From the Empirical Rule, we can deduce the following (select ALL that apply): (Note: IQR stands for the Interquartile Range and SD stands for Standard Deviation) – The IQR is always larger than the SD. – The IQR is usually larger than the SD for bell shaped distributions. – The SD is always larger than the IQR. – The SD is usually larger than the IQR for bell-shaped distributions. – The SD is equal to the IQR for bell-shaped distributions. – The SD is always equal to the IQR. 1 pts
The Correct Answer and Explanation is:
Correct answer:
✅ The SD is usually larger than the IQR for bell-shaped distributions.
Explanation:
The Empirical Rule applies specifically to bell-shaped, or normal distributions. It states:
- About 68% of data falls within 1 standard deviation (SD) of the mean
- About 95% falls within 2 SDs
- About 99.7% falls within 3 SDs
The Interquartile Range (IQR) is a measure of statistical spread based on the middle 50% of the data, calculated as Q3 – Q1. Unlike standard deviation, which considers all values and how they spread around the mean, IQR focuses on the central portion of the data and is resistant to outliers.
For a perfectly normal distribution, the IQR and SD have a predictable relationship. In such distributions:
- The IQR spans from the 25th percentile (Q1) to the 75th percentile (Q3), covering the middle 50%
- This middle 50% falls approximately between -0.674 SD and +0.674 SD from the mean
So, the IQR in a normal distribution is roughly equal to 1.34896 times the SD. But since this only covers the central part of the data and SD captures more of the distribution, the SD is typically larger in effect when comparing spread.
This makes the correct answer:
✔ The SD is usually larger than the IQR for bell-shaped distributions
Now let’s eliminate the other options:
- ❌ The IQR is always larger than the SD: False in many common distributions
- ❌ The IQR is usually larger than the SD for bell-shaped distributions: Also false
- ❌ The SD is always larger than the IQR: Not always true across all types of distributions
- ❌ The SD is equal to the IQR for bell-shaped distributions: Not exactly true
- ❌ The SD is always equal to the IQR: False; this never holds across all distributions
Thus, the only true and reliable statement is:
“The SD is usually larger than the IQR for bell-shaped distributions.”
