Question

1. below is an ogive of the commute times of a random sample of teachers.

teacher commute times
cumulative relative frequency
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
0
11
22
33
44
55
66
77
commute time in minutes
a) explain how you can find the median commute time of teachers in this sample. what is the median commute time?
b) explain why you cannot use the graph to find the mean of the data set.

Explanation:

Step1: Recall median - ogive relation

The median of a data - set from an ogive is found by locating the 50% cumulative relative frequency on the y - axis and then finding the corresponding x - value (commute time in this case).

Step2: Answer part a

To find the median commute time of teachers in the sample, locate 50% on the cumulative relative frequency (y - axis) of the ogive. Then draw a horizontal line from this point until it intersects the ogive curve. From the intersection point, draw a vertical line down to the x - axis. The value on the x - axis is the median commute time.

Step3: Recall mean - ogive limitation

The mean of a data - set is calculated as $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}f_{i}}{\sum_{i = 1}^{n}f_{i}}$, where $x_{i}$ are the data values and $f_{i}$ are their frequencies. An ogive only gives cumulative relative frequencies and does not provide the individual data values or their frequencies directly.

Step4: Answer part b

The ogive does not provide the individual commute times and their corresponding frequencies. Since the mean calculation requires the sum of the product of each data value and its frequency, we cannot use the ogive to find the mean of the data - set.

Answer:

a) Locate 50% on the y - axis (cumulative relative frequency), draw a horizontal line to the ogive, then a vertical line to the x - axis. The x - value is the median commute time.
b) The ogive does not give individual data values and their frequencies. Since the mean calculation needs these, we cannot use the graph to find the mean.