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Hello my name is Beth Dixon and this is a video series based on Vicki Borlaug's PowerPoint presentation
on sample mean, variance, and standard deviation
thanks to Vicki Borlaug for the use of her PowerPoint in making this video
in this video we will use the TI-84 calculator and its list of feature to calculate standard deviation by hand
and I've put quotation marks up in the air with my hands and when I say by hand
when asked to calculate the standard deviation by hand what we want is for the student to show the work involved in each step of the
process will be using the calculator to do the arithmetic
so in our case by hand means not using the calculator to get the final answer in one step but
to use the calculator to do the step by step process of finding the standard deviation
let's look at some data and review our formulas
for our first example we will use the following data: 4, 10, 19, 19, and 23
remember
that the formula for sample mean is x-bar equals the sum of the X's divided by n
the sample variance
is denoted by s squared
and s squared equals the sum
of x minus x-bar squared
divided by n - 1
and the sample standard deviation is w
equals the square root
of s squared the sample variance
the first thing we want to put in our calculator is the data itself
which is the x column or the x values the data is the x values
we will put those in list one
to do that we will press the stat key
then choose edit and edit under
the edit menu
and again we're putting the X's or the numbers under list one
we will then type each number or data item in turn pressing enter after each one
if you make a mistake you can delete an entry and you can also use the second INS
which is the insert button if you make a mistake
this is what your calculator should look like after you've entered your data
always
check your data at this point
because if you put the wrong data and you'll get the wrong answer out
what is the next astep well let's look at what we're trying to find we're trying to find the mean
the standard deviation and the sample variance and I did say those last two in the opposite order that will find them
but the first thing we want the fine is the mean
the mean is given by the sum of the x's divided by the n or the sample size
so the first thing we need to find is the sum of the X's
so we need the sum of that first list
so we want the sum of list one divided by five
the calculator has a command for that just like other things have an APP that the calculator has a command for that
but it's buried in a menu the first step is going to be to get out of the menu that you're in now out of the edit mode
to do so you will press
second
then the word quit which is above the mode key and then clear
this should remove you from that mode
now you'll press second
then list
over to math
and then choose sum
now we need to tell the calculator what we need
to sum which list of we want to sum and our data is in list one and list one is above the one key so we will press second L1
and close the parentheses
and then press enter
the calculator should then gave you
the following screen and then answer of 75
divide that by five
and we should get the mean of 15
so our sample has a mean of 15
moving to the next step we want to start the process of finding the sample variance
we must use the order of operations on our formula and we do what's inside the parentheses
first that means we need to subtract the mean from each
data value each
x
in list one
so we will find x minus x-bar
to do so we can give the calculator of command that will complete this task in one step
move the cursor so that the L2 is highlighted as shown on the screen
this will allow you to put in a formula
and we will type in list one that typing second L1 -15
notice that the formula will appear at the bottom of the screen
and this takes the x's from the list one and subtracts the mean of 15 from each value
when we press enter it'll and by using the magic of the calculator
you will suddenly get all of the list one
numbers subtract 15 ---- 4 -15 is -11, 10 -15 is -5, 19 -15 is 4, 19 -15 is four and 23 -15 is eight and it didn't take long for me to say that
but with certain numbers it'll be much easier to do
and calculate this way than to try to do that without making a mathematical error on your own
now back to the formula in the order of operations
we've computed the subtraction in the parentheses now are ready for the exponent the square
so we need to take the numbers we just got in list two and square each of those values
again not by hand but by using a formula
so list three will be x minus x-bar
quantity squared well we'll move a cursor so L3 is now highlighted this allows you to put in a formula
you will press second L2
and then the square button which looks like x squared this will take all of the L2 square each value
again notice that the formula appears at the bottom of the screen
when you press enter all of those numbers in L2 are squared and appear in L3
now back to our formula we've taken care of x minus x-bar then we squared it our last order of operation is the summation and
remember the summation acts as a grouping symbol as well telling us to do everything else first
and lastly to sum up those results
so now we need
the sum
of x minus x-bar squared
so we need the sum of list three
we will take the sum of list three and divide by n -1 and n is our sample size in our case we
have five numbers and five -1 will give us four so its be the sum of list three
divided by four
we again need to hit second quits to get out of any menu we may still be in
then second list to get to our sum
excuse me, second list
to get to the um list of menu over to math and then choose sum
second L3 to tell it to sum list three in to a close the parentheses and then press enter remember L3 will be above the
three key
that will give us 242
which needs to be divided by four
remember that we're not getting this number by itself we are plugging it back into the formula
and so will we take 242 in divided by four and press enter we get 60.5
and that is the sample variance we're still not done because it also wants us to find the standard deviation
and the easiest way to do that is to let the calculator do most of the work for us if we press second and then square root
and then instead of typing out which we could
if you go second and then ANS
and then close the parentheses that'll give you your previous answer
it shows up as a ANS but it takes your previous answer and then in this case will take the square root of that
and we'll get our answer as 7.778174593
so this a standard deviation is equal to the square root of the variance or the square root of 60.5
and depending on your teacher
is how you should round this Ms. Borlaug rounded to one more decimal place than her original data
however in each teacher has different rules for rounding so be sure to check with your instructor about how to proceed with rounding
so quick summary of our results
just so you can remember
we started out with our data
this is what your screen should look like
and these are the mean
the sample mean the sample variance and the sample standard deviation
that now we also have a very quick way to check the sample mean and the standard
sample standard deviation notice this does not check
the variance but if you get your sample standard deviation correct you should have your variance
your data should already be in list one you do have to complete that step
then you press second then quit and clear to clear every thing off
then press stat
calculate
choose one variables statistics which should already be highlighted
the enter there is to choose the one variables statistics
so if you press one you will not need to press enter
type in second L1
then you will press enter again
you will then get these results here notice x-bar is our sample mean and it gives is the 15 we got previously
and the symbol for standard deviation is s
so we'll look here and that matches up to our previous to answer as well
notice here the x just says that we did it on the x values
there are some other values that we could have done it all on this calculator
but we will not do those in this course so just the x there is just to remind you that you put that there we're looking for the
s for the sample standard deviation
I have reached the 15 minute mark so I'm going to conclude this video
I hope this was helpful if you need additional help be sure and stop by and see me in the
Math and Behavioural Social Science Building
in room 222 Thank you