Understanding and Calculating Standard Deviation in Science Labs

[Music] hi it's Mr Anderson and in this video I'm going to talk about standard

deviation when you're collecting data in a science lab the amount of data you collect is important so is the average

but another important statistic is going to be the standard deviation of your sample and so in this video I'm going to

show you what it is conceptually and then going to show you how to calculate standard deviation by hand and then

finally I'm going to show you how to calculate it using a spreadsheet and so first of all what is it well to

understand standard deviation you have to understand the normal distribution and so what does that mean well you're

it's a bell-shaped curve you might think of it like that and so in the United States most men are about 5' n in other

words that's the average right here that's the mean or in statistics that's the

xbar um but there's going to be a lot of men who obviously are taller than that and a lot who are shorter than that and

so the standard deviation is going to measure the spread or or the variation in this bell-shaped curve and so

basically if we were to go right over to here this dark area is going to be one standard deviation above and one

standard deviation below the mean or it's going to be below the average and there's something cool about that about

68% of the individuals are going to be in this area so one standard deviation above and below if we were to look at

this for example down here is two standard deviations and so 95% of indiv idual are

going to be within two standard deviations from that mean and then finally if we go way down here 99% of

individuals are going to be um within three standard deviations of the mean but the standard deviation is going to

vary depending on the data that you collect and so if we had two curves like this so if this is one

curve and then we had another curve that looked like this that data plotted plotted on the

same curve curve this one is going to have a smaller standard deviation than this one they're both going to have

standard deviations obviously they're going to have amounts where it's 68 95 and 99% of the people but this one down

here since it's more spread out is going to have a higher standard deviation and so how do we calculate that well the

equation is a little scary um the scary part it ends up being right here so students are a little scared by that the

summation symbol um but it's actually pretty straightforward it's not that hard to calculate the standard deviation

and so let me show you how to do that and so first thing you want to do is you want to have a data set so here's going

to be our data set right here and to make this easy let's say we just have four pieces of data 1 2 3 four and five

so you're collecting data and this is the data in your data table and you want to figure out the standard deviation of

that well to set that up we're basically going to take the square root of the summation of this divided by the degrees

of freedom so that sounds a little bit scary and so let's go to the scariest part to begin with so let's look at

what's going on on right here underneath that square root and so what this is so if we go like this the summation of x -

xar squared basically means for each of these data points that I have we're going to have to figure out what's right

here so x - xar and so the first thing we have to do is figure out what the average is we have to figure out what

xar is well basically if I add one two three four five together I get 15 and if I D divide that by n which is the total

number of data points so in this case n equals five so we have five data points over here so if I divide 15 by 5

hopefully you can figure out an average the average is going to be three and so the mean is three or the average is

three so what we have to do is we have to calculate this value for all five of

these data points what does that mean well right here we're going to use X and X for the first case is going to be 1 so

that's going to be 1 - 3 and then we're going to square that so what is that 1 - 3 and we square that is going to be -2

and if we square that so that's -2 2ar and if we square that that's four let's go to the next one well this is 2 - 3 so

that stays the same so that's -1 sared and so that's going to be -1 2ar or that's going to equal 1 if we go to the

next one that's easy that's 3 - 3^ 2ar equals 0 and if we Square 0 that's going to be 0 go to the next one that's going

to be 4 - 3 and that's going to be 1^ 2ar or equal to 1 and then finally if we go 5 - 3

Square it that's going to be 2^2 and that's going to equal four and so if you ever see the summation sign don't be

scared by that it's not scary at all it just means you got to do a lot of work so for each of these data points 1

through five I had to calculate what was in there and then I have to add it all up so I have to add 4 + 1 + 1 + 4 and if

I add all those up I get 10 and so what's going to be inside there it's simply going to be 10 so let's figure

out the rest of my standard deviation standard deviation is going to be the square root in this case we solve this

as equal to 10 and then we're going to divide that by nus 1 so what's n that's our sample size in this case it's five

and so we take n minus one and that's going to equal four and so what is our standard deviation it's the square < TK

of 10 / 4 which is 2.5 or if we take the standard deviation of uh the the square root of 2.5 that's going to be something

like 1.58 um and so you're going to have to use a calculator to figure that out well

what does that mean if we were to plot this data as a histogram for example this would be our standard deviation

1.58 and so it takes a while time a while to figure that out based on um doing it by hand and so if you want to

give it a try and so here's data set over here and so try to calculate the standard deviation using this data set

over here and try to do it by hand I'll put the answer down in the description below the video but I would give it a

try it's worth doing once on your own and again this is going to be our formula standard

deviation and so try to do that try to do that by hand and so I'll wait no I won't wait for you to do that pause the

video try to do this one and I'm going to show you how to calculate this really really quickly so I'm going to show you

the spreadsheet shortcut uh and so how did how do you do that in a spreadsheet it's pretty simple so what I'm going to

do is going to take this data and I'm going to switch over here to excel so here's the data right here 02 4 5 and 7

and so I've entered my data in into different cells and now I'm going to figure out the mean just to show you how

easy this is to figure out the mean I'm going to hit it equals here and then I'm just going to start typing so I'm going

to type in average because the spreadsheet is not going to use the word mean so I type in average and then I

select my data I hit a close parenthesis I hit end and it's going to give me my average which is going to be 3.6 so if I

wanted to know the average there it is if I wanted to know the median for example I could just type median and I

could go down like that so spreadsheets are super simple and so what are we looking for we're looking for the

standard uh deviation so how do I do that I just hit equals I then start typing stde V can you see how it pops up

right here standard DV iation parentheses then I'm going to select that and then I'm going to go like that

so what's the standard deviation 2.7 what does that mean we had a bigger spread in the second data set than we

did in the first set a higher standard deviation and if you did it by hand it should have looked something like that

so that's standard deviation and I hope that's helpful