Four values are missing from the shown data set. Given that the mode is 89, the median is 81, the range is 32, and the data is listed in ascending order, find the missing values.
So that I can actually work through this process logically, I’ve labelled each of our missing values. So we’ve got 𝑎, 𝑏, 𝑐, and 𝑑. The first bit of information that I’m actually going to use from the question is the fact that the range is 32. I’m gonna use this because the data is actually listed in ascending order. And we have the first term.
So therefore we can actually use the range to calculate the final term. And we can do this because as the range is 32, therefore we can say that 𝑑, so our final value, minus 59, which is our first value, is gonna be equal to 32. So therefore if we add 59 to each side, we get 𝑑 is equal to 91. So brilliant! We’ve found our first term.
Next, I am gonna take a look at this bit of information. And it tells us that the mode is 89. So therefore, if we take a look at the values are left which are 𝑎, 𝑏, and 𝑐, we need to decide which ones of these are actually going to be 89. Well as it’s the mode and the mode is the most common value, then therefore at least two of our values must be 89. So at least two of 𝑎, 𝑏, or 𝑐 must be 89.
To decide which values are going to be 89, we can use this other bit of information. The median is 81. Well as we know that the median is 81, we then look at right how many values do we have. Well, we actually have one, two, three, four, five, six values. And as we have six values, therefore it means our median is going to be the third and fourth values, so 𝑎 plus 𝑏, divided by two.
So therefore, we can say 𝑎 plus 𝑏 over two is equal to 81. However, if 𝑎 and 𝑏 were both 89, then 89 plus 89 over two is not equal to 81. So therefore, we can say that 𝑎 cannot be 89. Therefore, as 𝑎 is not equal to 89, then 𝑏 and 𝑐 must be equal to 89, because we said that least two values must be 89 to allow the mode to be 89.
Okay, fantastic! One more value to find, let’s move on and find 𝑎. Well, to calculate 𝑎, we can actually move back over to the right-hand side because we had that 𝑎 plus 𝑏 divided by two is equal 81. As we explained, that’s because the median is the middle value and we’ve actually got six values. So therefore, the middle two values must be added together and divided by two to find out our median.
Great! So now what we can do is actually substitute our value for 𝑏, which is 89, into this equation. So we get 𝑎 plus 89 over two is equal to 81. Then if we multiply each side by two, we get 𝑎 plus 89 equals 162. And then if we subtract 89 from each side, therefore 𝑎 is gonna be equal to 73.
So therefore, we can say that the four missing values are 73, 89, 89 and 91. And, therefore, the data set 59, 65, 73, 89, 89, and 91 fulfills the parameters where the mode is 89, and that’s correct because we have two values that are 89. So this is most common value. The range is 32 because 91 subtract 59 is 32. And the median is 81, because our middle two terms, 73 and 89, divided by two when they’re added together gives us 81.