x in place of µ in formulas.

Transcription

1 Algebra Notes SOL A.9 Statstcal Varato Mrs. Greser Name: Date: Block: Statstcal Varato Notato/Term Descrpto Example/Notes populato A etre set of data about whch we wsh to ga formato. The heght of every mddle school studet the U.S. sample A subset of data from a populato. We use samples to make fereces about populatos. The heghts of studets our class. The th elemet of data a sample I data set {, 4, 5, 0}, x = 4 x or populato. mea of a populato Represeted by symbol µ (Greek letter mu ). For our purposes, we wll use x place of µ formulas. mea of a sample Represeted by x (x bar) Symbol our calculator uses. devato Σ mea absolute devato varace stadard devato z-score The dstace a data pot s from the mea of the data. Ca be postve, egatve, or zero. The Greek uppercase symbol Σ (sgma) s used to dcate summato. The terms sde the sgma are added together based o a dexg scheme surroudg the Σ Mea of absolute values of the devatos of elemets from the mea of a data set. The average of the squared devatos from the mea. Varace s represeted by the lower case symbol sgma squared (σ ) The square root of the varace, deoted by σ. Also called stadard score; measure of posto that determes the umber of stadard devatos a elemet s above or below the mea of a data set. Fd by calculatg x - µ or x x 5 = ; 4 x = x 3 4 mea absolute devato = MAD = varace = x ( x µ ) µ σ = stadard devato = σ = ( x µ ) z-score = z = x µ σ

2 Algebra Notes SOL A.9 Statstcal Varato Mrs. Greser Page Populato vs. Sample populato: a well-defed group of objects about whch we wsh to ga formato. Examples: the heghts of every U.S. Presdet, the ages of every math teacher VA. sample: subset of a populato. It s ofte very o dffcult to obta data for every member of a populato o samples obtaed radomly or by other methods; may or may ot be based Dfferet otatos are used f referrg to samples or populatos. o Ex: µ: mea of populato; x : mea of sample o WE WILL ALWAYS USE POPULATION NOTATION AND FORMULAS We refer to a specfc elemet, the th elemet, of a sample or populato as x. Measures of Spread or Varablty Measures of varablty attempt to descrbe the spread of a set of data. Some measures of varablty we kow: rage ad terquartle rage. New: average dstace from the mea also measures spread o Example: ages of chldre a famly (take your ow otes o ths!) o A small average dstace from the mea dcates data that s close together (alke) o A large average dstace from the mea dcates data that s farther apart (less alke) Mea Absolute Devato (MAD): average of dstaces from the mea o Devato s the dstace a pot s from the mea: o ca be postve, egatve, or zero o foud by subtractg the mea (µ) from the data pot value (x): x - µ. o = the umber of pots a data set o Absolute Devato s the absolute value of devato: x - µ o We eed to take absolute values of devatos so that the average s t zero Example : S = {, 3, 4, 6, 7, 9} µ = 5 (add the umbers ad dvde by 6) Devato = x - µ for each x x x - µ 5 = = = = 7 5 = 9 5 = 4 x - µ MAD

3 Algebra Notes SOL A.9 Statstcal Varato Mrs. Greser Page 3 Example : Fd the MAD for the dataset the table Mea (µ) = Dstaces: MAD You Try: Fd the MAD for the dataset S = {, 9, 40, 50, 60, 65, 95} Mea (µ) = (umber of pots) = Dstaces: MAD Summato Notato The Greek uppercase letter Σ (sgma) s used to deote summato (addg). Examples: 5 = = 5 x = x x for S = {4 8 0 } Example above Traverses data = umber of data pots Subscrpt otato s postoal; NOT AN EXPONENT!! x refers to the th elemet Example: x3 refers to the thrd elemet; above t refers to the value 8 Above, the summato s askg as to sum all the data the data set. Evaluate the summato. You try: Evaluate the summatos 4 a) 3 b) 5 c) ( + ) d) x for { } e) Rewrte the formula for mea usg Σ otato. Re-wrte the MAD formula usg summato otato: MAD = sum of x - µ becomes Mea Absolute Devato (MAD) = x µ

