Means, variations and the effect
of adding and multiplying
Textbook, pp. 373-74
Adding and multiplying
• The text is on p. 373 is not
tremendously clear on what’s
happening here. Let’s see if we can
clarify by giving an example.
• You will need to use you TI 83 or 84
calculator. Detailed instructions are
presented on the slides. Make sure
that your answer matches those on
the display.
Outline
• We are going to add, subtract, and multiply a
series of six numbers.
• We will calculate the mean, standard deviation,
and variance of the series and compare the
results.
• Hopefully this will make the text material
clearer.
• The series is 4 6 7 8 10 13
• Starting on the next slide, we will enter this
series and then start manipulating it. The skills
in using the calculator are important for further
work in this course.
• As always, this PowerPoint is available on the
Garfield web site.
Quick review
(in case you’ve forgotten about lists!)
• Press the STATS button,
and you should get
something like this
• Press ENTER to get the
list
• Enter the data, pressing
ENTER between each
item, and each should
appear in the list
Add, subtract, and multiply X using
the list function of the TI 83/84
• L1 now contains 6 numbers that we’ll
use to compare mean (μ), standard
deviations (σ) and variance (var)
• First, let’s manipulate X (the
numbers you just entered) to show
what happens when we add to,
subtract from, or multiply a random
variable
Finding and clearing what’s already in
L2
1. Highlight L2 atop the
column using the cursor
keys. I’ve left some junk
in here to show you how
to clear it out.
2. Next, press the CLEAR
key to eliminate what’s
in L2. You won’t see it
disappear immediately,
but the cursor moves to
the bottom and is ready
to receive data.
Reminder on how to enter L1, L2, L3
etc. into formulas on the TI 83/84
TITI83+
TI
83+
83+
nd” key
•
Press
the
“2
TITI84+
TI 84+
84+
(yellow on TI 83, blue
on TI 84)
• While depressing the
“2nd” key, depress the
“1” key on the keypad.
• This gives you L1. To
get L2, L3, L4, etc.,
depress the 2, 3, or 4
key, respectively
L2: how to add 6
i.e., L2=X+6
• You can add 6 to an
existing list easily.
• Insert L1 by following
the instructions on the
previous slide
• Now, enter “+6” after
the inserted L1 and
you’re just about ready
to go
Getting to L2=L1+3
(in stat-talk, Y=X+6)
• Press the ENTER key
and you should see
everything in L2
replaced
• Check these values
against L1, and each
should be 6 more than
its corresponding entry
in L1 (green double
arrows)
L3: subtracting 3
i.e., X-3
• Let’s make List 3 (L3) three
less than L1, i.e., each value
in L3 is 3 less than the
corresponding entry in L1
• How to do: enter L3=L1-3
(repeat what we just did
with L2, except on L3)
• You should end up with
something that looks like
the bottom screen on the
left, with each entry being 3
less than the corresponding
data in L1
L4: multiply by 3
i.e.,3X
• Let’s make List 4 (L4)
three times L1; each value
in L3 is 3 times as large as
the corresponding entry
in L1
• Clear L4 as before and
enter L4=3*L1, and press
ENTER
• Here’s a comparison of L4
with L1, except I copied
L1 into L5. L4 should be 3
times larger than the
entries in L5
Now, to calculate μ, σ, and var
(mean, standard deviation, and variance)
• Push the STAT button to
get the Statistics menu
• Use the cursor buttons
to select the CALC
menu at the top
• As soon as you move
the cursor to CALC, you
will see the following
menu
Calculating statistics variables
• Press the ENTER key and
you will get a blank
screen like this
• Enter L1 to determine the
which list you want the
statistics on (press 2nd,
followed by the 1 key on
the keypad as entered
earlier
• Now press ENTER, and
we’ll see what we get on
the next slide
Output from 1-VAR STATS
• After pressing ENTER,
your screen should look
like this
• μ=8, which the TI lists
as x
• So where’s the VAR?
