Graphical Analysis In Exercises 21–24, you ar...

Question

Graphical Analysis In Exercises 21–24, you are asked to compare three data sets.

(c) Estimate the sample standard deviations. Then determine how close each of your estimates is by finding the sample standard deviations.

i. ii. iii.

Accepted Answer

Hello everyone. Welcome back. Our next problem says, the stem and leaf plot shows ages, and then we're given a stem and leaf plot. We'll come back to it and describe those numbers after we finish reading through. The problem says estimate the sample standard deviation and compare it to the actual value. We have four answer choices. A, estimated standard deviation, approximately 9.5, actual sample standard deviation, approximately 11.93. Choice B, estimated 8.5, actual 9.94. See estimated 11.94, actual 11.94. The estimated 11.94, actual 9.5. So let's go ahead and look at our Our plot and think about how we would calculate both estimated and actual standard deviation. So, we have something called the range rule. For estimating. Standard deviation. And that says that the estimated standard deviation, so I'm just putting ESD. I, we'll say approximately equal to the range of values divided by 4. So in that case, range of courses, the highest value. And then subtract the lowest value. So going to our stem and leaf pot, in our 10 places, we have rows of 2345, and 6. So, our highest number of uh our highest number will be in the 60s, and then there's only one value in the 60s, and that has a 0 in the ones place, so 60 is our highest value. And then let's go to our first row, where there's a 2 in the 10's place. There's 2 values there with one's value of 2 and 5. So that means 22 is our lowest value. So we have 60 minus 22 divided by 4. So that is going to give us approximately 9.5. Let's change this to approximately. So this is just a rule of thumb for estimating a standard deviation for a sample. And let me just scroll up here. So, on a test, it's always useful to maybe rule out some incorrect answers to give myself fewer choices. So, I can eliminate choice B, which has an estimated sample distribution of 8.5, eliminate choice C, which has an estimated standard deviation of 11.94, and eliminate choice D, estimated standard deviation of 11.94. So, you can see here how that approach is helpful. If I were on a test, I've now been able to solve this problem and get an answer with just a very easy, quick estimate of standard deviation and nothing further. Because I'll know my answer to choices A, that's the only one with the corresponding estimated standard deviation. But this is a practice problem, so we will go through how to get the actual value. Um, but just be aware of things like this when you're looking at multiple choice questions on a test. So what's our formula for standard deviation? So S is equal to. And then we have. A square root, and then underneath the square root sign, and the numerator, we have the sum. That sum is from I equals one. Through in And it's the sum of in parentheses X sub i. Minus X bar, and then close parentheses in that different squared. And that whole sum then in the new over in the denominator. Or divided by, and the dominator should say, N minus 1, and then that fraction, all under a square root sign. And let me just draw redraw my square root sign so that it Encompasses the whole fraction there. Here we go. So we Need to calculate the mean of our samples here, and then look at the difference or the difference between in each value between the value and the mean. And square those differences. So, it's helpful in that case to make a table. So, we would need to list all of our actual values in sort of more conventional format than the stem and leaf plot. So, let's go ahead and list them out. So, our first row, we have, as I said, 10s 2, and the ones place 2 and 5. So, our first two values, 22 and 25. Then, with 3 in the 10s place, we have 04, and 6 in the ones place. So we have 30, 34. I'm 36. Then we have the next row with a 4 in the 10s place, and then in the ones place, 02 and 5. So we have 40. 42 and 45. We have to scroll up a little bit to fit everything. Then, uh, with a 5 in the 10's place, the next row down, just 1 in the one place, there's a 3 in the one's place, only one value there, so 53. And then we already said with our 6 in the 10s place, we've got 60. So I'm going to turn this into a quick table, and our first row is going to be. X sub I, so the value itself, and then we're going to. So now you can see we've turned this into a table with 3 columns. So X sub I is the first column are Values themselves, and then our middle column will be X sub I, and then subtracting X bar. But we still don't have our mean itself. So, first we need to add up all of our values of X. So, sum up 22, 25, 30, 34, 36, 40, 42, 45, 53, and 60. So, we'll just add that to the bottom here. Blind Because we'll need the sum of the squares later. So, that sum of all those numbers is going to be 387. Put some down on the bottom. So Armine ex bar. We'll equal our sum of all our numbers, 387, divided by the number of samples. So how many do we have? 123456789, 10. So divided by 10. So that means our mean is 38.7. So, for each of our values here, we need to subtract. That's 38.7 from each of our individual values. So that's going to give us for 22. -16.7425, -13.7. 430-8.7. For 34-4.7. 436-2.7 44 1.3, or 42, 3.3, 45, 6.3. For 53 14.3 And 460, 21.3. And then we're just going to square each of these values in our 3rd column. So, from in the column for 22, we have 278.89 for 25, we'll add 187.69. For 30, we'll add 75.69. And then for 34, 22.09. 436, 7.29. For 41.69. 44210.89. 445, 39.69. 453204. 49 And for 60, 453. 69. So now, down at the bottom, we're going to take a sum of all of those, because we know we need the sum for our numerator and our standard deviation formula. So add all of these squared values together, and our sum is going to be. 1200. 81.1. So now we're ready to go back and fill in our standard deviation numbers. So we have that square root sign. And then, under the square root in the numerator, the sum of all of those squared differences. So, in the numerator, we have 1,281.1. The denominators, N minus 1, where N is the number of samples, which we said was 10, so the denominator will be 9. And take the square root of that. Division 1,281.1 divided by 9. And that's going to give us an answer of approximately. 11.93 For actual standard deviation. And sure enough, when we go to choice A, we see that that corresponds with what's here. So once again we were given a stem and leaf plot. With ages, and we were supposed to estimate the sample standard deviation and compare it to the actual value. So, we estimated the standard deviation using that range rule, where we took the range and divided by 4, giving us an estimated standard for standard deviation of approximately 9.5. And then we calculated the actual sample standard deviation. And so we took all our values. Calculated the difference between the value and the mean, and then squared those differences to add into our standard deviation formula, the square root of the sum of all the differences between the samples and the means, divided by N minus 1. And that gave us an actual sample standard deviation of 11.93. Hope to see you in the next video.

Graphical Analysis In Exercises 21–24, you are asked to compare three data sets.

(c) Estimate the sample standard deviations. Then determine how close each of your estimates is by finding the sample standard deviations.

i. ii. iii.

Key Concepts

Sample Standard Deviation

Estimating Standard Deviation

Data Set Comparison

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