Video Transcript
Liam took a sample of 100 balls from box A. He weighed each ball and recorded its weight. He used the information to draw the cumulative frequency graph shown on the grid. Use the cumulative frequency graph to find an estimate for the median weight of the balls. Option (A) 50 grams, option (B) 80 grams, option (C) 90 grams, option (D) 104 grams, or option (E) 120 grams.
In this problem, we are told that 100 balls have been selected and their weights have been recorded. And rather than seeing this information in a table, we can see that it has been presented as a cumulative frequency graph. And we need to determine an estimate for the median.
We can recall that the median of a data set is the middle value when the data is in ascending or descending order. Now, we don’t have all the individual weights of the balls. But we do have this cumulative frequency graph, and it represents the data in order. That’s because we know that the cumulative frequency, or ascending cumulative frequency, of a value 𝑎 indicates the frequency of values that are less than 𝑎. For example, we can see that the coordinates 20, four lie on the cumulative frequency curve. That’s a weight of 20 grams and a cumulative frequency of four. This tells us that four balls have a weight less than 20 grams. So how do we use the graph to find the median, which is the middle value?
Well, the first thing we need to do is find the median position. This is equal to the total frequency over two. We were given that 100 balls were recorded. And if we weren’t told this, we could also determine this from the highest point on the cumulative frequency graph. And as 100 is the total frequency, then the median position is half of this, which is 50. In context, this means that if we lined up all 100 balls in order of weights, the median would be at the halfway point at the 50th ball.
It’s worth noting at this point that usually when finding the median, we add one to the total frequency before dividing by two. But when using a graph to find the estimate, we don’t need to add one. We can just halve the total frequency. We then draw a line from the median position on the 𝑦-axis until it meets the curve. And we must be careful not to incorrectly draw this from the 𝑥-axis at 50 grams.
Drawing a vertical line from this point down to the 𝑥-axis will give us the estimate for the median, which we can read as the value 104. And so, we have found the answer that the estimate for the median is 104 grams, which is the answer given in option (D).
