Let's calculate the interquartile range (IQR) for the given data. The IQR is found using the formula:

$IQR = Q_3 - Q_1$

Step 1: Arrange the Data in Ascending Order

330, 334, 342, 364, 377, 383, 390, 393, 398, 409,
411, 442, 484, 486, 491, 496, 526, 539, 543, 549

Step 2: Find $Q_1$ and $Q_3$

• $Q_1$ (First Quartile): The median of the lower half (first 10 values)

  • Lower half: 330, 334, 342, 364, 377, 383, 390, 393, 398, 409
  • Median of lower half:
    $Q_1 = \frac{377 + 383}{2} = 380$

• $Q_3$ (Third Quartile): The median of the upper half (last 10 values)

  • Upper half: 411, 442, 484, 486, 491, 496, 526, 539, 543, 549
  • Median of upper half:
    $Q_3 = \frac{491 + 496}{2} = 493.5$

Step 3: Compute the IQR

$IQR = Q_3 - Q_1 = 493.5 - 380 = 113.5$

Final Answer:

$IQR = 113.5$

---

Would you like a breakdown of any specific step?

Related Questions:

1. How is the standard deviation calculated manually?
2. What does the interquartile range tell us about data distribution?
3. How does the presence of outliers affect the IQR and standard deviation?
4. What is the difference between IQR and range?
5. How can boxplots help visualize the IQR?

Tip:

A higher IQR indicates greater data spread within the middle 50%, making it useful for detecting variability in distributions!