Question Video: Calculating the Mean- Median- and Mode of a Set of a Data Physics

A student records seven measurements of time, 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?

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Video Transcript

A student records seven measurements of time, 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 given a table of seven measurements of time. And all of those measurements, as we can see from the head of the table, have been given in seconds. The seven measurements specifically are three seconds, six seconds, four seconds, three seconds, six seconds, three seconds, and eight seconds. And what we’ve been asked to do is to find the mean value of the measurements, the median value of the measurements, and the modal value of the measurements. So let’s start with the mean value.

To find the mean value of a list, such as the one that we’ve been given here — we’ve got a list of seven measurements of time — what we need to do is to firstly add up all of the quantities in that list and then, secondly, divide by the total number of quantities. And so, for our list of measurements, we can say that the mean is equal to, firstly, the sum of all of the quantities in our list. So that’s three plus six plus four plus three plus six plus three plus eight. And we mustn’t forget to include the unit of seconds here because, remember, each one of these seven quantities is representing a time in seconds.

And so, essentially, we can factor out the unit of seconds and just add up all of the numbers. And that will give us some added time, some total time, in seconds. And then to finish your finding our mean, we need to divide this entire sum by the total number of quantities, which we know are seven: one, two, three, four, five, six, seven. Therefore, we divide this whole thing by seven. And then, when we evaluate this whole fraction on the right-hand side of our equation, we find that our mean value is 4.714 dot, dot, dot seconds. But remember, we’ve been asked to give our answer to the nearest whole number. So we need to round our value to this position. In other words, this is the stuff that we’re going to keep.

But in order to work out what happens to this four, we need to look at the next number. This next number is a seven, which is greater than or equal to five. So our four is going to round up. And in doing so, our mean becomes five seconds. And therefore, the answer to the first part of our question is that the mean value of the measurements to the nearest whole number is five seconds.

Moving on then, we’ve been asked to find the median value of the measurements. Let’s recall that the median value of a list is the value that splits an ordered list into a lower half and a higher half. Now, there are a couple of things to note about this definition. Firstly, the list must be ordered, in other words, arranged from the smallest quantity to the largest quantity. And secondly, the median is the value that splits this list into a lower half and higher half. In other words, it’s the value right in the middle of our ordered list. So let’s first start by taking our list and ordering it from smallest to largest.

We can see that the smallest quantities, the smallest measurements in our list, are time(s) of three seconds. And this occurs three times in our list. So our list, without including the unit of seconds for now just to keep things simple, is going to have three lots of three at the beginning because once again we’ve got three measurements of three seconds. Then, the next smallest value available to us is four seconds, which only occurs once. So we’ll stick that in our list once. And then, we’ve got two measurements of six seconds. So we’ll put two counts of six in our list. And then finally, the only remaining value is eight, which will be the largest value in our list and the final value in our list.

So now, we’ve taken this randomly assorted list of seven measurements and arranged it from smallest to largest. To find the median then, we need to remember that there are seven different values in our list, seven different time measurements, just as we’ve been told in the very first sentence of the question. And to find the value exactly in the middle of this list, we need to record that if a list has 𝑛 values, then the value right in the middle of this list is going to be the 𝑛 plus one divided by twoth value.

In our particular case, the number of values in our list, which is 𝑛, is seven, as we’ve seen already. And so, the middle value is going to be 𝑛 plus one divided by two. Or in this particular case is going to be the seven plus one divided by twoth value in our list. Now, seven plus one is eight and eight divided by two is four. And so, we want to find the fourth value in our list. Starting from the smallest, which means this is the first value. This is the second. This one is the third. And then, this one is the fourth value. And that is our median value. Therefore, out of a list of values that are three, three, three, four, six, six, and eight, the median value is four.

And therefore, out of a list of values that consists of three seconds, three seconds, three seconds, four seconds, six seconds, six seconds, and eight seconds, the median value is four seconds because in an ordered list of seven values, the fourth value is going to be the median. And hence, we can say that the median value of the measurements is four seconds, which means we can move on to the final part of the question.

We need to find the modal value of the measurements. Now, let’s start by recalling that the modal value or the mode of a list is the value that occurs most frequently or most commonly in that list. In other words, in a list of three, three, three, four, six, six, and eight, the value that occurs most frequently or most commonly is three because it occurs three times. Whereas four only occurs once, six occurs twice, and eight only occurs once as well. And so, our value of three occurs the most number of times. It occurs three times, which means that the modal value of our measurements is going to be three seconds. And once again, we cannot forget that each one of these values is representing a time measurement. Hence, we must include the unit in our final answer.

So at this point, we found that the list we’ve been given has a mean value of five seconds to the nearest whole number, a median value of four seconds, and a modal value of three seconds.

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