Gagan and Tony are both practising for a shot put competition. They each made seven throws and the distances, in metres, are shown in the table. Gagan’s first throw was eight metres, his second was six metres, the third was 10, the fourth was nine, the fifth was three, the sixth was seven, and the seventh was eight. Tony’s first throw was five metres, his second throw was eight metres, third was six metres, fourth was nine metres, fifth was 10, sixth was seven, and seventh was seven. a) Whose throws were the most consistent? Justify your answer.
Before we answer this, let’s think about what we mean by most consistent. Well, if we turn it round, most consistent means least variable. So we’re looking for whether Gagan or Tony has the least variation in their throws. Well, what measure we’ve got for variability is the range.
The range of the values is equal to the highest value minus the lowest value. And in that case, that translates to the difference between the longest and the shortest throw for each person. Gagan’s longest throw was his third one at 10 metres, and his shortest throw was the fifth one at three metres. So the range of Gagan’s throws is 10 minus three. The difference between his longest and his shortest throw was seven metres.
Now Tony’s longest throw was also 10 metres, and that was his fifth throw. And his shortest throw was the first one, and that was only five metres. So the range of distances that Tony threw was 10 minus five. So we can write down our answer.
Tony’s throws were the most consistent because the difference between his longest and shortest throws was smaller than Gagan’s. He had a smaller range. A key part of that answer was recognising that the range of five metres was less than the range of seven metres.
The stem-and-leaf diagram shows the distances that 11 different snails travelled in a snail race. Part b) Rebecca says that the median distance is eight centimetres because when she puts the numbers in order, this is the number in the middle. Is Rebecca correct? Give a reason for your answer.
Now in this particular stem-and-leaf diagram, we can see that the stems represent the tens digits and the leaves represent the units digits. We know this because the key told us one slash one represents one ten plus one unit, 11 centimetres.
Stem-and-leaf diagrams are great because they give us information about every single individual piece of data, but they also give us a picture of the overall distribution of that data. So, for example, here we can see that not many snails travelled only a single-digit number of centimetres or more than 30 centimetres. Most of them travelled in the 10s or 20s of centimetres.
We’ve got 11 figures. And rather than writing them all out individually like this, we’ve represented them in the stem-and-leaf diagram. In fact, when we look at the numbers that they represent, we can see that somebody has very kindly put these in ascending order. This is in fact an ordered stem-and-leaf diagram.
The smallest distance that a snail travelled was three centimetres. The next smallest was seven centimetres, all the way up to 35 and 38 centimetres. Now Rebecca was looking for the median distance. And that simply means if you line all the distances from end to end, the smallest at the beginning, the longest at the end, which is the one that’s in the middle?
If we write all the distances down from smallest to largest, there are 11 numbers to write down. So which will be the middle number? Well, there are 11 positions. The middle position will be the average of the first and the last position. With 11 numbers, there are 11 positions. What’s the mean position in this set of numbers? One is the first; 11 is the last. The mean of those two numbers, one plus 11 over two, that’s 12 over two. The middle number is six. So the median itself is the sixth number, 18 centimetres. So we can see that Rebecca is wrong.
Now actually you don’t need to do all of that writing down that we’ve just done. There’s a much quicker way. Because we had an ordered stem-and-leaf diagram, we can just cross off the numbers in pairs, smallest and largest. Zero three is the smallest; 38 is the largest. Zero seven is the next smallest, and 35 is the next largest. 11 is the next smallest, and 29 is the next largest. 12 is the next smallest, and 26 is the next largest. 12 is the next smallest, and 22 is the next largest.
This leaves us with 18, which was in the middle. So our answer is that no, Rebecca is not correct. 18 centimetres is the median distance. In fact, she only wrote down the leaf and she ignored the stem. That’s where she went wrong.