IB MYP 4-5 Maths- Measures of dispersion- Study Notes - New Syllabus
IB MYP 4-5 Maths- Measures of dispersion – Study Notes
Standard
- Measures of dispersion
IB MYP 4-5 Maths- Measures of dispersion – Study Notes – All topics
Interquartile Range (IQR)
Interquartile Range (IQR)
The Interquartile Range (IQR) measures the spread of the middle $50\%$ of data.
It is calculated as:
\( \text{IQR} = Q_3 – Q_1 \)
Why use IQR?
- Not affected by outliers, unlike the full range.
- Shows how tightly data is clustered around the median.
Steps to Calculate IQR:
- Arrange the data in ascending order.
- Find \( Q_1 \) (lower quartile) and \( Q_3 \) (upper quartile).
- Compute \( IQR = Q_3 – Q_1 \).
Example:
Find the IQR for the data: 8, 10, 12, 14, 16, 18, 20, 22.
▶️Answer/Explanation
Step 1: Data is sorted (\( n = 8 \)).
Step 2:
\( Q_1 \) position = \( \dfrac{8+1}{4} = 2.25 \) → between 2nd (10) and 3rd (12): \( Q_1 = 10 + 0.25(12-10) = 10.5 \)
\( Q_3 \) position = \( \dfrac{3(8+1)}{4} = 6.75 \) → between 6th (18) and 7th (20): \( Q_3 = 18 + 0.75(20-18) = 19.5 \)
Step 3: IQR = \( 19.5 – 10.5 = 9 \).
Answer: IQR = 9.
Example:
The marks of 9 students are: 15, 18, 21, 24, 26, 30, 32, 35, 40. Calculate the IQR.
▶️Answer/Explanation
Step 1: Sorted data: (already sorted), \( n = 9 \).
Step 2:
\( Q_1 \) position = \( \dfrac{9+1}{4} = 2.5 \) → between 2nd (18) and 3rd (21): \( Q_1 = 18 + 0.5(21-18) = 19.5 \)
\( Q_3 \) position = \( \dfrac{3(9+1)}{4} = 7.5 \) → between 7th (32) and 8th (35): \( Q_3 = 32 + 0.5(35-32) = 33.5 \)
Step 3: IQR = \( 33.5 – 19.5 = 14 \).
Answer: IQR = 14.
Example:
The times (in minutes) taken by 10 students to complete an online quiz: 12, 14, 15, 18, 20, 21, 23, 25, 27, 30. Calculate the IQR.
▶️Answer/Explanation
Step 1: Data is sorted (\( n = 10 \)).
Step 2:
\( Q_1 \) position = \( \dfrac{10+1}{4} = 2.75 \) → between 2nd (14) and 3rd (15): \( Q_1 = 14 + 0.75(15-14) = 14.75 \)
\( Q_3 \) position = \( \dfrac{3(10+1)}{4} = 8.25 \) → between 8th (25) and 9th (27): \( Q_3 = 25 + 0.25(27-25) = 25.5 \)
Step 3: IQR = \( 25.5 – 14.75 = 10.75 \).
Answer: IQR = 10.75 minutes.
Example:
The daily temperatures (°C) in a city for 12 days: 18, 20, 22, 23, 24, 25, 27, 29, 30, 32, 33, 35. Calculate the IQR and interpret.
▶️Answer/Explanation
Step 1: Sorted data (\( n = 12 \)).
Step 2:
\( Q_1 \) position = \( \dfrac{12+1}{4} = 3.25 \) → between 3rd (22) and 4th (23): \( Q_1 = 22 + 0.25(23-22) = 22.25 \)
\( Q_3 \) position = \( \dfrac{3(12+1)}{4} = 9.75 \) → between 9th (30) and 10th (32): \( Q_3 = 30 + 0.75(32-30) = 31.5 \)
Step 3: IQR = \( 31.5 – 22.25 = 9.25 \).
Interpretation: The middle 50% of temperatures varied by 9.25°C, showing moderate spread.