Finding the Sample Mean and Standard Deviation for Grouped Dat...

Question

Finding the Sample Mean and Standard Deviation for Grouped Data In Exercises 39 and 40, make a frequency distribution for the data. Then use the table to find the sample mean and the sample standard deviation of the data set.

3 3 5 3 8 0 3 9 6 6 7 1 6 3 2 6 9 1 8 5 0 2 3 4 9
5 8 1 9 7 6 9 6 7 0 6 3 8 6 8 7 3 8 9 3 7 2 4 4 1

3 3 5 3 8 0 3 9 6 6 7 1 6 3 2 6 9 1 8 5 0 2 3 4 9

5 8 1 9 7 6 9 6 7 0 6 3 8 6 8 7 3 8 9 3 7 2 4 4 1

Accepted Answer

Welcome back, everyone. Given the following data set, construct a frequency distribution table and then use the table to calculate the sample mean and sample standard deviation of the data set. Now here we have a few values and we can tell that they repeat so it makes sense why we want to construct a frequency distribution table because that table will give us each value and the number of time it appears, that is its frequency. So let's construct that table, OK? So we're gonna have X each value, and then we're gonna have the frequency F for each value, OK? Let's go ahead and set that up. You know what? I think we could actually put the table over here, actually. No, you know what, let's leave it here. Let's keep it here. Now, how many different values are in our data set? Well, our data set goes from 0123456, and 7 all the way up to 8. OK? So now that we have all our values, then we can write down how many times each appears. Now notice that 0 appears one time. Uh, next, we can say that we, we can tell that one appears 3 times. OK, so let's mark them off as we go along. So 1 appears 3 times, 123. Next, notice that here 2 appears 123456 times. I should have actually uh got rid of the 1 here, but it appears 6 times, so the frequency of 2 is 6. Next, let's just get rid of all of these. Let's find a frequency of 3. It is here that we have 312 times, frequency of 3 is 2. And I think by now you get the idea. You just count and check for the rest to see how many times they appear. So if we continue that, the frequency of 4 would be 3, or 5 would be 5, or 6 would be 373, and 84. OK? So we have our frequency distribution table. Now that we have that, we can calculate the sample mean. Nor recall. OK, let me make some space down here. Recall that the sample mean x bar. OK, is equal to the sum of X values multiplied by their frequency divided by N from the frequency distribution table. OK. So now, if we can find a product of each value and its corresponding frequency, we can sum those up to use in the formula for the sample mean, OK? So, on our table, OK, let's create another column. And let's call that column X multiplied by F. And in this column, we're just gonna multiply the value by the frequency. So 0 multiplied by 1 is 01 multiplied by 3 is 32 multiplied by 6 is 12, 3 multiplied by 26, 4 multiplied by 312, 55 is 25, 63 is 1873 is 21, and 8 4s 32. So now that we have all of these values, OK. And now if we think about it here. In terms of the number of values that we have, notice that if we add the frequency up, OK, then we have 30 values. So now the sum of the frequencies multiplied by the values is going to be equal to 129. Thus, the sample mean Is going to be equal to 129. Divided by 30, which is equal to 4.3. OK, so that's a sample mean for our data set. Next, let's try to find a sample standard deviation. OK, so we're gonna find the sample standard deviation. Now here, the sample standard deviation from our frequency distribution table is going to be the square root of the sum of squared deviations, so that's x minus X by squared multiplied by our frequency F divided by the number of values n minus 1. So there's a lot to unpack here, but for starters, notice that we'll need to find the difference between each value and the sample mean, and then square it. Multiply it by the frequency and then sum all of those values to plug into our formula for the sample standard deviation. So again, let's just keep that to our table. Let's use our table to help us, OK? So now we know we already have X multiplied by F. so we can add a new column for our deviations X minus X bar. And then we can add a column for X minus X by squared, OK. And then we can add a column for X minus XB 2 multiplied by our frequency F, OK? Now let's continue. So, for example, for X being 0, OK, we already just found our mean to be 4.3, then X minus X will be 0 minus 4.3, which is 4.3. X minus X squared is gonna be 4.3 squared, which equals 18.49. And now, when we multiply that by our frequency, that's 18.49 multiplied by our frequency 1, which equals 18.49. So, essentially, to get the rest, we're going to continue the same pattern. So, for example, when X is 1, OK, it's deviation is -3.3, squared, that's 10.89, multiplied by the frequency, that's 32.67. If we continue our pattern, when X is 2, we're gonna get -2.3, 5.29, and 31.74. When X is 3, we're gonna get -1.3, 1.69, and 3.38. When X is 4, we're gonna get negative 0.3, 0.09, and 0.27. When X is 5, we get 0.7, 0.49, and 2.45. When X is 6, we get 1.7, 2.89, and 8.67. When it's 7, we get 2.7, 7.29 and 21.87. And when it's 8, we get 3.7, 13.69, and 54.76. So now we should be able to find our sum here of the squared deviations multiplied by the frequency. Because when we add all of those up, OK, just make some space here. Oops, didn't mean to get rid of everything. But when we add all of those values up, OK. Then we should find that it is equal to 174.3. And now that we have that value, plugging it into our formula for the sample standard deviation, there's gonna be the skirt of 174.3 divided by 30 minus 1, which, when we simplify. We get that standard deviation to be approximately 2.5. So what can we conclude? Well, now we know our sample mean is 4.3 and our sample standard deviation is 2.5. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Sample Mean

Sample Standard Deviation

Frequency Distribution

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