The puzzle club recorded how many days 𝐷 it took members to complete a jigsaw. The histogram shows information about the data they collected. There were 25 members with 20 is less than 𝐷 which is less than or equal to 40. There were 18 members with 55 is less than 𝐷 which is less than or equal to 70. There were no members with 𝐷 is greater than 90.
Part 𝑎) complete the histogram.
The formula we need to remember when it comes to histogram questions is this: frequency density is equal to frequency divided by class width. The biggest single mistake the students often make is to think that the height, the values on the 𝑦-axis, are the frequency, whereas in fact we need to perform some calculations to find the frequency density. There’s a rather large gap on our histogram. It corresponds to the 18 members who took more than 55 days but less than or equal to 70 days to complete the jigsaw. Notice how there were no members who took more than 90 days. That means this final gap doesn’t need to be filled with anything.
Before we can do anything with these 18 members, we need to find the scale on the frequency density axis. We’re going to use the fact that 25 members took between 20 and 40 days. We have the bar for this one shaded. The frequency for this group was 25, and the class with is the difference between 20 and 40; that’s 20. So the frequency density for this group is 25 over 20. At this point, we can type this value into our calculator. However, these questions often come up on noncalculator papers, so it’s useful to know what we could do next. We’re going to multiply this fraction to make the denominator 100. And to do that, we need to multiply the denominator by five. Remember whatever we do to the dominator we also do to the numerator.
So we’re also going to multiply the numerator by five. That tells us that 25 over 20 is equivalent to 125 over 100. Remember, that fraction line simply means divide. And when we divide by 100, we move the digits to the right two places. So 25 over 20 is equivalent to 1.25, and the height of our bar is 1.25. Let’s use this information to work out the scale of the frequency density axis. We can see that five large squares have a height of 1.25. We can find the value of one large square by dividing through by five. And that tells us that one large square is equal to 0.25. Let’s use this information to complete the scale on the frequency density axis. It’s also sensible to work out the value of one of the small squares on the axis.
Five small squares is the same as one large square, so five small squares is equal to 0.25. We can find the value of one small square by dividing through by five, and that tells us that one small square is equal to 0.05. We’ll need to use this information in a moment. To be able to complete the histogram then, we need to work out the frequency density for the 18 members who took between 55 and 70 days. The frequency of this group is 18, and the class width is the difference between 55 and 70; it’s 15. This time we’re going to simplify this fraction by dividing through by three. That tells us that 18 over 15 is equivalent to six-fifths or one and one-fifth. We know that one-fifth is equal to 0.2. So the frequency density for the group who took between 55 and 70 days is 1.2.
1.2 is 0.05 less than 1.25. We said that 0.05 was represented by one little square. So we need to draw the top of our bar here, and we’re done. We’ve completed the histogram showing the bar representing the members who took between 55 and 70 days
Part b) Work out whether there are more members with 𝐷 is less than or equal to 45 or more members with 𝐷 is greater than 45. You must show your working.
Let’s add in the cut-off point; that was 45 days. We need to work out the total number of members who took less than or equal to 45 days to complete the jigsaw and the number of members who took more than 45 days. We already know that 25 members took between 20 and 40 days, and 18 members took between 55 and 70 days. So how would we find the frequency from the other bars? Think back to our formula. Frequency density is equal to frequency divided by class width. We can multiply both sides of this equation by class width. And if we do, we see that frequency is equal to frequency density multiplied by class width. Another way of saying that is that the frequency is found by multiplying the height by the width. These bars are rectangles, and that’s actually the area of a rectangle.
So the frequency from a histogram is found by finding the area of each bar. Let’s find the area of the bar representing the number of people who took between zero and 20 days. The width of this bar is 20, and the height of it is three small squares above 0.25. That takes us to 0.4. 20 multiplied by 0.4 is eight. So eight people took between zero and 20 days to complete the jigsaw. The width of this third bar is five, and its height is three. Five multiplied by three is 15, so 15 people took between 40 and 45 days to complete that jigsaw. The bar representing the number of people who took between 45 and 55 days is 10 wide, and it’s got a height of 1.9; it’s two little squares below two. 10 multiplied by 1.9 is 19, so 19 people took between 45 and 55 days. The width of the very final bar is 20, and its height is 0.3. 20 multiplied by 0.3 is six, so six people took between 70 and 90 days to complete the jigsaw.
We can find a total number of people that took less than or equal to 45 days by adding eight and 25 and 15. That’s 48. The number of people who took more than 45 days is found by adding 19, 18, and six. That’s 43. So 43 people took more than 45 days. 48 is greater than 43, so there are more members who took the less than or equal to 45 days to complete the jigsaw.