In Exercises 5–20, find the range, variance, and standard devi...

Question

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.

Super Bowl Ages Listed below are the ages of the same 11 players used in the preceding exercise. How are the resulting statistics fundamentally different from those found in the preceding exercise?

41 24 30 31 32 29 25 26 26 25 30

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says to find the range variance and standard deviation for the given sample data. Include appropriate units such as minutes in your results. Listed below are the finishing times in minutes of the same 11 marathon runners, and we're given the values of 125, 140, 155, 160, 145. 138, 150, 152, 148, 142, and 157. We're given four possible choices as our answers. For choice A, we have the ranges equal to 25 minutes, the variance is equal to 108.7 minutes squared, and the standard deviation is equal to 10.04 minutes. For choice B, we have the ranges equal to 35 minutes. The variance is equal to 100.87 minutes squared, and the standard deviation is equal to 12.25 minutes. For choice C, we have the ranges equal to 35 minutes. The variance is equal to 100.87 minutes squared, and the standard deviation is equal to 10.04 minutes. And for choice D, we have the range is equal to 45 minutes. The variance is equal to 108.7 minutes squared, and the standard deviation is equal to 12.25 minutes. So for this problem, we need to find the range, the variance and standard deviation of our given sample data. So we're gonna start off by looking at the range. And recall that the range is going to be equal to our maximum value minus our minimum value. And so, if I look at our data set, we see that the maximum value is going to be 160, it'll have minus, and for a minimum value, we'll have 125. And so when to subtract these two values, this is going to be equal to. 35 minutes. That is going to be the range of our data. The next quantity that we need to calculate is the variance Squad. And here we call your formula for the variance, that's going to be equal to the sum of the quantity of X of I minus X in quantity squared, divided by the quantity of N minus 1 in quantity. Where X of pi is our individual data point, X bar is going to be the mean, and N is the number of data points, which is going to be 11 in this case. So, in order to use this formula, the first thing we need to do is to calculate our mean X bar, and we call it to calculate the mean that we just need to add up all of our data points and divide it by the number of data points. So X bar is going to be equal to the quantity of 125 plus 140. Plus 155. Plus 160. Plus 145. Plus 138. Plus 150. Plus 152 Plus 148 Plus 142, plus 157. And that whole quantity is going to be divided by the number of data points that we have, which in this case, is 11. So, evaluating this expression, this is going to be equal to. 146.55. Minutes. Here we just rounded to 2 decimal places. So, now we're gonna use this average to actually calculate our variance. And the easiest way to do this is to actually make a table, since we have a lot of data points to calculate. So, we're gonna make a table with 3 columns. In the first column, we'll have our data point X of by. In the second column, we'll have the quantity of X by minus X bar. And then in our final column, we'll have the quantity of X of I minus X bar in quantity squared. So we're gonna have to fill out this table for all 11 data points. So for the first data point, We're going to have 125. So in the second column, I'll have 125 minus 146.55. Which is going to be equal to. Minus 21.55. Then our next column will have the quantity of minus 21.55 in quantity squared. And this is going to be equal to. 464.40. Again, around just two decimal places. Now, for our next data point, we'll have 140. So, in the second column, we will have 140 minus 146.55, and this is going to be equal to. Minus 6.55. Then in our last column, we will have the quantity of minus 6.55 in quantity squared, and that it's going to be equal to. 42.90. The next data point that we're gonna look at is going to be 155. Then we will have 155. -146.55 in the second column, which is going to be equal to. 8.45. Then Lecom will have 8.45 squared. And that's gonna be equal to. 71.40. For the next data point, we will have 160. Then in the second column, we will have 160 minus 146.55. Which is going to be equal to. 13.45. Then in the last column, we will have 13.45 squared, which is going to be equal to. 180 090. For the next data point, we will have 145. And then in the second column, we will have 145 minus 146.55. Which is going to be equal to -1.55. The last column will have the quantity of -1.55. In quantity squared, and that's gonna be equal to. 2.40. For the next data point, we will have 138. In the 2nd column, we will have 138. -146.55. And that's gonna be equal to. Minus 8.55. Then last column, we'll have the quantity of minus 8.55. In quantity squared. And that's gonna be equal to. 73.10. For the next data point, we will have 150. So, in the second column, we will have 150 minus 146.55. Which is going to be equal to. 3.45, then in the last column, we will have 3.45 squad. Which is going to be equal to. 11.90. For our next data point, we will have 152. And in the second column, we will have 152 minus 146.55. Which is going to be equal to. 5.45. Then in our third column, we will have 5.45 squared. Which is going to be equal to. 29.70. For the next value, we will have 148. Then in Zon, we will have 148. -146.55. Which is going to be equal to. 1.45. Then in our third column, we will have 1.45. Squad, which is going to be equal to. 2.10. For the next data point, we will have 142, and so in the second column, we will have 142 minus 146.55. And that it's going to be equal to. Minus 4.55. So in our third column, we'll have the quantity of -4.55. In quantity squared. And evaluating that, it's going to be equal to. 20.70. And for our final data point, we will have 157. And so in that column, we will have 157 minus 146.55. Which is going to be equal to. 10.45. And so in our 3rd column, we will have 10.45. Squared, which is going to be equal to. 109.20. So now we need to do is to add up everything in the 3rd column. So, for that sum. When we add up all those values, we get for the sum. A value of 18. 0.7 So now, putting this back into our formula for the variants. We will see that Squared. is equal to 18.7. Divided by the quantity of 11 minus 1. And evaluating this expression, this is going to be equal to. 100.87, and this has units of minutes squared. Since we are squaring both, since we're squaring the quantity of X by minus X bar, and both of those quantities had units of minutes. So when we square those quantities, we also have to square the units. So this is the the value of the 2nd quantity that we needed to find. Now, for the 3rd quantity, we need to find our standard deviation, and that's related to the variance. So our standard deviation S, is going to be equal to the square root of a variants as squared. So this is going to be equal to the square root. Of 100.87. Which, when we evaluate that expression, it's going to be equal to. 10.04, and that has units of minutes. So, here we just again rounded to two decimal places. And we get the units of minutes, since we're taking the square root of Squad, and we take the square root of value, we also need to take the square root of the units. So we're taking the square root of minutes squared, which gives us units of minutes. So that's the third quantity that we needed to find. So now that we have calculated all three of the quantities, we can now compare those values to our answer choices, and we see that the correct answer choice actually corresponds to answer choice D. So, I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Range

Variance

Standard Deviation

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