4 Algebra Notes SOL A.9 Statstcal Varato Mrs. Greser Page 4 Stadard Devato Aother way to esure that devatos are always postve s to square them. We ca sum the squares of the devatos ad dvde by the umber of devatos to get varace, deoted by σ : x µ ( ) Varace = σ = The square root of varace s a closer approxmato of the devato of pots from the ceter. We call the square root of varace, σ, stadard devato. Stadard Devato = σ = ( x µ ) Example revsted: S = {, 3, 4, 6, 7, 9}, µ = 5. Fd the varace ad stadard devato are as follows: x x - µ 5 = = = = 7 5 = 9 5 = 4 (x - µ ) varace = σ = stadard devato = σ = (square root of varace) You try: Fd the MAD, varace, ad stadard devato of S = {,, 3, 4, 5} Example: Fd the stadard devato for the ages of kds a famly {0, 0,, 4, 5, 0}, µ = (approxmately) x x - µ (x - µ ) varace = σ = stadard devato = σ = We prevously calculated the MAD for ths data: MAD = 4.7 How do the MAD ad stadard devatos for the data compare? Why do you thk ths s so? Usg the Calculator It s a lot of work to fd these descrptve statstcs by had! Fortuately, the calculator comes to our rescue. Use the calculator gude hadout to fd the mea absolute devato (MAD), varace, ad stadard devato of the followg data sets: a) {0, 0, 30, 40, 50, 60} b) {0,, 7,, 0, 7, 6, 8, 9} c) {0, 9, 0, 04, 0, 49}

5 Algebra Notes SOL A.9 Statstcal Varato Mrs. Greser Page 5 Z-Scores (Stadard Scores) We use stadard devatos to assess average dstace from the mea for a set of data. A z-score tells us how may stadard devatos a elemet s above or below the mea of the data set. Stadard Score = z-score = z = x µ σ postve z-score: the pot s that may stadard devatos above the mea egatve z-score: the pot s that may stadard devatos below the mea Example: A set of values has a mea of 85 ad a stadard devato of 6. Fd the z-score of the value 76. x µ z-score = = = =.5 σ 6 6 Ths tells us that 76 s.5 stadard devatos below the mea. Example: The followg umbers are metabolc rates of 7 members who took part a study of detg (uts are calores per 4 hours - these are the same calores used to descrbe the eergy cotet of foods) How may stadard devatos above or below the mea s 460 calores? µ = σ = z-score How may pots above are wth stadard devatos from the mea? How may pots are wth.5 stadard devatos from the mea? Example: Joh weghs 0 lbs; hs dog Fdo weghs 90 lbs. If huma males wegh a average of 60 lbs wth a stadard devato of 0 lbs, ad all dogs of Fdo s breed have a average weght of 80 lbs wth a stadard devato of 5 lbs, how do Joh ad Fdo compare, relatve to ther populatos, wth respect to weght? Joh s z-score = Fdo s z-score = Cocluso?

6 Algebra Notes SOL A.9 Statstcal Varato Mrs. Greser Page 6 Example: Gve a set of data wth a stadard devato of 3.0 ad a mea of 9.0, what would the value of a elemet be f ts z-score s.5? You try: a) Fd the z-score for a data value of 5 f a set of data has a mea of 75, ad a stadard devato of 5. b) How may stadard devatos from the mea s 80 f a set of data has a mea of 70 ad a stadard devato of.5? c) Fd the value of a elemet a dataset wth a mea of 8, a stadard devato of, ad a z-score of.5. d) Test A has a mea of 50 ad σ = 0. Test B has a mea of 0 ad σ = 5. Whch s better: a score of 65 o test A or a score of 9 o test B?