Find the inter-quartile range of 1, 3, 4, 5, 8, 9, 10, 11, 12, 14, 16

Analyze the question: We need the inter-quartile range (IQR) = Q3 − Q1 for the data set, which is already arranged in ascending order.

Concept: The quartiles divide ordered data into four equal parts. IQR = Q3 − Q1.

Solution:

Data (n = 11): 1, 3, 4, 5, 8, 9, 10, 11, 12, 14, 16

• Median (Q2) = 6th value = 9
• Lower half (below the median): 1, 3, 4, 5, 8. Q1 is the middle value = 4
• Upper half (above the median): 10, 11, 12, 14, 16. Q3 is the middle value = 12

IQR = Q3 − Q1 = 12 − 4 = 8

Check with the position formula: Q1 is at position (n+1)/4 = 3, the 3rd value = 4. Q3 is at position 3(n+1)/4 = 9, the 9th value = 12. This gives the same result, 12 − 4 = 8.

Tip: For an odd number of data points, drop the median, split the data into two halves, and find the median of each half. Q1 and Q3 are those medians. Always write the data in order first.