Answer :
Certainly! Let's walk through the steps to find the median and interquartile range (IQR) of the delivery distances using the cumulative frequency table provided.
### Step-by-Step Solution:
a) Drawing the cumulative graph:
1. Plot the Points:
- Plot cumulative frequency of 10 at the upper bound of the first interval, 5 km.
- Plot cumulative frequency of 30 at the upper bound of the second interval, 10 km.
- Plot cumulative frequency of 50 at the upper bound of the third interval, 15 km.
- Plot cumulative frequency of 80 at the upper bound of the fourth interval, 20 km.
2. Join the Points:
- Once the points are plotted on the graph paper, join the points with straight lines to form the cumulative frequency diagram.
You now have a cumulative frequency graph which you can use to estimate the median and quartiles.
b) Estimating the median and interquartile range from the graph:
1. Finding the Median (50th percentile):
- The total number of deliveries (N) is given by the last cumulative frequency: 80.
- The median is the value at the 50th percentile, which is at [tex]\( N/2 = 80/2 = 40 \)[/tex].
- On the cumulative frequency graph, draw a horizontal line from 40 on the cumulative frequency axis until it intersects the cumulative frequency curve.
- From this intersection point, draw a vertical line down to the distance axis to find the median delivery distance.
2. Finding the First Quartile (25th percentile):
- The first quartile (Q1) is the value at the 25th percentile, which is at [tex]\( N/4 = 80/4 = 20 \)[/tex].
- On the cumulative frequency graph, draw a horizontal line from 20 on the cumulative frequency axis until it intersects the cumulative frequency curve.
- From this intersection point, draw a vertical line down to the distance axis to find the first quartile delivery distance.
3. Finding the Third Quartile (75th percentile):
- The third quartile (Q3) is the value at the 75th percentile, which is at [tex]\( 3N/4 = 3 \times 80/4 = 60 \)[/tex].
- On the cumulative frequency graph, draw a horizontal line from 60 on the cumulative frequency axis until it intersects the cumulative frequency curve.
- From this intersection point, draw a vertical line down to the distance axis to find the third quartile delivery distance.
4. Calculating the Interquartile Range (IQR):
- IQR is calculated as [tex]\( Q3 - Q1 \)[/tex].
Results:
- Median Delivery Distance: The median delivery distance is found to be 7.5 km.
- Interquartile Range (IQR): The IQR is found to be 1.67 km.
So, the estimated median delivery distance from the cumulative frequency table is 7.5 km, and the interquartile range is 1.67 km.
### Step-by-Step Solution:
a) Drawing the cumulative graph:
1. Plot the Points:
- Plot cumulative frequency of 10 at the upper bound of the first interval, 5 km.
- Plot cumulative frequency of 30 at the upper bound of the second interval, 10 km.
- Plot cumulative frequency of 50 at the upper bound of the third interval, 15 km.
- Plot cumulative frequency of 80 at the upper bound of the fourth interval, 20 km.
2. Join the Points:
- Once the points are plotted on the graph paper, join the points with straight lines to form the cumulative frequency diagram.
You now have a cumulative frequency graph which you can use to estimate the median and quartiles.
b) Estimating the median and interquartile range from the graph:
1. Finding the Median (50th percentile):
- The total number of deliveries (N) is given by the last cumulative frequency: 80.
- The median is the value at the 50th percentile, which is at [tex]\( N/2 = 80/2 = 40 \)[/tex].
- On the cumulative frequency graph, draw a horizontal line from 40 on the cumulative frequency axis until it intersects the cumulative frequency curve.
- From this intersection point, draw a vertical line down to the distance axis to find the median delivery distance.
2. Finding the First Quartile (25th percentile):
- The first quartile (Q1) is the value at the 25th percentile, which is at [tex]\( N/4 = 80/4 = 20 \)[/tex].
- On the cumulative frequency graph, draw a horizontal line from 20 on the cumulative frequency axis until it intersects the cumulative frequency curve.
- From this intersection point, draw a vertical line down to the distance axis to find the first quartile delivery distance.
3. Finding the Third Quartile (75th percentile):
- The third quartile (Q3) is the value at the 75th percentile, which is at [tex]\( 3N/4 = 3 \times 80/4 = 60 \)[/tex].
- On the cumulative frequency graph, draw a horizontal line from 60 on the cumulative frequency axis until it intersects the cumulative frequency curve.
- From this intersection point, draw a vertical line down to the distance axis to find the third quartile delivery distance.
4. Calculating the Interquartile Range (IQR):
- IQR is calculated as [tex]\( Q3 - Q1 \)[/tex].
Results:
- Median Delivery Distance: The median delivery distance is found to be 7.5 km.
- Interquartile Range (IQR): The IQR is found to be 1.67 km.
So, the estimated median delivery distance from the cumulative frequency table is 7.5 km, and the interquartile range is 1.67 km.
