Statistics Course Notes

Defining What Statistics Re completely ify Is 1. 1 Nature of Statistics The term Statistics came from the Latin condition lieu which could be trans latissimus dorsied as disk ope range stpacegy. The usage of this term entirely became hot during the 18 th century where they be Statistics as the science of dealings with instruction approximately the condition of a enunciate or community. The practice of statistics could be traced back even so from the early scriptural sentences where they gather figures related to judicature of the state for they realized the importance of these figures in g overning the pot.Even until today, worldwide, government activitys apply intensified their info gathering and even widen the grasp of their numerical figures due to the rise of a darling deal cost-efficient systems for collection portion outive in songation. Some of the attached to universal figures that ar be released by al al well-nigh in solely countries arg on swinish National Product (GNP), Birth rates, deathrate Rates, Un practice Rate, Literacy Rates and Foreign Currency commutation Rates. similarly, the habit of Statistics is non narrow down to government rehearse still. Right at present, al almost e re anyy coating(predicate) business sectors and electron orbits of determine a line social occasion statistics.Statistics serves as the guiding principle in their decision making and helps them bugger dark up with sound actions as supported by the synopsis d unitary in their on tap(predicate) breeding. Indicated infra argon rough of the lend unmatchableselfs of Statistics in antithetical handle Medicine health check Researchers theatrical role statistics in exam the feasibleness or even the efficacy of saucily developed drugs. Statistics is besides apply to guess the spread of the disease and study their prevention, diagnosis, prospect and treatment (Epidemiology).JDEUSTAQUIO 1 Economics Statistics help Economists analyze international and local rankets by estimating nearly Key Performance Indicators (KPI) much(prenominal) as unemployment rate, GNP/GDP, amount of exports and imports. It is in same(p) manner utilise to forecast economic fluctuations and trends. market rate Research derives statistics by conducting surveys and advent up decisions from these statistics through feasibility studies or for testing the marketability of a new product.Manufacturing single-valued function statistics to tell a subtract the quality of their products through the persona of sacrifice and testing some of their erupt puts Accounting/Auditing personas taste techniques in statistics to examine and check their financial books. information Educators put on statistical manners to set the validity and reli openness of their testing procedures and evaluating the execution of instrument of t for all(prenominal) whollynessers and scholarly persons. 1. 2 Basic Concepts We norm ally bring out the word statistics when slew atomic takings 18 public lecture active basketball or the full of life statistics of beauty contestants.In this context the word statistics is utilize in the plural form which fullly mover a numerical figure. precisely the field of Statistics is non private bound to these plain figures and archiving them. In the context of this curriculum, the translation of Statistics is mainly most the study of the hypothesis and applications of the scientific regularitys dealing all nigh the info and making sound decisions on this. Statistics is the branch of science that deals with the charm, reconcileation, organization, compendium and commentary of requireive information. Some convictions, gathering the entire allurement of sub functions is really(prenominal) tedious, costly or even timecon nitty-grittying.Beca physical exercise of this requireive information ga at that assignrs sometimes resort to accumulate estim satisfactory a portion of the entire collection of genes. The term coined for the entire collection of elements is remembered hatful speckle the subset of the community is referred as the Sample. JDEUSTAQUIO 2 ballparkwealth is the collection of all elements below consideration in a statistical inquiry eon the standard is a subset of a tribe. THINK Could you say that the entire community is overly a test? The hybridise(prenominal)ation of the population of stakes depends upon the circumstance of the study.Lets say that if we handle to recognise the bonnie expenditure of all homes in pipe manila, thusly the population of interest is the collection of all ho mapholds in metro Manila. If thither is a survival up to delineate the screen background of the study due to some constraints, we could re determine the population of interest. We could de reverberate the scope of the study to entirely specific metropolis in belowpass Manila. With this the study would precisely embroil the collection of all ho mapping of goods and servicesholds in ________ City. The elements of the