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Mathematical Order of Operations document sample

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DEFINITIONS

Mathematical Order of Operations
Actual
Parentheses Exponent Multiplication Division Addition Subtraction
Mnemonic Device, or Memorizing Technique
Please Excuse My Dear Aunt Sally

Example: 3-1*2+2^(2/4)

Correlation                      = Measure of common relation between two ranges
Answers are between -1 and 1
Six combinations: HI NEG, MED NEG, LO NEG, LO POS, MED POS, HI POS

Variance                         = Used in determining standard deviation
Standard deviation               = Square root of variance
In normal dist                    67% of information is one SD before and after mean
95% of information is two SD's before and after mean

Exponential
Raised to the 2nd power           Squared               32
Raised to the 3rd power           Cubed                 33
Raised to the 4th power                                 34
…                                                 …
Raised to the nth power                                 3n

Raised to the 1 over 2 power     Square-root
Raised to the 1 over 3 power     Cube-root
Raised to the 1 over 4 power
…
Raised to the 1 over n power

PRODUCT                            Finds the product of a range 1,2,3,4,5
POWER                              Finds the power of a number 3 to the 2 power
SQRT                               Finds the squareroot         the square root of 3

AVERAGE FUNCTIONS
Mean                               Adds values in a range, and divides by the number of value
Median                             Puts values in the range in numerical order, and finds the middle value in the range --
if even number of values, finds the mean of the two middle values
Mode                               Most frequent value in a range

Why Mean, Median or Mode?
MEAN: most common; use if values are similar and continuous, and not skewed
EX: use to find average salary in a country where there are not many very rich people and not many very poor people
-- where there are about the same amount of people in every salary category
MEDIAN: useful if values are not similar, but continuous
EX: use to find average salary in a country where there are a few very rich and many poor people
MODE: most useful for \"qualitative\" measurements, non-continuous
EX: use to find favorite primary color (say, blue=1, red=2, yellow=3)

GEOMEAN                          = Multiplies "n" numbers in a range, and takes the nth root
Used with non-negative, non-zero numbers; usually, percentages (such as Annual
Rate of Growth)
MEAN V. GEOMEAN?          From http://www.math.utoronto.ca/mathnet/questionCorner/geomean.html
Mean                     If all the numbers in the range were the same, what would the value (average) have
to be in order to achieve the same total ?

Geomean                  If all the numbers in the range were the same, what would the value (average) have
to be in order to achieve the same product ?

LN                      = Natural Log: What root of n gives e?
EXP                     = Exponent of e: e raised to n gives what answer?
LOG                     = Log: What root of n gives a number? Number is assumed to be 10 if missing.

Annual Rate of Growth
Simple
Using natural logs
Long-range
IN EXCEL

NEG, LO POS, MED POS, HI POS

, and finds the middle value in the range --
the two middle values

ich people and not many very poor people
ry

and many poor people

; usually, percentages (such as Annual
questionCorner/geomean.html
me, what would the value (average) have

me, what would the value (average) have
?

er is assumed to be 10 if missing.
PrtScr: The whole screen
Alt-PrtScr: The active window
NUMBER1          1   98%
NUMBER2          2   78%
NUMBER3          3   57%
NUMBER4          4   79%
NUMBER5          5   82%
NUMBER6          6   68%
NUMBER7          7   55%
NUMBER8          8   42%
NUMBER9          9   95%
NUMBER10        10   40%
COUNT
PRODUCT
MEAN
GEOMEAN

GEOMEAN,
using PRODUCT
and COUNT
MEDIAN
MODE
SQRT
e                             =                 2.7183

LN                        e raised to x gives n? -- we are given "e" and "n", we find "x"
e^x=n
To find: e^x=2.718282                                                     [same as log(2.718282,2.718282)]
Question: What is the natural log of 2.718282?

To find: e^x=7.389056                                                        [same as log(7.389056,e)]
Question: What is the natural log of 7.389056?

EXP                       e raised to n gives x? -- we are given "e" and "n", we find "x"
e^n=x
To find: e^1=x
Question: What is the exponential of 1?

To find: e^2=x
Question: What is the exponential of 2?

EXP(LN(N)=N
Question: What is the exponential of the natural log of 99?

LN(EXP(N)=N
Question: What is the natural log of the exponential of 99?

LOG                           y raised to x gives n? -- we are given "y" and "n", we find "x" -- y is assumed to be 10 if missing.
y^x=n
QUESTION                      FORMULA
What is the logarithm
(log) of 8, with base 2?      2^x=8
What is the log of 16, with
base 2?                       2^X=16
What is the log of 27, with
base 3?                       3^X=27
What is the log of e, with
base e?                       e^X=2.718282
[same as ln(e)]

What is the log of 10?        10^x=10
What is the log of 10, with
base 10?                      same
What is the log of 3?         10^x=3
What is the log of 3, with
base 10?                      same

What is the log of:                        1
10
100
1000
same as log(2.718282,2.718282)]

same as log(7.389056,e)]

n", we find "x" -- y is assumed to be 10 if missing.
SIMPLE                USING NATURAL LOGS
Annual Rate                  Annual Rate
of Growth-  Annual Rate      of Growth-  Annual Rate
Years     GDP      C               GDP         of Growth-C      GDP         of Growth-C
1999     74.7           51
2000     76.3          51.6          2.14%
76.5          51.1          0.26%
83.6          57.8          9.28%
84.4          58.9          0.96%
89.4          60.9          5.92%
96.6           66           8.05%
91.3          62.5         -5.49%
Mean Growth                               3.02%
Median Growth                             2.14%
Mode Growth                                #N/A
Geomean Growth                           #NUM!

But, suppose we only have data for years 1999 and 2006 (T1 and T2)?
Or, suppose we are interested in long-term annual growth, not each year, and we don't want to use simple or ln?

Long-range Annual Rate of Growth Formula
GDP                                                                    C
ValueT2 =           ValueT1 * (1 + g)^(T2-T1)                          ValueT2        =
want to use simple or ln?

ValueT1 * (1 + g)^(T2-T1)

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