A student recorded seven measurements of mass, as shown in the table. What is the mean value of the measurements to the nearest whole number? What is the median value of the measurements? What is the modal value of the measurements?
Okay, so in this question, we’ve been told that a student has recorded seven different measurements of mass. And these seven measurements are shown in this table here. Firstly, we can see that each mass measurement is shown in grams. And secondly, we can see that those seven mass measurements are 12 grams, 18 grams, 16 grams, 15 grams, 38 grams, 55 grams, and 18 grams. Now, the first part of the question asks us to find the mean value of these seven measurements. And we need to find the mean value to the nearest whole number. So to answer this, let’s first start by recalling what we mean when we say “mean.”
Now, the mean value of a set of measurements is a kind of average, found by firstly summing all of the measured values or, in other words, adding up all the quantities measured and then by dividing by the number of measured values. In other words, if we wanted to find the mean of this set of measurements, then we would firstly have to add up all of the measured quantities. So we’d have to add 12 to 18 to 16 to 15 to 38 to 55 to 18 whilst remembering that each one of these quantities is representing a mass measurement. And so also remembering to include the unit of grams in this addition. And then, we would divide this whole thing by the number of quantities that we’ve measured. In other words, there are one, two, three, four, five, six, seven measured quantities. And so we divide this whole sum by seven. And when we evaluate this fraction, we find that the mean value of this set of measurements is 24.5714 dot dot dot grams.
However, we’ve been asked to find the mean value to the nearest whole number. And so we need to round this quantity to the nearest whole number. In other words, we’re going to need to round here. But in order to understand what actually happens to this number here, whether it rounds up or stays the same, we need to look at the next value. Now, that value is a five. And five is greater than or equal to five. Therefore, the previous significant figure, this four, is going to round up. In other words then, to the nearest whole number, the mean value of the measurements is 25 grams, which means we can look at the next part of the question. “What is the median value of the measurements?”
Now, we can recall that the median value of a set of data is the value that separates the sorted dataset into the higher half and the lower half. In other words, it splits the sorted dataset right down the middle into two equal halves. But the important thing is that it is a sorted dataset. In other words, all of the values in the dataset are arranged from smallest to largest. So let’s start by sorting our dataset into values from smallest to largest. We can see first of all that the first value in this dataset, 12, is indeed the smallest value. And so we can write 12 first. And then, we need to look at all the values remaining and find the next smallest value. And in this particular case, that value is 15. So 15 goes next in our sorted list.
Then, we see that 16 is only slightly larger. And then, we see that we’ve got one mass measurement that’s 18 grams. And we’ve got another mass measurement that’s 18 grams. We need to remember to include both. And then moving on, we’ve got 38 grams. And then, 55 grams is the final value in our list, which means that, at this point, we’ve got a sorted dataset, the smallest values here. And as we move towards the right, the values increase until we get to the largest value.
At this point then, we can find the median value of our measurements because that value is the one that splits our dataset right down the middle. And in this particular case, the value happens to be 18 because this is right in the middle of our list. And the reason for this is that, in our dataset, there are three quantities to the left of it and three quantities to the right. Notice how we didn’t necessarily say that there’re three quantities smaller and three quantities larger because, in one particular instance, the value of 18 here is the same as our median value in our dataset. But the important thing is that the median splits the dataset right down the middle.
Now, with a small dataset, it’s relatively easy to spot which value is the one right in the middle. However, if we had a very large dataset with many measurements, so let’s say we had 77 measurements, rather than just seven like we have here. Then visually spotting the value in the middle of the dataset would be quite difficult. So instead, what we can do is to remember that if we’ve got an ordered dataset, with 𝑛 number of quantities, where this is the first quantity, this is the second, this is the third, this is the fourth, and so on and so forth, all the way up till the 𝑛th quantity, which is the last one in the dataset. Then the median quantity is the 𝑛 plus one divided by two-th quantity in the dataset.
In other words, for our dataset here, where 𝑛 is equal to seven because there’re seven quantities, one, two, three, four, five, six, seven, our median quantity is the seven plus one divided by two-th quantity. And seven plus one is eight. Eight divided by two is four. So in this dataset, the fourth quantity is our median. And we can see that that’s exactly right. We’ve got the first quantity, second quantity, third, and then the fourth one. And we said the fourth one was the median, which is 18 once again.
If, instead, we had 𝑛 is equal to 77, a much larger dataset, then the median quantity from that dataset would be the 77 plus one divided by two-th quantity. In other words, it would be the 39th quantity in the dataset. But if we had something like 𝑛 is equal to 78, for example, if we had an even number of entries in our dataset, then our median would be the 78 plus one divided by two-th quantity in our dataset. Or, in other words, the 39 and a half-th quantity in our set, which basically means that it’s the value exactly in between the 39th quantity and the 40th quantity. And so, for a dataset containing an even number of values, to find the median, we need to find the value in between two quantities in our dataset. But anyway, so in this particular instance, we’ve got seven quantities in our dataset. And as we’ve seen, the fourth one in the sorted dataset is the median value, which happens to be 18 grams. Once again, we mustn’t forget the unit because each of these measurements is representing a mass.
At which point, we can move on to the last part of the question. “What is the modal value of the measurements?” Well, we can recall that the mode of a dataset is defined as the value occurring most frequently in the dataset. So let’s start by writing our dataset out once more. And in order to find the mode, by the way, it doesn’t need to be a sorted dataset, in the sense that we don’t need to have the smallest value all the way to the largest value. But it does help, because what we’re looking for is the value that occurs most frequently. Well, in our dataset, 12 only occurs once. 15 also only occurs once. And the same is true for 16. However, 18 occurs twice. So at the moment, 18 is the value that occurs most frequently in our dataset. Moving on, we see that 38 only occurs once and so does 55. And at this point, we’ve reached the end of our dataset.
So the value that occurs most frequently is 18 because it occurs twice in our dataset. And so we can say that we found the modal value of our measurements. It’s 18 grams. Once more, we cannot forget the unit because we’re dealing with a set of measurements of mass. At which point, we’ve reached the end of our question.