Understanding Measures of Central Tendency Ratio

This question asks about the relationship between the mean, median, and mode of a data set, specifically when given a ratio involving the mean and median. These three values are known as measures of central tendency and describe the center point of a data set.

In statistics, for a moderately skewed distribution, there is an empirical relationship between the mean, median, and mode. This relationship is often expressed by the formula:

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

Given Information: Ratio of Mean and Median

We are given that the ratio of the mean and median is 4 : 5. This can be written as:

\(\frac{\text{Mean}}{\text{Median}} = \frac{4}{5}\)

To work with this ratio, we can assume that the Mean and Median are specific values based on this ratio. Let's represent them using a constant \(k\):

• Let Mean = \(4k\)
• Let Median = \(5k\)

Here, \(k\) is a positive constant.

Calculating the Mode using the Empirical Formula

Now, we can substitute these expressions for Mean and Median into the empirical formula relating the three measures of central tendency:

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

Substitute \(4k\) for Mean and \(5k\) for Median:

\(\text{Mode} \approx 3 \times (5k) - 2 \times (4k)\)

\(\text{Mode} \approx 15k - 8k\)

\(\text{Mode} \approx (15 - 8)k\)

\(\text{Mode} \approx 7k\)

So, based on the given ratio of Mean and Median, the Mode is approximately \(7k\).

Finding the Ratio of Mean and Mode

The question asks for the ratio of the mean and mode. We have the expressions for Mean and Mode in terms of \(k\):

• Mean = \(4k\)
• Mode \(\approx\) \(7k\)

Now, let's find the ratio of Mean to Mode:

\(\frac{\text{Mean}}{\text{Mode}} = \frac{4k}{7k}\)

Since \(k\) is a positive constant, we can cancel \(k\) from the numerator and denominator:

\(\frac{\text{Mean}}{\text{Mode}} = \frac{4}{7}\)

Thus, the ratio of the mean and mode is 4 : 7.

Summary of the Solution Steps

1. Identify the given ratio of Mean to Median as 4:5.
2. Use the empirical formula relating Mean, Median, and Mode: Mode \(\approx\) 3 Median - 2 Mean.
3. Represent Mean and Median as \(4k\) and \(5k\) respectively.
4. Substitute these values into the empirical formula to find Mode in terms of \(k\).
5. Calculate the ratio of Mean to Mode using the expressions in terms of \(k\).
6. Simplify the ratio to get the final answer.

Final Answer Derivation

Given \(\frac{\text{Mean}}{\text{Median}} = \frac{4}{5}\). Let Mean = \(4k\), Median = \(5k\).

Using the empirical relationship: Mode \(\approx\) 3 Median - 2 Mean

Mode \(\approx\) \(3(5k) - 2(4k)\)

Mode \(\approx\) \(15k - 8k\)

Mode \(\approx\) \(7k\)

Ratio of Mean to Mode = \(\frac{\text{Mean}}{\text{Mode}} = \frac{4k}{7k} = \frac{4}{7}\)

The ratio of its mean and mode is 4 : 7.

Revision Table: Key Statistical Concepts

Understanding the definitions and relationships between mean, median, and mode is crucial for solving such problems.

Additional Information: When the Empirical Formula Applies

The empirical formula connecting the mean, median, and mode (Mode \(\approx\) 3 Median - 2 Mean) is an approximation. It holds true for distributions that are moderately skewed, meaning they are not perfectly symmetrical but also not extremely lopsided. For perfectly symmetrical distributions (like the normal distribution), the mean, median, and mode are all equal. For heavily skewed distributions, this empirical formula might not provide an accurate approximation of the mode. It's a useful rule of thumb for many real-world data sets found in areas like economics or social sciences.