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 Jersey Numbers Listed below are the jersey numbers of the 11 offensive players on the starting roster of the New England Patriots when they won Super Bowl LIII. What do the results tell us?

12 26 46 15 11 87 77 62 60 69 61

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. The following data represents the duration in minutes of workout sessions completed by a group of individuals at a fitness center over a week, and we're given them the values of 30, 45, 60, 50, 40, 55, 70, 35, 65, 50, and 42. What do the results tell us about the variation in workout durations? Now we're given 4 possible choices as our answers. For choice A, we have the range is equal to 30 minutes, the variance is equal to 145.25 minutes squared, and the standard deviation is equal to 11.20 minutes. And the workout durations show high variability, meaning that workout times differ significantly from person to person. For choice B, we have the range is equal to 40 minutes, the variance is equal to 155.8 minutes squared, and the standard deviation is equal to 12.48 minutes. And the workout durations show moderate variability, meaning that they are somewhat spread out but not extremely different. For choice C, we have the range is equal to 40 minutes, the variance is equal to 155.8 minutes squared. The standard deviation is equal to 13.25 minutes, and the workout durations have low variability, indicating that most workout times are close to the average. And for choice D, we have the ranges equal to 30 minutes, the variance is equal to 175.40 minutes squared, and the standard deviation is equal to 11.20 minutes. And there is no variation in workout durations, as all the values are nearly the same. So, the first quantity that we're going to calculate is going to be our range. I recall that the range. is equal to our maximum value. Minus our minimum value. And so, if we look at our data set, we see that the maximum value is going to be 70, and the minimum value is going to be 30. That means our range is going to be equal to 70 minus 30. Which is equal to 40, and that has units of minutes. And so that's one of the quantities that we needed to calculate. For the next quantity, we need to calculate our variance, which is going to be Squad. And recall that the variance is going to be equal to the sum. Of the quantity of X of I minus X bar. In quantity squared. Divided by the quantity of N minus 1. For X of by here is one of our data points, XR is our mean, and N is the number of data points that we have. So, to use this formula, the first thing we'll need to calculate is our sample mean, X bar. And recall that this is just going to be equal to the sum of all of our data points divided by the number of data points that we have. So, we'll have for export, that's going to be equal to the quantity of 30. Plus 45 Plus 60 + 50. Plus 40. Plus 55. Plus 70. Plus 35. Plus 65 Plus 50. Plus 42 in quantity, and all of that is divided by N, which is going to be our number of data points, which in this case is going to be 11. And so when we evaluate this expression, we get that X bar, it's going to be equal to. 49.27 minutes. So now we can use this mean to actually calculate our variance. And since we need to calculate, The difference from the mean for each of our data points, and then squaring that quantity, this is best done in a table. So we're gonna create a table. Where the first column is our workout duration, which is our X ofI, and it has the values of 30, 45, 60, 50, 40, 55, 70, 35, 65, 50, and 42. The next column is going to be Xabi minus X bar. And so, when we calculate this quantity for each of our data points, we get the values of -19.27, minus 4.27, 10.73, 0.73, minus 9.27. 5.73, 20.73 minus 14.27. 15.73, 0.73, and minus 7.27. Then in the next column, we're going to calculate the quantity of X by minus X bar in quantity squad, so we're just squaring the previous column, and when we do so, we get for our values 371.3329, 18.2329, 115.1329, 0.5329. 85.9329. 32.8329. 429.7329. 203.6329, 247.4329, 0.5329, and 52.8529. And we'll actually need a sum of all that. And so we sum our last column, we get a value of 1,558.1819. So we can now substitute that value back into our variance formula, so we get that Squared is going to be equal to 1,558.1819 divided by the quantity of 11 minus 1. And when we calculate this, this is going to be equal to. 155. 8. So here we just round it to one decimal place. And this actually has units of minutes squared. And so this is the next quantity that we are actually looking for. And our final quantity that we need to calculate is our standard deviation S. And that is just going to be equal to the square root of Squared. Which is equal to the square root. Of 155. 0.8 And when we evaluate that expression, we get that S is going to be equal to. 12.48, and this has units of minutes. Since when we, when we took the square root of Squared, we also need to take the square root of the units. So taking the square roots of minutes squared, we get units of minutes. And so this is the last quantity that we needed to calculate. Now we need to interpret our results to answer the last part of this question. Now, if you look at the range, we have a range of 40 minutes, and this indicates a moderate spread in workout durations. And the variance in standard deviation suggests some variability in workout times, meaning participants' workout durations differ significantly but are not extremely scattered. So, based upon that interpretation, as well as the values that we calculated, when we look at our answer choices, we see that the correct answer choice actually corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Range

Variance

Standard Deviation

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