The vertical line chart shows the ages of a group of students. Part a) How many students are in the group altogether?
There is also a part b) and c) that we will look at later. The 𝑥-axis shows the ages of all the students. The youngest was 11 and the oldest was 18. The 𝑦-axis represents the number of students from zero to seven. Each square on the 𝑦-axis represents one student. The first vertical line goes up to three. Therefore, there are three 11-year olds. The second bar goes up to two. Therefore, there are two 12-year olds. The highest line goes up to seven. This tells us there are seven 13-year olds. There are five 14-year olds, three 15-year olds, six 16-year olds, no 17-year olds as there is no line at 17, and three 18-year olds. We need to work out how many students there are altogether. This means we need to add the eight numbers: three, two, seven, five three, six, zero, and three. Adding these eight numbers gives us an answer of 29. There are 29 students altogether in the group.
The second part of the question says the following. b) What is the median age of the group of students?
The median is the middle value when the data is listed in numerical order. One way of doing this is to write down the ages of all 29 students in order and find the middle value. We will do this first. However, we will also look at a quicker method when we have a large group as in this case. There were three 11-year olds in the group. Therefore, we write the number 11 three times. There were two 12-year olds. So we write 12 twice. There were seven 13-year olds. So we write 13 seven times. We continue this for the 14-year olds, 15-year olds, 16-year olds, and 18-year olds. There are five 14s, three 15s, six 16s, and three 18s. Note that there are no 17s as there are no 17-year olds in the group.
As we’re looking for the middle value, we begin by crossing off the lowest number and the highest number. We then cross off the next highest and the next lowest. We repeat this process until we reach the middle. We have now crossed off 14 numbers, starting with the lowest number 11. And we’ve also crossed off 14 numbers, starting with the highest number 18. The number that is left is 14. Therefore, the median age of the group of students is 14.
Whilst this method will always work out the median number, it is quite long-winded if we have a large group of numbers. It is also quite easy to make a mistake when crossing out the numbers. As a result, we can use a little formula that makes it easier to find the median. The median position can be calculated using the formula 𝑛 plus one divided by two, where 𝑛 is the number of students in the group in this case. As there were 29 students in the group, we need to calculate 29 plus one divided by two. 29 plus one is equal to 30. Dividing this by two gives us 15. Therefore, the median person in the group is the 15th.
We now need to work out what age the 15th person in the group is when they’re in numerical order. There were three students who were 11 years of age and two that were 12 years of age. Adding these numbers gives us five. Therefore, five students were 12 or younger. This is less than 15. So we also need to add the next number. This is called the cumulative or running total. We need to add three, two, and seven. Three plus two plus seven is equal to 12. Therefore, there are 12 students 13 years of age or younger.
We have still not reached the 15th student. The next number in our line chart was five. We need to add five to three, two, and seven. This gives us a total of 17. Therefore, 17 students are 14 years of age or younger. As 17 is greater than 15, we know that the 13th, 14th, 15th, 16th, and 17th students are all 14 years of age. As the 15th student is 14 years of age, we’ve once again proved that the median age of the group of students is 14.
The final part of the question says the following. c) In the UK, it is required by law that you must be 17 or older to drive a car. In this group, what is the age of the youngest student who is legally allowed to drive a car?
There are no students that were 17 years of age in this group. All the students that are younger than this — those 16, 15, 14, and so on — are too young to legally drive. This means that the only students who are legally allowed to drive a car in the UK are 18 years of age. The three students that are 18 are legally allowed to drive a car.
The answers to the three parts of the question are 29, 14, and 18.