BASIC STATISTICS. Discrete. Mass Probability Function: P(X=x i ) Only one finite set of values is considered {x 1, x 2,...} Prob. t = 1.

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1 BASIC STATISTICS 1.) Basic Cocepts: Statistics: is a sciece that aalyzes iformatio variables (for istace, populatio age, height of a basketball team, the temperatures of summer moths, etc.) ad attempts to extract coclusios based o the behavior of these variables. Statistics is oe of the scieces that allow us to kow, or at least to uderstad, the reality i which we live. Through statistics, we ca obtai very valuable iformatio that will help us to make decisios regardig ay aspect of our life. The purpose of statistics is to aalyze past iformatio to help us make decisios for the future Radom variable: A set of the differet umerical values that adopt a quatitative character. It is the piece of data susceptible to acquire differet values i differet ad specific circumstaces. Statistics is the quatitative study of variables, therefore, these values may be cosidered as the raw material for statistic studies. Ay variable that has a specific probability law associated; each of the values that may take has a correspodig specific probability. The variables may be qualitative or quatitative. Qualitative variables (or categorical): those variables that are ot i umerical form, but appear as categories or attributes (geder, professio, eye color). Quatitative variables: those variables that may be expressed umerically (temperature, salary, umbers of goals i a soccer match, etc.). Quatitative variables may be defied, accordig to the type of values that represet as: Discrete: Those values that represet isolated values (atural umbers) ad that caot take ay itermediate value betwee two established cosecutive values. For istace; umber of goals, umber of childre, umber of bought records, umber of heartbeats... Discrete 1-Prob Quiebra Oly oe fiite set of values is cosidered {x 1, x,...} Mass Probability Fuctio: P(X=x i ) Prob t = 0 t = 1 No 1

2 Cotiuous: Those values that represet ifiite values (real umbers) i a give iterval, so that they ca represet ay itermediate value, i theory at least, i their rage of variatio. For istace; size, weight, blood pressure, temperature... Cotiuous Ay value of a iterval may be cosidered Desity fuctio f(x): F(x)= f(x) 0 x - f ( t) dt f(t)dt =1 Statistic Distributio Possible Values Frequecy: Number of times a datum is repeated. There are two types of frequecies: Absolute frequecy: the absolute frequecy of a statistical variable is the umber of times that value of the variable appears i the sample. Relative frequecy: Absolute frequecy is a measure iflueced by the size of the simple. Icreasig the sample size also icreases the absolute frequecy. This correlatio makes it a measure ot useful to compare. That is why it is ecessary to itroduce the cocept of relative frequecy, or the quotiet obtaied dividig the absolute frequecy over the sample size. The followig cocepts have to be cosidered whe studyig the behavior of a variable: (Compoets of a Statistical Study) Populatio: is the set of all the elemets that possess certai properties ad are the elemets desired to study a particular pheomeo (homes, umber of screws maufactured yearly i a plat, flippig a coi, etc.). Statistic populatio or uiverse is the referece set used to make the observatios. Idividual: a statistical uit, or idividual, is each of the elemets that make up the statistic populatio. The idividual is a observable etity that does ot have to be a perso. It ca be a object, a livig beig ad eve a abstract cocept.

3 Sample: is the populatio subset uder study used to extract coclusios regardig the characteristics of the populatio. The sample must be represetative, i the sese that the coclusios obtaied from it must be applicable to the etire populatio. Samples ca be probabilistic or o probabilistic. A probabilistic sample is chose by meas of mathematical rules, ad therefore the probability of selectig each of the uits is kow i advace. A o probabilistic sample is ot ruled by mathematical probability rules, ad therefore, while it is possible to calculate the size of the sample error whe workig with probabilistic samples, it is ot possible to do so with the o probabilistic samples. The more basic probabilistic sample is the simple radom simple, i which all the compoets or uits of the populatio have the same opportuities to be selected. Cesus: We say we are coductig a cesus whe we are observig all the elemets that make up the statistic populatio. Parameter: is a characteristic of a populatio, summarized for its study. It is cosidered a true value of the characteristic uder study. 3

