In our adventure learning statistics, we need to know how to use statistical
software. Of course, learning something new is easier when you get to practice
the skill on a task, so I'm going to begin explaining the usefulness of
statistical software with the most basic task of statistics - summation. Summation
means to add up a set of numbers. Summation is the first step in many
statistical calculations. When you have numbers in a data set, you could add them, or
subtract them, multiply, or square them or divide them, and then you can add them
all up. Calculating the sum of a set of numbers is called summation, and we will
begin doing summation by hand then learn to do it much more easily with
statistical software. The symbol for summation is the Greek letter Sigma. When
Sigma is followed by a variable name such as X, it means to add up all of the
X values. Now the full form of this notation is read sum all of the X sub
i's from i equals 1 to i equals n. This mathematical notation gives us exquisite
ability to specify which values in a data set will be included in the
summation, but it is really more than we need right now. So because we are just
learning statistics, I'm going to put on the training wheels and simplify the
equation. The full equation will be useful in a mathematically based
statistics class, but we will use only the parts that we need and simplify this
equation to the sum of X. Now just so you know, the symbol for multiplication is
the Greek letter capital pi. PI X would mean to multiply each of the X values in
the data set instead of adding them. We won't be using this notation but it's
still handy to know about. Sometimes, values will be squared or added
before being summed, sometimes afterward. So let me clarify what I mean by the sum
of X. Here is a small data set consisting of the numbers two, four, and eight. The
sum of two, four, eight is 14. Now of course saying some of 2, 4, 8 is quite a
mouthful, and as our data set got larger it would be very difficult to do. So,
let's make it simpler. We'll put all of these numbers into a table. We will
collectively call the numbers in the column X, and let each individual number
be an x value. We can now say that the sum of X is 14. If we were to add new
numbers to the column, we could still call the sum the sum of X. This is much
easier to talk about than listing every number. Ok now let's get really fancy,
let's create a whole new column based upon our X values. We will call this
column x squared. This column comprises all of the X values after squaring them.
To create this column, square each x value, and record the result in the new
column, then add up the x squared values. We can now say that the sum of x squared
is 84. We can go on creating as many columns as we like each time doing
something to the X values, and using the result to populate a new column. We could
multiply each x value by a corresponding Y value with y being a new column of
numbers related to X. We could subtract the mean of X from each of the X scores,
we could even create a sum of X first and then later square that value. So
let's practice. Using the provided x and y values, create four new columns and
then summarize each one. I have already started the x squared column I took the
x value 10 squared it 100 and then I wrote 100 in the x squared column. Here
is how the rest of the column would look 15 squared is 225, 12
squared is 144, 9 squared is 81, and 10 squared again is 100. Now you try.
Complete the y squared column, stop the video, do the math and then come right
back and check your work. The Y squared values are 9, 16, 1, 1, and 9. The next column
is called X minus y. Any idea how you will create it? You will subtract each Y
value from the corresponding x value. We always work across rows, so don't
subtract the one from the 10 because the math is easier work across columns. So 10
minus 3 is 7. 15 minus 4 is 11. 12 minus 1 is 11. 9 minus 1 is 8, and 10 minus 3 is 7.
Now that leaves the XY column. Whenever we see two variable names next to each
other with no mathematical notation, we multiply. To create this column, we
multiply each x value by its corresponding Y value. Stop the video,
finish the x times y column, and then come right back and check your work.
The values for x times y are 30, 60, 12, nine, and thirty. Our next step will be to
summarize or add up each column. Here, I'm going to use some notation so you can
become familiar with it. The first value that we are going to create is the sum
of X. This means you add up all the numbers in the X column. 10 plus 15 plus
12 plus 9 plus 10 equals 56. That is the sum of X. Can you calculate the sum of Y?
It's 12. how about the sum of x squared? Add up all of the values in the x
squared column to get 650. The sum of Y squared is 36. The sum of X minus y is 44,
and the sum of XY is 141. Once we have the sum of each column, we can also work
some math on the sums. We could square the sum of X. This is called the sum of X
quantity squared (to distinguish it from the sum of x squared). The sum of X we
already know is 56, so 56 squared is 3136. Can you figure out how to calculate
the sum of Y quantity squared? 36 squared is 144. Now do the sum of X minus y
quantity squared. It is 1936. And finally, calculate the sum of x times the sum of
Y. 56 times 12 is 672. You now know how to do summation but we can make summation
easier. I'm going to introduce you to the Excel spreadsheet software, and then
teach you how to simplify the summation math using formulas and functions in
Excel.