Formula showing the relationship between mode, median, and mean: Mode = 3 Median - 2 Mean

The image contains the equation:

\[
\text{Mode} = 3 \cdot \text{Median} - 2 \cdot \text{Mean}
\]

This equation is a well-known empirical relationship in statistics, often referred to as Pearson's mode skewness formula. It provides an approximate relationship between the three measures of central tendency: the mode, median, and mean. Let's break down the solution and explanation step by step.

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1. Understanding the Components

- Mode: The value that appears most frequently in a dataset.
- Median: The middle value when the data is arranged in ascending or descending order. If there is an even number of data points, the median is the average of the two middle values.
- Mean: The arithmetic average of the dataset, calculated as the sum of all values divided by the number of values.

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2. Empirical Relationship

The given equation:

\[
\text{Mode} = 3 \cdot \text{Median} - 2 \cdot \text{Mean}
\]

is an empirical formula used to estimate the mode based on the median and mean. This relationship is particularly useful when the mode is not directly observable (e.g., in grouped data) or when you want to infer the mode from other summary statistics.

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3. Derivation and Context

This formula is derived from the concept of skewness in a distribution:
- In a perfectly symmetrical distribution (like the normal distribution), the mode, median, and mean are equal.
- In skewed distributions:
- For a positively skewed distribution (longer tail to the right), the mean is typically greater than the median, which is greater than the mode.
- For a negatively skewed distribution (longer tail to the left), the mean is typically less than the median, which is less than the mode.

Pearson observed that for many real-world datasets, the relationship between these three measures can be approximated by the formula above.

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4. Solving the Problem

The problem does not provide specific numerical values for the median and mean. However, if such values were given, you could directly substitute them into the formula to find the mode. For example:

#### Example:
Suppose:
- Median = 50
- Mean = 40

Using the formula:

\[
\text{Mode} = 3 \cdot \text{Median} - 2 \cdot \text{Mean}
\]

Substitute the given values:

\[
\text{Mode} = 3 \cdot 50 - 2 \cdot 40
\]

Calculate step by step:

\[
\text{Mode} = 150 - 80 = 70
\]

Thus, the mode would be 70.

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5. Final Answer

If no specific values are provided, the solution remains the formula itself. However, if values were given, you would follow the steps above to compute the mode.

\[
\boxed{\text{Mode} = 3 \cdot \text{Median} - 2 \cdot \text{Mean}}
\]