Video Transcript
Which of the following is the correct definition of the median value of a set of 11 measurements of the values of an experimental variable? Some values occur more than once, but these are all unusually large or unusually small values. (A) The measured value that occurs the most times in the set of measurements. (B) The difference between the largest value and the smallest value. (C) The sum of the measured values divided by 11. (D) The value that has five values smaller than it and five values larger than it. For (E), the largest value divided by the difference between the largest value and the smallest value.
So in this question, we’re talking about a set of 11 measurements of the values of an experimental variable, and we need to find the correct definition of the median value of these measurements. Let’s start by thinking about what we mean by experimental variable.
Well, a variable is any quantity that can take different values. So, an experimental variable is just a variable that’s in an experiment. Common examples of experimental variables might be the temperature of a beaker of water, the electrical current flowing in a wire, or the amount of force applied to a test object. When we carry out an experiment, we can collect data which usually takes the form of measurements of the values of the different experimental variables. And in this question, we’re asked to consider a set of 11 measurements of an experimental variable. So to make this easier to visualize, let’s just make up some numbers.
Let’s imagine that we took 11 measurements of the values of an experimental variable, and we obtained the values 37, 28, 26, 39, two, 42, 95, 23, 32, 43, and 95. If these were real measurements of the values of an experimental variable, then they could be measurements of temperature in degrees Celsius or measurements of speed in meters per second, or measurements of any other quantity. But for this question, it doesn’t matter what they’re measurements of. We just have 11 values.
The question specifies that some values occur more than once in our set of measurements, but that these are all unusually large or unusually small values. So, our set of 11 values contains two measurements of 95, which is unusually large compared to the other measurements. Okay, so, this question asks us to find the correct definition of the median value of our 11 measurements. So, let’s take a look at the different options that we have.
Option (A) is that the correct definition of the median value is the measured value that occurs the most times in the set of measurements. This option is actually the definition of a different value that describes our data. The measured value that occurs the most times in a data set is called the modal value, or the mode. If we look at our made-up data set, we can see that each value occurs exactly once, except 95 which occurs twice. So, because 95 occurs more than any other measurement, we can say that the modal value or the mode in this case is 95. But this is not the same as the median value, so we can cross out option (A).
Option (B) is that the definition of the median value is the difference between the largest value and the smallest value. In our data set, the largest value is 95, which occurs twice, and the smallest value is two. The difference between these is 95 minus two, which gives us 93. So, what we’ve actually done here is calculate the size of the range over which the data is spread. So, option (B) is actually not the definition of the median, but the definition of the range. So, option (B) is not the correct answer either.
Option (C) is that the definition of the median value is the sum of the measured values divided by 11. So, once again, this option tells us how to calculate a value that describes our data, but it isn’t the median. If we take the sum of the measured values and then divide the sum by 11, so in our case, we would do 32 plus 28 plus 26 plus 39 plus two plus 42 plus 95 plus 23 plus 32 plus 43 plus 95 and then divide that by 11. Then, we would be taking the sum of the measurements and dividing it by the number of measurements that we have. This is actually how we calculate the mean. For our data set, if we add up all of these values, we get 462. And if we divide this by a number of measurements, 11, we find that the mean is 42. However, this isn’t the median. So, it means (C) is not the correct answer either.
Option (D) is that the correct definition of the median value of our set of 11 measurements is the value that has five values smaller than it and five values larger than it. To help us visualize this, let’s write down all of our values in ascending order. Now, we can see that the number in the middle, 37, has five values smaller than it and five values larger than it. In this case, this is actually the same as the median value.
Usually to calculate the median of a set of values, we would write down all of the values in ascending order and then look for the value in the middle. Often, we can do this by crossing out the lowest number then the highest number then crossing out the second lowest number then the second highest number, and so on. And if we had an odd number of values to start with, we’re left with one number in the middle. This number is the median value.
If we started with an even number of values, then we’d be left with two numbers in the middle. And in this case, the median value would be the number halfway in between. But since we have an odd number of values in our data set, we’re just left with one number in the middle. And because we have exactly 11 values, that means we have five values that are smaller than the median and five values that are bigger than the median.
So, option (D) doesn’t give us a standard definition of the median that we can use in all cases. But in this specific case, it does define the median value. The wording in the second sentence in the question is actually really important here. It tells us that some values occur more than once, but these are all unusually large or unusually small values. To show why this is important, let’s see what happens if we don’t just have repetition in unusually large or unusually small values.
So, for example, let’s take a value close to the middle, 39, and change it to 37. So, now, the value 37 occurs more than once. But this isn’t an unusually large or unusually small value. Now, if we calculate the median value, which again we can do by crossing out the smallest value then the biggest value then the second smallest value and the second biggest value, and so on until we’re left with just one value in the middle, we can see that the median is unchanged. It’s still 37. But actually, the definition given in option (D) no longer applies. There are still five values smaller than the median. But because 37 occurs twice, there are actually only four values that are larger than the median.
So, once again, this shows how option (D) is not the standard definition of the median. It only defines the median value for the specific set of values described in the question. So, it looks like (D) is the correct answer. But let’s just check option (E) to make sure. Option (E) is that the correct definition of the median value is the largest value divided by the difference between the largest value and the smallest value. So, for our made-up data, the largest value is 95 and the difference between the largest value and the smallest value is actually something we’ve already calculated. It’s the range.
So, the value we’re calculating here would be the largest value, 95, divided by the range, 93, which gives us an answer of about 1.02. So, dividing the largest valued by the range isn’t something that we’d usually do to come up with a value to describe our data. And the value produced by this method doesn’t actually have a name. And we know for sure that this isn’t the median because the median is 37. So, that means that option (E) is not the correct answer either, which leaves us with the correct answer of option (D).
If we have a set of 11 measurements of an experimental variable where some values occur more than once but these values are all unusually larger or unusually small, then the definition of the median value is the value that has five values smaller than it and five values larger than it.