4 Probability: Is the set of possibilities that a evet occurs or ot at a give time. These evets may be measurable i a scale from 0 to 1 (the scale ca also be expressed i percetages ragig from 0% ad 100%), where the evet that caot occur has a assiged probability of 0 ad a evet that ca occur with certaity has a assiged probability of 1, ad the remaiig evets will have assiged probabilities betwee cero ad oe, that will be greater the greater the probability of occurrece is. Example: Whe flippig a coi we wish to kow what is the probability of it fallig as heads or as tails, that is, there is a 0,5 (50%) of it beig heads or of 0,5 (50%) beig tails. The experimet must be radom, that is, several results may occur withi a possible set of solutios, ad this must be true eve whe doig the experimet uder the same coditios. Therefore, we do ot kow a priori which evet will occur. Example: Christmas lottery. There are experimets that are ot radom ad therefore the laws of probability caot be applied to them. Probability distributio model: specificatio of the values of the radom variable with their respective probabilities..) Measures of radom variables Oftetimes it is more expeditious, easy ad precise, to study a variable usig umerical values tha the visual descriptio of a variable by meas of tables ad graphics, sice umerical values give us a idea of the locatio or of the ceter of the data (positio measures), ad usig quatities that iform us about the cocetratio of the observatios aroud said ceter (dispersio or variability measures) a) Measures of cetral tedecy: Carry iformatio about the middle values of the data series. A cetral tedecy measure is a value represetative of a set of data ad that teds to be positioed, accordig to its magitude, i the ceter of the data set a Mea: is the weighted average value of the data set of values that the statistical value represets. The mea is the sum of all the variables divided over the total umber of available data. The mea is calculated utilizig the followig formula; 4

5 x1 + x + x x X = 1 + x = i= 1 x i If the xi value of the X variable is repeated a i umber of times, this is expressed i the arithmetic mea formula as: x i i X = Where x i are the variables, i the times the variable x i appears, ad N the sum of all the i. That is; N = Σ i The arithmetic mea is also called the distributio s ceter of gravity. Media: Is oe of the most represetative calculatios of the sample. The media is the value of the itermediate elemet oce all the elemets have bee ordered. The media is calculated orderig the data i icreasig order ad takig the value positioed i the middle, that is, the value that has 50% of observatios o the left ad 50% o the right. Its locatio is established dividig the umber of values by : Whe there is a odd umber of values for the variable, the media will be, precisely, the cetral value, the value whose cumulative absolute frequecy coicides with the expressio. Therefore the media coicides with oe value of the variable. The problem arises whe there is a eve umber of values for the variable. If the result of is a value lower tha the cumulative absolute frequecy, the value of the media will be the variable which absolute frequecy fulfils the followig coditio: N i 1 < N i Me = x i. 5

6 N However, if the value is: = N i, to obtai the media we will have to use x + +1 the followig formula: = i x Me i Leged: media Mode: Is the most frequet value of the statistical variable ad the highest value of the histogram. Example: The mode for the set,,5,7,9,9,9,10,10,11,1 ad 18 is = 9. Example: The set 3,5,8,10,1,15 ad 16 does ot a mode. Example: The set,3,4,4,4,5,5,7,7,7 ad 9 has two modes, 4 ad 7. It is a bimodal set. A distributio with oe sole mode is called uimodal. Leged: Mode, Media, Mea 6

7 b) Other measures: These measures carry iformatio regardig the maer of distributio of the remaiig values of the data series. Los Quatiles (quartiles, deciles, percetiles) are localizatio measures. They carry iformatio regardig the value of the variable that will occupy the positio (expressed as a percetage) we are calculatig withi the whole set of variables. We ca say that the quatiles are positioig measures that divide the distributio i a give umber of parts so that each of them cotais the same value of the variable. The more importat are: QUARTILES, divide the distributio i 4 equal parts (three divisios). Q 1,Q,Q 3, correspodig to 5%, 50%,75%. DECILEES, divide the distributio i 10 equal parts (9 divisios). D 1,...,D 9, correspodig to 10%,...,90% PERCENTILEES, whe they divided the distributio i 100 parts (99 divisios). P 1,...,P 99, that correspod to 1%,...,99%. There is a value i which the quartiles, the deciles ad the percetiles coicide whe they are equal to the Media, such as: 4 5 = = 100 Quartiles: The quartiles are the three values that dived the ordered data set i four, percetually equal, parts. There are three quartiles, usually represeted by Q1, Q, Q3: The first quartile, Q1, has the lowest value that is greater tha a oe fourth of the data; that is, the variable s value that is greater tha 5% of the observatios ad is smaller tha the 75% of the observatios The secod quartile, Q, (that coicides, it is idetical or similar to the media, Q = Md), is the lowest value that is greater tha half of the data, that is 50% of the observatios have a greater value tha the media ad 50% have a lower value. The third quartile, Q3, has the lowest value that is greater tha three fourths of the data, that is, the value of the variable that has a value greater 75% of the observatios ad of a lower value tha 5% of the observatios. 7

