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## Integrated Reasoning - Cumulative Graphs

Coming back to Integrated Reasoning question types, let’s discuss a cumulative graph today. They are usually a little trickier than your usual line/pie/bar graphs since you have to focus on not the data points but ‘the change’ from one data point to another.  Every subsequent data point will be either above or at the same level as the previous data point.

Let’s try to understand what a cumulative graph is before we look at questions on one.

Here, the X axis gives the hourly wage and the y axis gives us the cumulative number of people. This means that the point giving the number of people with hourly wage of \$30 actually gives the number of people whose hourly wage is UP TO \$30. The point giving the number of people with hourly wage of \$40 gives the number of people whose hourly wage is up to \$40 and hence includes the number of people whose hourly wage is up to \$30. That is the reason the graph will always be flat or will have a positive slope.

If we want to focus on the people whose hourly wage varies from \$30 to \$40, we need to look at the slope of the graph in between these two points. The difference between these two points gives us the number of people with hourly wage in the range of \$30 – \$40. This implies that if the slope is steep, many people lie in this range.

Set 1: 200 people were surveyed to find out their hourly wage. 100 people had college degrees while other 100 were those who had not completed high school. The following graph gives the cumulative number of people and their hourly wages.

Question 1: True/False: As per the given graph, the average hourly wages of people who did not complete high school is higher than the average hourly wages of people who have college degrees.

Question 2: The median wage of the people with college degrees lies between ___________ while the median wage of people who did not complete high school lies between ___________ (select two options, one for each blank)

(A)   \$10–\$20 per hr

(B)   \$20–\$30 per hr

(C)   \$30–\$40 per hr

(D)   \$40–\$50 per hr

(E)    \$50–\$60 per hr

Question 3: Approximately what percentage of ‘people with college degrees’ have hourly wage in the range \$50 – \$60?

(A) 25%

(B) 35%

(C) 45%

Solutions:

Solution 1: False

Just because the graph of ‘people who did not complete high school’ lies above the graph of ‘people who have college degrees’, it doesn’t mean that average hourly wage of people who did not complete high school is higher. Since the y axis gives the cumulative number of people, having a higher graph early on implies that many people have lower salaries. For example, about 45 people who did not complete high school have hourly wage up to \$10. On the other hand, only 18 people with college degrees have hourly wage up to \$10. Looking at the data, we can say that the average hourly wage of people with college degree will be much higher than the average hourly wage of people who did not complete high school.

Solution 2: The median wage will be the average of the wage of the 50th person and 51st person. The thick black line shows the range in which these two lie.

Let’s first look at people who have college degrees. The 50th and 51st people will have wages lying in the range \$50 – \$60. Answer (E)

What about people who did not complete high school? The 50th person and the 51st person lie in the range \$10 – \$20. Answer (A)

Solution 3: Number of people with hourly wage up to \$50 is 45. Number of people with hourly wage up to \$60 is 80. Hence number of people whose hourly wage lies in the range \$50 – \$60 is about 80 – 45 = 35. Since 100 people were surveyed, the required percentage is about 35%. Answer (B)

Ensure you understand these graphs well.