The mean and standard deviation of a normal distribution are Rs.60 and Rs.5 respectively. Find the Inter Quartile Range and Mean Deviation of the distribution

To solve this problem, we'll first find the Inter Quartile Range (IQR) and then calculate the Mean Deviation (MD) for the given normal distribution.

### Given:
- Mean (\(\mu\)) = Rs. 60
- Standard Deviation (\(\sigma\)) = Rs. 5

### 1. **Inter Quartile Range (IQR):**

The Inter Quartile Range is the difference between the 75th percentile (\(Q_3\)) and the 25th percentile (\(Q_1\)) of the distribution.

For a normal distribution:
- \(Q_1\) is located at the 25th percentile.
- \(Q_3\) is located at the 75th percentile.

Using the Z-score table for the normal distribution:
- \(Q_1\) corresponds to a Z-score of approximately \(-0.675\)
- \(Q_3\) corresponds to a Z-score of approximately \(+0.675\)

Using the formula for converting a Z-score to a value in the distribution:
\[
Q_1 = \mu + (Z_1 \times \sigma) = 60 + (-0.675 \times 5) = 60 - 3.375 = 56.625
\]
\[
Q_3 = \mu + (Z_3 \times \sigma) = 60 + (0.675 \times 5) = 60 + 3.375 = 63.375
\]

Thus, the Inter Quartile Range (IQR) is:
\[
IQR = Q_3 - Q_1 = 63.375 - 56.625 = 6.75
\]

### 2. **Mean Deviation (MD):**

The Mean Deviation (MD) of a normal distribution is given by:
\[
MD = \frac{2 \times \sigma}{\sqrt{\pi}} = \frac{2 \times 5}{\sqrt{\pi}} \approx \frac{10}{1.772} \approx 5.64
\]

### **Summary of Results:**
- **Inter Quartile Range (IQR):** Rs. 6.75
- **Mean Deviation (MD):** Rs. 5.64

Would you like more details or have any questions about these calculations? 

Here are some related questions to consider:
1. What does the Inter Quartile Range tell us about a distribution?
2. How is the Mean Deviation different from Standard Deviation?
3. What is the significance of the Z-score in a normal distribution?
4. How would the IQR change if the standard deviation increased?
5. How does Mean Deviation compare to Median Absolute Deviation (MAD)?

**Tip:** When dealing with normal distributions, always remember that the area under the curve represents probabilities, and key percentiles correspond to specific Z-scores.

Learn how to calculate the Inter Quartile Range (IQR) and Mean Deviation (MD) of a normal distribution with mean Rs. 60 and standard deviation Rs. 5. Understand the significance of these measures in statistics and their calculation methods. Suitable for advanced high school students and above.

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