8 Deciles: The deciles are ie umbers that divide the successio of ordered data i te, percetually equal, parts. They are also a particular case of percetiles, sice a decile ca be defied as a percetile i which the value that idicates its proportio is a multiple of te. Percetile 10 is the first decile; percetile 0 is the secod decile, etc. The first decile D1: idicates there is oly a 10% probability for the variable s value to be below said figure. The fifth decile D5, also called Base Case, also idicates there is 50% probability for the value to be above as for the value to be below this figure. It represets the Media of the distributio. Percetiles or cetiles: The percetiles are, perhaps, the most utilized measures for locatio or classificatio purposes (i the case of people whe the characteristics are weight, height, etc.). The percetiles are umbers that divided the successio of ordered data i oe hudred, percetually equal, parts. These are the 99 values that divided i oe hudred equal parts the set of ordered data. The Percetile is, simple, the value of the trajectory of a variable, that ecompasses a specific proportio of the populatio. The percetiles (P1, P,... P99), read as first percetile,..., percetile 99, show the variable that leaves behid a cumulative frequecy equal to the percetile s value: The first percetile is greater tha oe percet of the values ad lower tha the remaiig iety-ie. The percetile 60 is the value of the variable that is greater tha 60% of the observatios ad lower tha 40% of the observatios. The 99 percetile is greater tha 99% of the data set ad is lower tha the remaiig 1%. c) Dispersio measures: Those measures that allow us to relate the distace of the variable s values to a give cetral value, or that allow us to idetify the cocetratio of data i certai sector of the trajectory of the variable. They study the distributio of the values of the series, aalyzig if said values are more or less cocetrated or more or less disperse. Rage: Measures the amplitude of the sample s values. It is calculated as the differece betwee the highest ad the lowest value. R e = x max - x mi 8

9 Variace: Measures the distace betwee the values i the series ad the mea. It is the sum of the square of the differeces betwee each value ad the mea, multiplied by the umber of times each value has repeated. The result obtaied is the divided over the sample size. S x = σ x = r i= 1 ( x x) i N i The variace will always be greater tha zero. The closest it is to zero the more cocetrated are the values of the data series aroud the mea. The greater the variace, the more dispersed the values. Stadard deviatio: is the square root of the variace. It expresses the dispersio of the distributio ad it is expressed i the same uits of measuremet as the variable. The stadard deviatio is the most utilized measure of dispersio i statistics. σ = std ( X ) = + var( X ) 3.) Distributios of probability As metioed before, a radom variable is the variable that ca represet differet values, or set of values, with differet probabilities. Radom variables have importat characteristics: its values ad the probabilities associated to these values. A table, graphic, or mathematical expressio that shows the probabilities each radom variable has of adoptig differet values is called a probability distributio of the radom variable. The statistical iferece (that is, the process doe by the Riskmeter ) relates to the coclusios that may be extracted from a populatio of observatios based o a observatio simple; i this case we wish to kow somethig about a distributio based o a radom sample of said distributio. I this maer we see we are workig with radom samples of a populatio that is larger tha the obtaied simple; said isolated radom simple is othig more tha oe of the may differet samples that could have bee obtaied through the selectio process. That is why usig the distributios of probability has such relevace. 9

