The lengths of several poles in feet are 30, 44, 60, 64, and 34. If a pole with a length of 28 is added to this list, which of the following statements would be true? The mean would increase. The range would decrease. The median would decrease. The median would increase.
So, here, we’re given some values of poles in feet. Let’s start by writing these poles in order of size. So, we’d have 30, 34, 44, 60, and 64. We’re then told that a pole with a length of 28 is added to this list. And we’re asked some questions about the mean, the range, and the median. So, let’s think about these logically and check by calculations.
We’ll start with the statement in option A regarding the mean. We can find the mean of a set of values by finding the total sum of the values and dividing by the number of values that we have. So, we’re being asked to compare the mean of our original set of values with the mean when we add 28. Adding 28 to our values in order, the 28 would come first because it’s a lower value. We’d expect logically that our mean would also be a lower value. But let’s check with a calculation.
To find the mean, then, of our original values without the 28, we would add 30, 34, 44, 60, and 64 and then divide by five since there were five values. The values in our numerator simplify to 232. So, we have 232 over five. And performing the division, we’ll get 46.4.
To find the mean of our values when 28 is one of our values, we add together our values 28, 30, 34, 44, 60, and 64. This time we’ll divide by six since there are now six values. Simplifying our answer, we have 260 over six. We can check our numerator easily since 232 plus 28 gives us 260. Performing the division then will give us a mean of 43.3 recurring. That’s 43.333 and so on.
Comparing our two means then, with and without our value 28 in the list of values, we can see that the mean would decrease, which confirms what we had thought logically. It means that option A is incorrect. So, let’s clear our workings and have a look now at the range of our values.
Again, we’re going to compare the range of our values without 28 and then with 28. We can recall that the range of a set of values is the largest value subtract the smallest value. So, considering that the range is effectively the difference between the largest and the smallest, then when we add in a 28 to our values, then the difference will change.
Notice that if 28 had been in the interval between 30 and 64, then it wouldn’t have changed. So, thinking about this logically then. Even though 28 is smaller than the other values, the difference will be larger. In fact, we can probably guess that it will be two larger since the difference between 28 and 30 is two. Let’s see if we can confirm this with a calculation.
The range of our original values without 28 will be 64 subtract the smallest value, which was 30, giving us an answer of 34. The range of our values with 28 added will be 64, since that hasn’t changed, subtract 28, giving us a value of 36. Therefore, the range of the values when we add 28 has increased, meaning that option B is incorrect.
Now, let’s have a look at the median of our values and see if option C or D is true. We can remind ourselves that the median of a set of values will be the middle value. So, without working out the specific values yet, if we take the median of five numbers and then we add in a lower number to give us six values, we’d expect the median to shift downwards, meaning that it would decrease. But let’s check.
So, looking at our values without 28, the median in this case is fairly easy to see. At the halfway point, it’s the value 44. So, that’s our median. Next, to find the median when we include 28 in our values, this middle value will lie between 34 and 44. We can think of this in two ways, either by considering that the difference between these is 10. Therefore, a halfway between must be five on from 34, giving us 39. The alternative is to add our two values of 34 and 44 and divided by two, which is 78 over two, or 78 divided by two, giving us the same medium value 39, which we saw a minute ago.
So, now, we check our median before and after we add the value 28. We can see that the median has decreased, confirming what we thought earlier. We can, therefore, conclude that option D is incorrect and option C is the true statement. So, our answer is the median would decrease.