Approximating the Standard Deviation from Data of a Histogram

[00:00:00]a hi class this is Professor sniff I'm gonna be doing a short movie on approximating the standard deviation using the histogram we don't know the values of the original data we just know for example that there were five values between 700 and 750 by between 750 and 850 in 880 50 and so on so what we do is we said well let's figure out what we think they could be and so between 700 and 750 on average that would represent 725 so make a little tick mark at 725 and so I would say oh there's about 5 values that would fall in that region from 750 to 750 775 now let's say that there's five of them and I'll do the same here and so be halfway between 750 and 800 with me 775 and then here this was 725 and then we noticed something kind of cool we noticed that if you go from 700 to 750 that has a width of 50 and they notice the distance from 725 at 775 is also 50 but each one of these midpoints is you just add 50 so the next one would be 825 at 58 75f 50 again and we would be at 925 I'm going to create a table showing you the values the midpoints and the frequencies so now I've opened an Excel spreadsheet to show the table of the values that we've accumulated 725 was the first midpoint and we had 5 775 had 5 8 25 had 15 and 875 had 20 and 9 25 had 5 so we can use that information to come up with the mean we don't know what the exact values are but we're gonna assume that there were five at 725 so we're gonna multiply these two numbers together and then we're going to add them all up and then divide by the total and just to do a double check you add up these frequencies 5 Plus 5 is 10 plus 15 is 25 and then 25 + 25 or 50 and the original problem there was a sample of 50 bulbs that they stated in the problem that they were quickly so let's go ahead and use the Aleks calculator to find the mean and then I'm going to show you a neat thing that you can do with mine under deviation so now I've entered in the values so I have five for the first frequency times the first midpoint of 725 side 5 times 725 I've as the next frequency times 775 15 times the next midpoint of 825 20 that frequency times the midpoint of 875 and then the last one had a frequency of 5 times the midpoint of 925 so if we divide that by 50 and notice I double clicked and it highlighted the everything inside the Aleks calculator and hit divide by 50 and once I do that that will give me the mean so the means around 840 and if we look at my data the mean you can think of it as a balancing point and it looks like 840 if I had a little folk from here at 8:40 that would balance those blocks so this week but 840 looks about right so if I got a mean of 1,000 that would be suspect it's not even on the axes so 840 looks about right now to find the standard deviation this is the cool part you're going to select undo and the standard deviation is the measure of spread and so what we're going to do is we're going to figure out how far each of the data values are from 840 so if I take 725 - 840 and then I want to square that because if you just do the deviations they're going to always balance out to give you zero and so we want to square them to eliminate that zeroing effect of the deviation so I'm going to square that one subtract 840 and then square subtract 840 again and then square subtract 840 and then square ooh this is an error of that 840 I messed up there you slide that over just a smidgen all right so I'm going to use my arrow cursor key to cursor over to 840 and I put a parenthesis of a there and then we're going to 8:40 again and then square and then subtract 840 one more time so I'm using the arrow key to cursor over minus eight forty and then square so now I have all the squared deviations and since this is a sample because it says a sample of 50 bulbs when we find the sample standard deviation we had to do that little correction factor of subtracting a 1 and so now I almost have it this is the variance to find or part me the sample variance so now to find this standard deviation I'm gonna double click again that a highlight the whole expression and hit the square root key and that's going to give me the formula for the standard deviation we're going to hit equals well let's just double check to make sure everything right 725 - 840 squared 775 - 840 squared 825 - 840 squared 875 - 840 squared + 9 25 - 840 squared and then I made sure that the 50 minus 1 is underneath the radical and then we hit equals and so then we have fifty five point five five eight and they want us to round your answer the intermediate before but we didn't do any intermediate we do it at all in one fell swoop swoop because we have this neat calculator let the found the final answer they say rounded to one decimal place so at least one we're going to do fifty five point six so that would give us the estimated of the standard deviation let's check to make sure all right all right let's do one more real quick all right so the first thing we need to do is find the mean so it's going to be nine that's the first frequency and halfway between zero and ten is five next we see is nine and then to get the next midpoint I'm going to add the width and from 10 to 0 is 10 so add 10 to 5 and you get 15 or you can visually see that 15 is halfway in between 10 and 20 plus the next frequency of 18 times 25 a 6 times 35 and then we're going to double click and divide by and here it says there's 42 patients and if we add up those frequencies 9 plus 9 is 18 18 and 18 is 36 and 36 and 6 it's 42 so we get that 42 well that's gonna help us find the mean so we get whoo something's wrong they usually give you nice numbers ah it's 18 which is so nice that they give us nice numbers let me know if we made some error yeah that's better 20 so we get a mean of Quentin I'm glad I made that mistake so we get 18 excuse me got a mean of 20 so I'm gonna hit undo and then I'm gonna subtract that 20 from each one of the values basically the midpoint so I'm gonna subtract 20 and then I'm going to square it minus 20 and use my cursor subtract 20 square and then subtract 20 and then square and then remember for the sample standard deviation we have to add in that correction factor subtracting that one so it's 42 minus one or you can write 41 double click that highlights the whole expression and then take the square root then we hit equals and we get 9.1 and then want us to round our final answer to at least one decimal place so that gives us an answer of 9.1 I hope this movie on approximating the standard deviation of a data set given a histogram it has been helpful let's check to make sure I'm correct and we are thank you so much bye-bye