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Calculate the Interquartile Range (IQR) for the given frequency distribution of travel times to work.
ScanSolve AI Answer
Step-by-step solution
3 stepsTo find the Interquartile Range (IQR) for the given frequency distribution, we need to calculate the first quartile (Q1) and the third quartile (Q3).
First, let's create a table with cumulative frequencies and class boundaries:
| Time to travel to work | Frequency (f) | Cumulative Frequency (cf) | Class Boundaries |
|---|---|---|---|
| 1 - 10 | 8 | 8 | 0.5 - 10.5 |
| 11 - 20 | 14 | 8 + 14 = 22 | 10.5 - 20.5 |
| 21 - 30 | 12 | 22 + 12 = 34 | 20.5 - 30.5 |
| 31 - 40 | 9 | 34 + 9 = 43 | 30.5 - 40.5 |
| 41 - 50 | 7 | 43 + 7 = 50 | 40.5 - 50.5 |
The total number of data points (N) = 50.
The class width (c) = 10.5 - 0.5 = 10.
The formula for a quartile (Qk) for grouped data is:
Qk = L + [((k × N)/4 - cf_b) / f] × c
Where:
L = lower boundary of the quartile class
k = 1 for Q1, 3 for Q3
N = total frequency
cf_b = cumulative frequency of the class before the quartile class
f = frequency of the quartile class
c = class width
Calculate Q1 (First Quartile)
-
Position of Q1: (1 × N)/4 = (1 × 50)/4 = 12.5th value.
-
Identify Q1 class: The 12.5th value falls in the 11 - 20 class, as its cumulative frequency (22) is the first to exceed 12.5.
- Lower boundary of Q1 class (L₁) = 10.5
- Frequency of Q1 class (f₁) = 14
- Cumulative frequency before Q1 class (cf_b₁) = 8
- Class width (c) = 10
-
Calculate Q1: Q1 = 10.5 + [(12.5 - 8) / 14] × 10 Q1 = 10.5 + [4.5 / 14] × 10 Q1 = 10.5 + (45 / 14) Q1 = 10.5 + 3.21428... Q1 ≈ 13.71
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Calculate Q3 (Third Quartile)
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Position of Q3: (3 × N)/4 = (3 × 50)/4 = 150/4 = 37.5th value.
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Identify Q3 class: The 37.5th value falls in the 31 - 40 class, as its cumulative frequency (43) is the first to exceed 37.5.
- Lower boundary of Q3 class (L₃) = 30.5
- Frequency of Q3 class (f₃) = 9
- Cumulative frequency before Q3 class (cf_b₃) = 34
- Class width (c) = 10
-
Calculate Q3: Q3 = 30.5 + [(37.5 - 34) / 9] × 10 Q3 = 30.5 + [3.5 / 9] × 10 Q3 = 30.5 + (35 / 9) Q3 = 30.5 + 3.8888... Q3 ≈ 34.39
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Calculate the Interquartile Range (IQR)
IQR = Q3 - Q1
IQR = 34.3888... - 13.7142...
IQR = 20.6746...Rounding to two decimal places:
IQR = 20.67
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