Using and Interpreting Concepts

Question

Finding and Discussing the Mean, Median, and Mode In Exercises 17–34, find the mean, the median, and the mode of the data, if possible. If any measure cannot be found or does not represent the center of the data, explain why.

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Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says the dot plot below shows the ages and years of participants in a workshop. And below texts were given a dot plot. Where on the X-axis, we have the years of 21, 22, 23, 24, 25, 26, 27, and 28. And above 21, we have 1 dot, above 22, we have two dots. Above 23, we have 1 dot. Above 24, we have 1 dot, above 25, we have 3 dots. Above 26, we have 1 dot, above 27, we have no dots, and above 28, we have 1 dot. We need to calculate the mean, median, and mode of the ages, which measure best describes the center of the data. So the first thing we need to do is determine our data set based upon the dot plot. And so we'll start off with our lowest value of 21, and it only appears once since it has one dot above it, we will have a value of 21. Then if we look at the next value, we'll have 22, but it has two dots above it, so that means it appears twice, so we'll have 22, then 22 again. The next value is gonna be 23, it only has one dot above it, so it only appears once in our data. So the next value is just gonna be 23. Same thing with 24, 24 only has one dot above it, so it only appears once, so the next value is gonna be 24. If we look at the next value, that's gonna be 25, but it has 3 dots above it, so that means it appears 3 times in our data set, so we'll have 25. Then 25 again, and then 25 for a third time. The next value is going to be 26, and it only has one dot, so it appears once, will have 26. 27 has no dots above it, so it doesn't appear up in our data set. And then our final value is gonna be 28, and it only appears once, since it has a single dot above it. So we'll next uh write down 28. So, our data set consists of the values of 21, 22, 22, 23, 24, 25, 25, 25, 26, and 28. So, we need to calculate the mean, median, and mode of this data set. So we're gonna start off by calculating the mean, and we call that the mean is equal to the sum of all of our data divided by the number of data points that we have. That means our mean is going to be equal to and here in the numerator, we'll have the quantity of 21 plus 22, plus 22 plus 23 plus 24. Plus 25 + 25. Plus 25. Plus 26. Plus 28. And that whole quantity is divided by the number of data points that we have. Which in this case is 10. So, evaluating this expression, we'll get that our mean is going to be equal to. 241 divided by 10, which is equal to 24.1. So that is the first quantity that we were asked to find. Now for the next quantity, we need to determine the median and recall that the median is going to be the value that shows up in the middle of our data set once we have written it in ascending order. And so here we already have our data set ordered, and since we have 10 data points, we need to actually find the value that is the midpoint between the two values in the middle, since we have an even number of data points. And so the two values that show up in the middle are going to be 24 and 25. So recall to find a midpoint, we just need to add those two values together. So we'll have 24 plus 25, and then we'll divide that quantity by 2. That means that our median is going to be equal to. 24.5. So that is the value for our second quantity that we are asked to find. Now the last quantity that we need to find is going to be the mode. And recall that the mode is the value or values that appear with the highest frequency. And here we just have one value for our mode, since we have a value of 25, showing up 3 times, which is the most that any number actually occurs. That means that our mode is going to be equal to 25. So that is the value of the last quantity that we're asked to find. Now we just need to answer the last part of the question and decide which measure best describes the center. Now our data here is fairly symmetric with no extreme outliers. And if we look at our values, our values for the mean median mode are all very close together. So we have the value of 24.1 for the mean, 24.5 for the median, and 25 for the mode. So, in this case, we're actually want to prefer the median, since it is often preferred for small data sets with repeated values. Even though all three are perfectly reasonable. So, the answer to that last part of the question is going to be our median. So just to recap, we have a mean with a value of 24.1, a median with a value of 24.5, and a mode with a value of 25, and the measure that best describes the center of the data is going to be the median. So those are going to be our answers for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Mean

Median

Mode

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