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.

Body Temperatures Refer to Data Set 5 “Body Temperatures” in Appendix B and use the body temperatures for 12:00 AM on day 2.

Body Temperatures Refer to Data Set 5 “Body Temperatures” in Appendix B and use the body temperatures for 12:00 AM on day 2.

Accepted Answer

Welcome back, everyone. In this problem, we want to use the given data to determine the range. We want to then apply the range rule of thumb to estimate the standard deviation and compare this estimate with the actual standard deviation calculated from the entire data set. Our data set is heights in inches, and they are 67, 72, 69, 65, 74, 71, 68, 66, 70, 73, 64, and 75 respectively. For our answer choices, A says the estimated standard deviation is exactly equal to the actual standard deviation. B says the estimated is close to the actual value but slightly lower. C says it's significantly higher than the actual value, and the D says it's significantly lower than the actual value. Now to help us solve our problem, let's first start by finding the range. Recall that the range is the difference between the maximum and the minimum values, OK? So now in this case, we have to look for the maximum and minimum values in our data set. Notice that 75 is the maximum value and 64 is the minimum value. So our range is 65 minus 75 minus 64, which equals 11 inches. Now notice that the next part of our problem tells us to apply the range rule of thumb to estimate the standard deviation. What does that mean? Well, that rule basically tells us that our estimated standard deviation, OK, which we can write as estimated, is approximately equal to the range divided by 4. OK. In this case, that would be 11 divided by 4, which equals 2.75 inches. So this is the estimated standard deviation for our data set. Now, we want to compare this estimate with the actual standard deviation. So that means we need to calculate the standard deviation. Now, what do we know? Well, recall, OK, that this formula for sample standard deviation tells us that it's equal to the square root of the sum of the squares of the differences between each data point and the mean squared, well, as I said, the squares, sorry, all divided by the sample size N minus 1, OK? So to figure out our actual standard deviation, we'll need to calculate the, the differences for each of our data points, square them, sum them all up, and plug them into our formula. Now before we can do any of that, we'll need or a mean, OK? And in this case, that would be X bar. The mean of our data set would be equal to the sum of all the data points, OK, divided by the sample size N. No, when we all those add all of those values uh in our data set. OK. Then we should get a sum of, we should get a sum of 834, OK. And now we're gonna be dividing that by 12, which gives us 69.5 inches, OK? So this is the mean number, or sorry, the mean height from our data set. Now that we have the mean. We can make a table to help us figure out the sum of all the differences between each data point and the mean. So, let's first start. Let's write our data value, our data point X1, OK. And then we're gonna show the deviation from the mean X1 minus X bar, and then we're gonna square that deviation, Xi minus X bar squared, OK? Now if I make some space here for all our values. So we're gonna have here, just quickly make our table. No, for each data point, as we said, we know there are 12 data points, 67, 72, 69, 65, 74. 71 68 66. Make some more space here. And then we have 70. 73, 64 and 75. Now, when our data point is 67, the deviation is gonna be 67 minus 69.5, and that gives us negative 2.5. And when we square that, we get it to be 6.25. Essentially, we're just gonna repeat the same process for the rest of our data points. So when it's 72, the deviation is 2.5, and the square of that is 6.25. When it's 69, we get negative 0.5 and 0.25. When it's 65, we get -4.5 and 20.25. When it's 74. We have 4.5 and 20.25, OK. When it's 71, we have 1.5 and 2.25. When it's 68, we have negative 1.5 and 2.25. When it's 66, -3.5 and 12.25. 70, 0.5 and 0.25. Uh, 73. 3.5, OK, and 12.25. 64, we get negative 5.5 and 30.25. And then for 75, we have 5.5 and 30.25, OK? So now that we have the square of all the deviations, we just need to sum them up and plug them into our Formula 4 the standard deviation. So now when we do that. OK, then the sum of all of these values is going to be equal to 143. This is going to be 143 divided by the sample size 12 minus 1. I know when we calculate here, we should get our standard deviation, our actual standard deviation, to be 3.61 inches. So what does this tell us? Well, no, notice that the estimated standard deviation using the range rule of thumb was 2.75 inches, while the actual standard deviation is 3.61 inches. So the range rule of thumb provides a rough estimate, but tends to be lower than the actual standard deviation. So that shows us then that this method is useful for quick approximations, but it might not always be precise. Therefore, when we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Range Rule of Thumb

Standard Deviation

Data Set Comparison

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