Clear infographic displaying the formulas for calculating Mean, Median, and Mode in statistics.

The image you've uploaded presents the formulas for Mean, Median, and Mode in statistics. Let's go through each one, explain what they mean, and how to use them.

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🔹 1. Mean

Formula:
$$
\text{Mean} = \frac{\text{Sum of Observations}}{\text{Total Number of Observations}}
$$

Explanation:
- The mean is the average of a set of numbers.
- You add up all the values (sum of observations) and divide by how many values there are (total number of observations).
- Example:
If data is: 2, 4, 6, 8
Sum = 20, Number of observations = 4
Mean = $ \frac{20}{4} = 5 $

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🔹 2. Median

The median is the middle value when the data is arranged in ascending order.

#### ✔ Case 1: When 'n' is odd
$$
\text{Median} = \left( \frac{n+1}{2} \right)^{th} \text{term}
$$

- Here, $ n $ = total number of observations.
- You find the position of the middle term using this formula.
- Example:
Data: 3, 5, 7, 9, 11 → $ n = 5 $ (odd)
Position = $ \frac{5+1}{2} = 3^{rd} $ term → Median = 7

#### ✔ Case 2: When 'n' is even
$$
\text{Median} = \frac{ \left( \frac{n}{2} \right)^{th} \text{term} + \left( \frac{n}{2} + 1 \right)^{th} \text{term} }{2}
$$

- You take the average of the two middle terms.
- Example:
Data: 2, 4, 6, 8 → $ n = 4 $ (even)
$ \frac{4}{2} = 2^{nd} $ term = 4, $ \frac{4}{2} + 1 = 3^{rd} $ term = 6
Median = $ \frac{4 + 6}{2} = 5 $

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🔹 3. Mode

Formula (for grouped data):
$$
\text{Mode} = L + h \cdot \frac{(f_m - f_1)}{(f_m - f_1) + (f_m - f_2)}
$$

Where:
- $ L $ = Lower limit of the modal class (the class with highest frequency)
- $ h $ = Size of the class interval
- $ f_m $ = Frequency of the modal class
- $ f_1 $ = Frequency of the class before the modal class
- $ f_2 $ = Frequency of the class after the modal class

Explanation:
- The mode is the most frequently occurring value.
- This formula is used for grouped data (data in intervals), not raw data.
- It estimates the mode within the class that has the highest frequency.

Example (illustrative):
Suppose we have:

| Class | Frequency |
|-------|-----------|
| 10–20 | 5 |
| 20–30 | 10 ← Modal class
| 30–40 | 7 |

Then:
- $ L = 20 $
- $ h = 10 $
- $ f_m = 10 $
- $ f_1 = 5 $
- $ f_2 = 7 $

$$
\text{Mode} = 20 + 10 \cdot \frac{(10 - 5)}{(10 - 5) + (10 - 7)} = 20 + 10 \cdot \frac{5}{5 + 3} = 20 + 10 \cdot \frac{5}{8} = 20 + 6.25 = 26.25
$$

So, the estimated mode is 26.25.

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📌 Summary Table:

| Measure | Formula | Used For |
|--------|--------|---------|
| Mean | $ \frac{\text{Sum of Observations}}{n} $ | Average value |
| Median | $ \left( \frac{n+1}{2} \right)^{th} $ if odd; average of $ \frac{n}{2} $ and $ \frac{n}{2}+1 $ if even | Middle value |
| Mode | $ L + h \cdot \frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)} $ | Most frequent value (grouped data) |

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✔ These formulas are essential for descriptive statistics and help summarize data sets effectively.

Let me know if you'd like an example problem using these formulas!