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Mathematical Order of Operations document sample
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)