Question

which of the following is true?
i. the mode number of barrels sold is equal to the median number of barrels sold.
ii. the median number of barrels sold is equal to the mean number of barrels sold.
iii. the mode number of barrels sold is equal to the range of the number of barrels sold.

Explanation:

Step1: Find the number of students

From the graph, the number of students for each number of barrels:

• For \(0\) barrels: \(6\) students
• For \(1\) barrel: \(2\) students
• For \(2\) barrels: \(3\) students
• For \(3\) barrels: \(3\) students
• For \(4\) barrels: \(7\) students
• For \(5\) barrels: \(9\) students

Total number of students \(n=6 + 2+3+3+7+9=30\)

Step2: Calculate the mode

The mode is the value that appears most frequently. Here, the number of barrels with the highest frequency (number of students) is \(5\) (since \(9\) students sold \(5\) barrels).

Step3: Calculate the median

Since \(n = 30\) (even), the median is the average of the \(\frac{n}{2}=15^{th}\) and \((\frac{n}{2}+ 1)=16^{th}\) ordered - values.
Cumulative frequencies:

• Up to \(0\) barrels: \(6\)
• Up to \(1\) barrel: \(6 + 2=8\)
• Up to \(2\) barrels: \(8+3 = 11\)
• Up to \(3\) barrels: \(11 + 3=14\)
• Up to \(4\) barrels: \(14+7 = 21\)

The \(15^{th}\) and \(16^{th}\) values are in the \(4\) - barrel group. So the median is \(4\)

Step4: Calculate the range

The range is the difference between the maximum and minimum values. The maximum number of barrels is \(5\) and the minimum is \(0\). So the range \(R=5 - 0=5\)

Answer:

III. The mode number of barrels sold is equal to the range of the number of barrels sold.