Video: AQA GCSE Mathematics Higher Tier Pack 3 • Paper 2 • Question 16

Here is a box plot. One of the following statements is true. Tick the true statement. The value of the median is halfway between the lowest value and the highest value. The value of the median is halfway between the values of the lower quartile and the upper quartile. The value of the median is a third of the way between the values of the lower quartile and the upper quartile. The value of the median is a third of the way between the lowest value and the highest value.

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Video Transcript

Here is a box plot. One of the following statements is true. Tick the true statement. The value of the median is halfway between the lowest value and the highest value. The value of the median is halfway between the values of the lower quartile and the upper quartile. The value of the median is a third of the way between the values of the lower quartile and the upper quartile. The value of the median is a third of the way between the lowest value and the highest value.

Let’s begin by recalling what we actually know about a box plot, sometimes called a box and whisker plot due to its shape. There are five values we can read from a box of whisker plot. If we lined all of our pieces of data up in order from smallest to largest, the very first piece of data is represented by this first line and the very last piece of data is represented by this last line. They are the lowest and highest possible values in that dataset.

The left side of the box represents the lower quartile. Again, if we imagine all of our data is lined up in order from smallest to largest, this is the piece of data that is one-quarter of the way through all of the data. The median is halfway through the data. And the upper quartile, or UQ, is three-quarters of the way through that dataset.

And box and whisker plots are often favoured as they can show us the spread of the data. So each of these statements has information about one of these figures. So let’s work them out.

The scale is a little tricky here. It’s five small squares for every 10. So if we divide through by five, we see that one small square is two. We can work out the lowest value in our dataset by drawing a vertical line until we meet the horizontal axis. The lowest value is 80. We’ll repeat this for the lower quartile. The lower quartile is exactly halfway between 100 and 110. So it’s 105. We’ll then find the value of the median — that’s 120 — and the upper quartile — that’s 150.

The highest or largest value in our dataset is one small square above 160. We said that one small square is worth two. So that means the highest value in our dataset is 162. And now we have enough information to decide which of the statements is true.

Statement one says the value of the median is halfway between the lowest value and the highest value. Now you might be tempted to think that this is correct. The median is halfway through the dataset. But we’re looking to find the value of the median, the lowest value and the highest value. The lowest value is 80 and the highest value is 162. So let’s find the number that is halfway between these.

And to do this, we can essentially take the average of the lowest and highest value. We add these numbers together and we divide by two. 80 plus 162 is 242. And 242 divided by two is 121. The median is 120. So this statement is false. It’s not true.

We’ll now consider the second statement. This says the value of the median is halfway between the values of the lower quartile and upper quartile. We’ll repeat the process from before. We’ll add these values together and divide by two. 105 plus 150 is 255. This is an odd number. So when we divide it by two, we’re going to get a decimal. Our median is an integer. So we actually don’t need to work out 255 divided by two. We know it simply cannot be equal to 120. So this second statement is also not true.

What about the third statement? The value of the median is a third of the way between the values of the lower quartile and the upper quartile. We’ll have to calculate this a slightly different way. We’ll find the difference between the two values and divide this by three. And we can then add that on to the value for the lower quartile. The difference between the lower quartile and upper quartile is 45. 45 divided by three is 15. We’ll add this to the value of the lower quartile. And that’s 120. So a third of the way between the values of the lower quartile and the upper quartile is 120. That’s the same as our median. So the second statement is indeed correct.

We will check the last statement to be sure. The value of the median is a third of the way between the lowest value and the highest value. This time, we’ll find the difference between the lowest value and highest value and divide that by three. 162 minus 80 is 82.

Now actually we don’t need to perform the full calculation. 82 is not divisible by three. And here’s how we know. We add the digits of the number together. If their sum is divisible by three, then that means the original number is also divisible by three. And it’s important to know that only works for the divisibility by three. So eight plus two is equal to 10. And 10 is not divisible by three without a remainder. It’s not a multiple of three. So we know that 82 divided by three is not going to be an integer value. And if we were to add a decimal value onto the lowest value, which is 80, we will end up with a decimal value. Our median is 120. So we’ve confirmed that this last statement cannot be true.

The third statement is true. The value of the median is a third of the way between the values of the lower quartile and the upper quartile.

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