10 Discrete distributios: Are those distributios i which the variable ca adopt a specific umber of values. The most oteworthy distributios amogst the existig oes are: Berouilli; the model followed by a experimet that is doe oly oce ad ca have two solutios: true or false: Whe the solutio is true (success) the variable equals 1 Whe the solutio is false (failure) the variable equals 0 Because there are oly two possible solutios they are complemetary evets: The probability of success is called "p" The probability of failure is called "q" Whe: p + q = 1 The Berouilli distributio is the applied to experimets that are doe oe time oly ad have two possible results, failure or success, ad hece the variable ca oly have two values: 1 or 0. Example: flippig a coi. Biomial; the biomial distributio is based o the Berouilli distributio. It is applied whe the Berouilli experimet is doe a "" umber of times, each of the assays beig idepedet from the previous oe. The variable the ca adopt values betwee: 0: if all the experimets have bee failures : if all the experimets have bee successes Example: flippig a coi repeatedly. Poisso; the Poisso distributio is based o the biomial distributio. The Poisso distributio is applied i the cases whe usig a biomial distributio the experimet is doe a high "" umber of times ad the probability of success "p" per assay is low. The followig coditio must be met: " p " < 0,10 " p * " < 10 Example: umber of errata per page i a book 10

11 Cotiuous distributios: Are those that preset a ifiite umber of possible solutios. Types of distributios: Uiform; a distributio that may adopt ay value withi a iterval (all the values have the same probability). 0,5 0,0 0,15 0,10 0,05 0, ,0% -,50,50 Characteristics: All the possible values the variable may adopt, located betwee the maximum ad miimum quatities, preset the same possibilities of beig reached. The etrepreeur idetifies a value rage for the variables. Exogeous variables. Fuctio parameters the etrepreeur ca idetify ad quatify. Normal; It is used to measure ad represet may variables such as weight, height, exam scores..., i which the distributio is symmetrical from a cetral value, aroud which it takes values with a great probability of existig with hardly ay extreme values. It is the most used distributio model. The importace of the ormal distributio is maily due to the may variables associated to atural pheomea that follow the ormal distributio model (sizes, weights, breadth, cosumptio of a give product, exam scores, degree of adaptatio to a eviromet, etc.). This distributio is also characterized by the arragemet of the values i a bell shape called Gaussia distributio, aroud a cetral value that coicides with the middle value of the distributio. Of the total values, 50% are located to the right of this cetral value ad the other 50% are to the left. 11

12 This distributio is defied by two parameters: X: N ( ) represets the mea value of the distributio ad it is precisely the value at the ceter of the curve (i the Gaussia distributio). : is the variace. It idicates if the values are more or less ear the cetral value: a low variace value idicates the values are close to the mea; if it is high it idicates the values are very dispersed. Whe the distributio mea equals 0 ad the variace equals 1 is called a "stadard ormal distributio". The advatage of usig this distributio is that there are tables with the cumulative probability for each poit of the curve of this distributio. Characteristics: Pre-established miimum Pre-established maximum All values betwee the miimum ad maximum values of the distributio are equally probable. Triagular; the triagular distributio is useful for a iitial approximatio i situatios i which we do ot have reliable data. It allows us to estimate the duratio of the activities of a Project usig three estimatio degrees: optimistic, very pessimistic ad pessimistic. 0,45 0,40 0,35 0,30 0,5 0,0 0,15 0,10 0,05 0, ,0% 5,0% 90,0% -1,709 1,709 1

13 Characteristics: A distributio fuctio commoly applied to sales variables ad market costs. Edogeous variables, the etrepreeur ca use them to egotiate. The etrepreeur ca idetify ad quatify the fuctio parameters. Practical example: Value of the variable (X i ) Absolute frequecy Relative frequecy Cumulative Frequecy /0 = 5% /0 = 0% /0 = 15% /0 = 35% /0 = 5% 0 Total 0 100% Measures of cetral tedecy: Mea X = (1.0*5)+(1.7*4)+(.35*3)+(.01*7)+(0.94*1) = Media Orderig the set: Med.= 1.7 Mode Mode =.01 (the most ofte repeated value cosiderig it has the highest frequecy) 13

14 Other measures (Q 3 ) Percetile P 75 = 3 * = 3 * 0 = P 75 = Third quartile Observig the table of cumulative frequecy we realize that for X i =.01, 75% observatios are below ad 5% are above. Dispersio measures: Variace Stadard deviatio = [( ) * 5]+[( ) * 4]+ + [( ) *1] = Rage R= =