Finding the Mode from Mean and Median

The question asks us to find the mode of a distribution given its mean and median. In statistics, for moderately skewed distributions, there is an empirical relationship between the mean, median, and mode. This relationship is often expressed by the formula:

Mode \(\approx\) 3 \(\times\) Median - 2 \(\times\) Mean

This formula is particularly useful when the distribution is not symmetrical but also not extremely skewed, as it provides a good estimate for the mode.

Applying the Empirical Formula

We are given the following values:

• Mean = 12
• Median = 15

Now, we can substitute these values into the empirical formula to calculate the estimated mode:

Mode \(\approx\) 3 \(\times\) Median - 2 \(\times\) Mean

Mode \(\approx\) 3 \(\times\) 15 - 2 \(\times\) 12

Mode \(\approx\) 45 - 24

Mode \(\approx\) 21

Based on the empirical formula, the estimated mode of the distribution is 21.

Comparing with Options

Let's compare our calculated mode with the given options:

1. 21
2. 24
3. 15
4. 18

Our calculated value of 21 matches Option 1.

Summary of Calculation

We used the empirical relationship between the measures of central tendency to estimate the mode.

Given:

• Mean = 12
• Median = 15

Formula: Mode \(\approx\) 3 * Median - 2 * Mean

Calculation:

Mode \(\approx\) 3 * 15 - 2 * 12

Mode \(\approx\) 45 - 24

Mode \(\approx\) 21

Therefore, the mode is approximately 21.

Revision Table: Mean, Median, and Mode Relationship

Additional Information: Measures of Central Tendency

Mean, median, and mode are three common measures of central tendency that represent a typical value in a dataset. Understanding their relationship is important in statistics.

• Mean: The average of all values. It is sensitive to extreme values (outliers).
• Median: The middle value in a dataset that has been ordered from least to greatest. It is not affected by extreme values.
• Mode: The value that appears most frequently in a dataset. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode.

The empirical formula Mode \(\approx\) 3 * Median - 2 * Mean is based on observed data and is a rule of thumb, not a strict mathematical identity that holds for all distributions. It works best for distributions that are moderately skewed. In a perfectly symmetrical distribution, the mean, median, and mode are all equal.

• For a positively skewed distribution (tail on the right), typically Mean > Median > Mode.
• For a negatively skewed distribution (tail on the left), typically Mode > Median > Mean.

The formula used helps estimate the mode's position relative to the mean and median in such skewed distributions.