Estimating Standard Deviation with the Range Rule of Thumb. In...

Question

Estimating Standard Deviation with the Range Rule of Thumb. In Exercises 29–32, refer to the data in the indicated exercise. After finding the range of the data, use the range rule of thumb to estimate the value of the standard deviation. Compare the result to the standard deviation computed using all of the data.

Audiometry Use the hearing measurements from Data Set 7 “Audiometry.” Does it appear that the amounts of variation are different for the right threshold measurements and the left threshold measurements?

Accepted Answer

Welcome back, everyone. In this problem, we're referring to the data set given below of athletes running speeds. First, we want to determine the range of the data set, then apply the range rule of thumb to estimate the standard deviation, and finally compare this estimate with the actual standard deviation computed from the entire data set to figure out if the variation in speeds appear to differ between male and female athletes. In our data set, the male athlete's running speeds in meters per second are 8.5, 9.2, 9.8, 8.9, 10.1, 9.5, 8.7, 9.3, 9, and at 9.7. While the female athletes running speeds in meters per second. are 7.4, 7.8, 8.1, 7.9, 8.3, 7.6, 8, 7.7, 7.5, and 8.2. For our answer choices, A says the estimated standard deviation is significantly lower than the actual standard deviation, suggesting that the data set has a non-uniform spread with substantial variation, and that male athletes exhibit greater variation. B says the estimated standard deviation closely matches the actual standard deviation, meaning the range rule of thumb provides a reasonable approximation, and in both groups show approximately equal variation. She says the estimated standard deviation is significantly higher than the actual standard deviation, indicating that the data set contains an olayer affecting the range, and female athletes have greater variation. While this says the estimated and actual standard deviations differ slightly, but both values suggest a similar level of variation in the data set where the variations are indistinguishable due to sample size limitations. Now there's a lot to unpack in this problem, so let's start with the first part of the problem. Let's determine the range of each data set. No, we know that the range is the difference between the maximum and the minimum values for a data set. So for the male athletes, OK, the range in blue is going to be the difference between the maximum value 10.1 and the minimum value 8.5. 10.1 minus 8.5 equals 1.6 m per second. Now what about the range rule of thumb to help us estimate the standard deviation? What is that? Well, that rule of thumb basically tells us that the estimated standard deviation is a quarter of the range. So in this case for the males, the estimated standard deviation will be approximately equal to 1.6 divided by 4, which is equal to 0.4 m per second, OK? So that's the estimated standard deviation. Now let's figure both of these out for the female athletes. Their range is gonna be equal to the difference between the highest value 8.3 and the least value 7.4. 8.3 minus 7.4 equals 0.9 m per second. Which means then that the standard deviation for the female sample will be 0.9 divided by 4 or 0.225 m per second, OK? So now we have the range and the estimated standard deviation for the male and female athletes. Now, in the next part of our problem, we're told that we want to compare this estimate with the actual standard deviation computed from the entire data set. So we're gonna take this data set and find the standard deviation. Now what do we know about it, about the standard deviation that is how to calculate it. Well, recall. That the variance, which is the standard deviation squared is equal to the sum of the square of the differences between each data point and the mean for our respective samples, OK, divided by the sample size N minus 1. And since that's the case, that must mean then that the computed standard deviation will be the square root of that expression. So now, if we're going to figure out our standard deviation for each, then we'll need to find the differences between each data point and the mean, square them, find their sum, and then plug them into this formula to find the standard deviation for each respective data set, OK? So, let's break it up on two sides. Let's put the males on the left and the few males on the right, OK? No, let let me just draw a line here to keep them separate. And now for the meals, we're going to make a, a, a table here, OK? So we're gonna have our data point, OK? And then we're gonna have our differences squared. Or yeah, we're gonna have our differences squared. And then, well, no, let's put our difference first of all, OK, between the data point and the mean. And then we'll have the square of those differences, OK? And then we'll find what each of those are, and then find their some at the end of the table, OK? And then we, we basically repeat the process for our female data set. Now for males, we know that their data points are 8.5, 9.2, 9.8, 8.9, 10.1. Uh, 9.5, 8.7, OK, um, 9.3. 9 and 9.7, OK. So let's go ahead and do that for the males, and then we'll repeat the process to compute the standard deviation for the females, OK? Now of course, we'll need the mean for the male group and mean is going to be the sum of all the data points, OK, divided by the sample size. OK. So in this case, that's gonna be equal to 92.7 divided by 10, which equals 9.27. No, here, that means we're gonna subtract 9.27 from each data point and square it. So, for example, 8.5 minus 9.27, OK, is equal to -0.77 and 0.77 squared. is equal to 0.5929. Now essentially we're just going to repeat that process for the rest of data points. So for 9.2, the difference is 0.0 negative 0.07 and its square is 0.0049. For 9.8, it's 0.53 and 0.2809. For an 8.9, 0.37, oops. Let me put that properly here. 0 0.37 and 0.1369. Next, we have 0.83 and 0.6889. 9.5, it's 0.23, and uh 0.0529. For 8.7, it's negative 0.57 and 0.3249. It's for 9.3, it's 0.03 and 0.0009. For 9, it's negative 0.27 and 0.0729. And for 9.7, it's 0.43 and 0.1849. OK? Know that we have all of that, then we can find our total here. For our squares, OK, which is gonna be equal to 2.341, OK? So now this is a very important value because now we can use this to find our standard deviation for the mails because it's gonna be equal to the square root of that sum, 2.341, OK, divided by N, which is 10 minus 1. And we get then our standard deviation to be 0.51 m per second for the males. No, to find it for the females, essentially we're gonna repeat this process. Write all the data points, find the differences, and then sum them. So now for the men, when it comes to the females, again, we sum all the data points, so it's gonna be 78.5 divided by 10, which equals 7.85. And now in the interest of time, I'll allow you to calculate them on your own, OK? But I'm gonna take a minute and work them and then you can compare them to what you find. OK, so when you complete it, you should get a set of answers like this. So you can take a quick minute, pause the video, make sure that you have all the these uh responses or all these results, sorry, for the female athletes data set. Know that we have the square of all the differences, then we can go ahead and try to find the total, OK? And you should find that you get the total to be equal to 0.82 5. That means the standard deviation for the females. Is gonna be equal to the square root of 0.825 divided by 10 minus 1, which is gonna be equal to 0.3. So that's 0.3 m per second, OK? So, the standard deviation for the meals are 0.51 m per second. That is the computed standard deviation. Well, the computed standard deviation for the females is 0.3 m per second. No, remember That the estimated standard deviation for the males, if we go up, was 0.4 m per second, while for the females it was 0.225 m per second. So we, when we compare the estimated and computed standard deviations, the estimated standard deviation is significantly lower than the actual standard deviation, suggesting that the data set has a non-uniform spread with substantial variation. In other words, if the standard deviation for male athletes is larger, larger, it means their speeds are more spread out, showing greater variation, while if it's larger for female athletes, it suggests they have a wider range of running speeds. But if both values were similar, we conclude that male and female athletes exhibit comparable variation in running speeds. So since the male athletes exhibit greater variation, OK, that means answer choice A is the correct answer. The estimated standard deviation is significantly lower than the actual standard deviation, suggesting that the data set has a non-uniform spread with substantial variation, where the male athletes exhibit greater variation. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Range Rule of Thumb

Standard Deviation

Comparative Analysis of Variability

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