's a useful rule of thumb, but not a mathematical identity that holds true for all distributions.