• Even if we scroll down,
we only find median,
max, min and IQR data
Relation between standard
deviation (σ) and variance (var)
• VAR = σ2
• For TI calculators, the standard
deviation calculated by 1-VAR STATS
is the Sx variable
• Here, σ = Sx = 3.16227766
• σ2 in this cases therefore equals 10
(3.1622772)
• (Note that 10 is therefore a good
approximation of π)
What next?
• We are going to be
making a table to
compare for each of L1,
L2, L3, and L4
• You need to set up a table
that looks like the one to
the right, with
appropriate labels
• Make a table like this one
in your notes, and we’ll
get the relevant statistics
for you to fill in
X
μ
σ
VAR
X+6
X-3
3X
Running 1-VAR STATS on L2
Do the same as we did on
L1
•
•
•
Select 1-VAR STATS
after going to STAT and
CALC
Press ENTER while
outlining 1-VAR STATS
Add “L2” in the blank
screen (press “2nd” key
and “2” on the keypad
simultaneously)
Results for running 1-VAR STATS on L2
• And you should get the
equivalent of what we got
before (see screen at left).
• Enter the data for μ, σ
(Sx), and VAR in the table
you created in your notes
• Remember to calculate
VAR by squaring σ (Sx in
the TI)
• (Notice what has changed
and what hasn’t)
Results for L3 and L4
• Repeat for L3 and L4.
• You should get screens
like this for L3....
• ….and like this for L4
• Be sure to copy all the
relevant data into the
table, and don’t forget
to square Sx to get VAR
Comparison of results of calculations
on L1, L2, L3, and L4
• What stays the same
as X is reduced or
increased by
addition or
subtraction?
• What happens when
we multiply? (e.g.,
L4)
• How does this
correlate with what
you read in the text?
μ
σ
VAR
(σ2)
X
(L1)
X+6
(L2)
X-3
(L3)
3X
(L4)
8
14
5
24
3.16.. 3.16..3.16.. 9.48..
10
10
10
90
Explanation of calculations
on L1, L2, L3, and L4
• Increasing or
decreasing X does
not change σ
• Multiplying X by 3
triples σ
• Multiplying X by 3
increases the VAR
by 9, or 32
(variance increases
by the square of
multiplier)
μ
σ
VAR
(σ2)
X
(L1)
X+6
(L2)
X-3
(L3)
3X
(L4)
8
14
5
24
3.16.. 3.16..3.16.. 9.48..
10
10
10
90
Summary of adding and multiplying X,
means, standard deviation, and variances
• E(X±c)=E(X)±c
X
(L1)
X+6
(L2)
X-3
(L3)
3X
(L4)
8
14
5
24
• Var(X±c)= Var(X)
μ
• E(aX)=aE(X)
σ
3.16.. 3.16..3.16.. 9.48..
• Var(aX)=a2Var(X)
VAR
(σ2)
10
10
10
90
Let’s put the equations into words
before we move on
1. E(X±c)=E(X)±c
2. Var(X±c)= Var(X)
1. If we change a random
variable simply by
adding or subtracting a
constant, we simply
add the constant to
the mean of X
2. Having changed a
random variable in this
way, the variance
DOES NOT CHANGE
Let’s put the equations into words
before we move on
3. E(aX)=aE(X)
3. If we change a random
variable by multiplying it
by a constant, the new
mean is simply the
constant times the original
mean before
multiplication
4. Var(aX)=a2Var(X)
4.
When we change a
random variable by
multipying by a constant,
the variance increases by
the square of the constant
So why do we even care?
Answer: to save ourselves a LOT of work
• If we add or subtract from X, we don’t have to
recalculate ANYTHING.
– The variance doesn’t change (nor does the
standard deviation, since σ=VAR½
– The mean increases by whatever we added or
subtracted
• If we multiply X by a constant a, the new mean
is just a times the original mean, and the VAR
is now the old VAR times the square of the
constant a (i.e., multiply VAR by a2)
• THIS MEANS WE DON’T HAVE TO DO ANY
LABORIOUS CALCULATIONS!
</end of slide show>