a sample of twelve commuter student from a college, the following data gives the typical one way commuting times (in minutes) from home to college for these 12 students.

Question

29,14,39,17,7,47,63,37,42,18,24,55.
1.find the value of the three quartiles,(Q1,Q2,Q3,)
2.WHERE DOES THE COMMUTING TIME OF 47 FALL IN RELATION TO THE THREE QUARTILE?
2.find the inter quartile range

Answer 1

To find the three quartiles (Q1, Q2, Q3) and the interquartile range (IQR) for the given commuting times, we can follow these steps:

Step 1: Sort the Data

First, let's sort the data in ascending order: \[ 7, 14, 17, 18, 24, 29, 37, 39, 42, 47, 55, 63 \]

Step 2: Find the Quartiles

1. Q1 (First Quartile): This is the median of the first half of the data.

   • First half: \[ 7, 14, 17, 18, 24, 29 \]
   • Median of the first half (Q1): The median of these 6 numbers is the average of the 3rd and 4th values.
   • \( Q1 = \frac{17 + 18}{2} = 17.5 \)

2. Q2 (Second Quartile): This is the median of the entire dataset, which divides the data into two halves.

   • The complete sorted list has 12 numbers.
   • Median of the full data (Q2): The median is found by taking the average of the 6th and 7th numbers.
   • \( Q2 = \frac{29 + 37}{2} = 33 \)

3. Q3 (Third Quartile): This is the median of the second half of the data.

   • Second half: \[ 39, 42, 47, 55, 63 \]
   • Median of the second half (Q3): The median of these 6 numbers is the average of the 9th and 10th values.
   • \( Q3 = \frac{42 + 47}{2} = 44.5 \)

Summary of Quartiles

• Q1 = 17.5
• Q2 = 33
• Q3 = 44.5

Step 3: Position of 47 in Relation to Quartiles

The value 47 is greater than \( Q3 (44.5) \). Therefore, it falls in the upper quartile, which means it is in the upper 25% of the data.

Step 4: Calculate the Interquartile Range (IQR)

The interquartile range (IQR) is calculated as: \[ \text{IQR} = Q3 - Q1 \] \[ \text{IQR} = 44.5 - 17.5 = 27 \]

Summary of Results

1. Quartiles:
   • Q1 = 17.5
   • Q2 = 33
   • Q3 = 44.5
2. Position of 47: Above Q3 (in the upper quartile).
3. Interquartile Range: IQR = 27