Constructing Data Sets In Exercises 5– 8, con...

Question

Constructing Data Sets In Exercises 5– 8, construct the described data set. The entries in the data set cannot all be the same.

Mean and median are the same and the data is bimodal.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says to select the data set where the mean and median are the same, and the set is bimodal. And we give 4 possible choices as our answers. For choice A, we have 579, 1113, and 15. For choice B, we have 488, 1212, and 16. For choice C, we have 2266, 14, and 18. And for choice D, we have 369, 1215, and 18. So we need to select the data set out of our answer choices where the mean and median are the same, and the set is by modal. So, we're gonna focus on that bimodal condition first. So recall that the mode of a data set is a value or values that occur most frequently. So for a bimodal data set, we need to have two values that occur most frequently. So, if we look at choice A, we see that each value in that data set only appears once, and therefore, it cannot be a bimodal data set. So that means that choice A is not one of our correct answers. If you look at choice B, we see that we have two values that appear twice, 8 and 12, and since we have two values that appear most frequently, that means that we do have a bimodal data set for choice B. If you look at choice C, again, we have two values that appear twice, 2 and 6, and so we have two values that appear most frequently, and so, therefore, answer choice C is a bimodal data set. And if we look at choice D, again, we have Each value only appearing once in our data set, and so that means it is not a bimodal dis uh data set, and so that means that choice D cannot be the correct answer. So now what we need to do is calculate the mean and median for our data sets for choice B and for choice C, and see if the mean is equal to the medium for either of those two choices. So, we're gonna begin with choice B. And here we'll first calculate the mean, and recall that the mean is equal to the sum of all of our data values, divided by the total number of data values that we have. That means that our mean is going to be equal to the quantity of 4 + 8 + 8. Plus 12 plus 12 + 16 in quantity, and that is going to be divided by the total number of data points that we have, and so we have 6 data points here, so that's just going to be divided by 6. So, evaluate this expression, this is going to be equal to 60, divided by 6. Which is equal to 10. So next, we need to calculate our median. As you recall that the median is the value of our data set that appears in the middle of our data set, once we have um ordered our data values from the smallest to the largest. And here we actually have an even number of data points since we have 6, so we're actually going to be finding the midpoint between the two values that show up in the middle, so the 3rd and 4th data points. In our data set here. And so, we're gonna be looking for the midpoint between 8 and 12. So, that means our median is going to be equal to, and here we call how to calculate the midpoint between two values, and so we're just going to add the two values together, so we'll have 8 plus 12, and we'll divide that quantity by 2. So that means that this is going to be equal to 20 divided by 2, which is equal to 10. So here our mean is equal to our median, that means that choice B is going to be our correct answer. But let's go ahead and check voice C, to see if it also has an equal mean or also has a mean equal to the median. So, we'll begin by calculating the mean. And so that is going to be equal to. The quantity of 2 + 2 + 6 + 6 + 14 + 18 in quantity, and then it's going to be divided by 6. And so, that is going to be equal to when we evaluate this expression, that's going to be 48 divided by 6. Which is equal to Eat So next we'll need to calculate the median. And here, again, recall that that is going to be the midpoint between our 3rd and 4th values in our order data set. So that's gonna be the midpoint between 6 and 6. Now, since both of those values are the same, that means that our median is just going to be equal to 6. And so for choice C, we see that the mean is not equal to the median, so it is not the correct answer choice, and so that leaves us with only the correct answer choice being answer choice B, since it is bimodal and the mean is equal to the median, which both had a value of 10. So that is going to be the correct answer for this problem, gonna be answer choice B. So, I hope that this explanation has been useful, and I'll see you in the next video.

Constructing Data Sets In Exercises 5– 8, construct the described data set. The entries in the data set cannot all be the same.

Mean and median are the same and the data is bimodal.

Key Concepts

Mean

Median

Bimodal Distribution

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