If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is app

Q: Why is this formula called 'empirical'?

The term 'empirical' means it is based on observation and experience rather than a strict mathematical derivation. It is an approximation that holds true for moderately skewed, unimodal distributions. It is not a universal identity, but it is a powerful tool in statistics for estimating the mode when you only have the mean and median.

Q: Can I use this formula for any frequency distribution?

No, you cannot. This formula is specifically designed for moderately skewed distributions. If a distribution is highly skewed or has multiple peaks (multimodal), this empirical relationship will not provide an accurate result. Always check if the distribution is unimodal and moderately skewed before applying it.

Q: Why is the order $\text{Mean} < \text{Median} < \text{Mode}$ in this case?

In a negatively skewed (left-skewed) distribution, the long tail on the left pulls the mean towards it, making the mean the smallest value. The median is less affected by extreme values, so it sits between the mean and the mode. The mode, being the peak of the distribution, remains at the highest point, making it the largest of the three in this specific scenario.

Question

If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

Options:

(A) 22.0

(B) 20.5

(C) 25.5

(D) 24.0

Sigma Percentile · Accepted Answer

### The Heartbeat of Data: Understanding Central Tendency

Welcome, aspiring physicist! Today, we are not just solving a statistics problem; we are uncovering the hidden geometry of data.

Statistics is the language of uncertainty, and understanding how the mean, median, and mode dance together is essential for any JEE aspirant. Imagine you are looking at a frequency distribution curve; it is not just a collection of numbers, but a physical shape.

### The Anatomy of Skewness

We are given a mean of $21$ and a median of $22$. Right away, your intuition should trigger.

In a perfectly symmetric distribution, the mean, median, and mode would all coincide at the same point. But here, $	ext{Mean} < 	ext{Median}$.

This is a crucial observation! It tells us that our distribution is negatively skewed, or left-skewed. The data has a long tail stretching towards the smaller values, which pulls the mean down, away from the peak.

### The Empirical Bridge

To find the mode, we turn to a classic tool in our statistical arsenal: Karl Pearson's empirical formula. This formula is a beautiful, albeit approximate, bridge between the three measures of central tendency.

It states that for moderately skewed distributions, the relationship is defined as:

$$	ext{Mode} = 3 	imes 	ext{Median} - 2 	imes 	ext{Mean}$$

This formula is a favorite of examiners because it tests your ability to recognize the relationship between these values without needing the raw data set.

### The Execution

Now, let us perform the calculation with precision. We have $	ext{Median} = 22$ and $	ext{Mean} = 21$.

Substituting these into our formula, we get:

$$	ext{Mode} = 3(22) - 2(21)$$

Let us break this down. First, $3 	imes 22 = 66$. Next, $2 	imes 21 = 42$.

Finally, we perform the subtraction: $66 - 42 = 24$.

### The Final Picture

Our result, **$	ext{Mode} = 24$**, is the $x$-coordinate of the peak of our distribution.

Look at the final order: $	ext{Mean} (21) < 	ext{Median} (22) < 	ext{Mode} (24)$. This confirms our earlier deduction about the negative skewness.

You have successfully navigated the relationship between these three pillars of statistics. Keep this geometric intuition in mind, and you will find that even the most complex statistical problems become clear and manageable.

Animated Solution for Mathematics - Statistics: If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

Select Answer:

Visualized Solution

Given Data

Distribution Skewness

Empirical Relationship

Substituting Values

Calculating $3 \times Median$

Calculating $2 \times Mean$

Final Subtraction

Final Answer

The Sigma Insight: Measures of Central Tendency (Mean, Median, Mode)

The Heartbeat of Data

The Anatomy of Skewness

The Empirical Bridge

The Execution

The Final Picture

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