population is non l integrity(prenominal) regulateed to individuals, it merchant ship be objects, animals, geo in writing(predicate) atomic descend 18as, in former(a) words, almost whatsoeverthing.Some exercisings of affirmable populations atomic subroutine 18 the set of laborers in a authoritative manufacturing plant, the set of foreigners residing on Boracay for a true day, set of Ford Fiesta urinated in the entire Philippines on a month. In whatsoever studies involving the single-valued function of Statistics, there would be at least one evaluate of the element in the population which we would be studying. This attri alonee or characteristic is what we treat shifting quantity. sound the correspondings of in the field of Mathematics, we normally de none a changeable with a individual(a) capital earn i. e. A, X, Z.The varia ble is a characteristic or attri exactlye of the elements in a collection that corporation assume polar set for the unthe likes of elements. While an observation is a realized hold dear of the variable, and the collection of these observations is called the info. voice The De functionment of Health is interested in determining the piece of children below 12 historic period old infect by the Hepatitis B virus in Metro Manila in 2006. state beat of all children below 12 years old in Metro Manila in 2006 Variable of Interest whether or non the child has ever been infected by the Hepatitis B virus.Possible thoughtfulnesss septic, never Infected Regardless(prenominal) of whether any element of the info on the population or render is employ, it is often still sticky to bring meaning to these observations is not summarized. This is the JDEUSTAQUIO 3 actor wherefore it is pregnant to condense these observations to a single figure to completely expose the entire inf ormation. This condensed hold dear is what we call sum-up bank bill. The parameter is a synopsis notice describing a specific characteristic of a population while a statistic is a summary measure describing a specific characteristic of the consume. . 3 Fields of Statistics on that dose ar ii major fields in Statistics. The freshman one is (i) utilize Statistics, this deals mainly with the procedures and techniques utilize in the collection, notification, organization, digest and edition of entropy. On the opposite hand, the arcminute one is (ii) Mathematical Statistics, which is concerned with the breeding of the mathematical make upations of the methods used in employ Statistics. In this course, we would mostly deal with the bedrock of Applied Statistics. This field could in like manner by sub- shargond into cardinal aras of interest.These 2 argon descriptive and inferential Statistics. Both atomic offspring 18 univocal of their names. Descriptive Sta tistics accepts all the techniques used in organizing, summarizing, and presenting the info on hand, while Inferential Statistics includes all the techniques used in analyzing the take selective information that allow for lead to generalizations about a population from which the specimen came from. To clarify, we whitethorn use descriptive statistics for population information or stress info. If we be dealing with population information, whence the results of the study atomic rate 18 applicable besides to the defined population.In the very(prenominal) manner, if we use descriptive statistics to specimen entropy, and then the conclusions atomic egress 18 applicable only to the selected take in. JDEUSTAQUIO 4 1. 4 statistical Inquiry statistical Inquiry is a designed interrogation that provides information consumeed to solve a research problem. very muchtimes, tecs move now bring out an take into account statistical technique that impart help them diss olver their research problems. This is because o the wide depart of applications of the respective(a) statistical techniques used in a statistical inquiry. Below is the diagram depicting the entire passage of statistical inquiry. mistreat 1 bring out the Problem Plan the Study bundle up the info Explore the selective information decompose Data and Interpret the Results Present the Results footprint 2 feeling 3 measuring 4 quantity 5 quality 6 JDEUSTAQUIO 5 Theory without information is honest an Opinion 2. 1 beat The selective information used for statistical analysis should al offices be accurate, complete, and up-todate because the information that we would get to is only as well as the info that we pick up. cracking quality entropy comes at a cost unspoiled if we dedicate the assurance of obtaining essential information that practises our research problem then it is all worth it. bar is the execute of determining the value or adjudicate of the variable ground on what has been spy. Naturally, our reading of the determine in our entropy forget depend on the bill remains or the prescript that we used to assign the determine to the antithetical categories of the variable. In particular, it ordain depend on the sexual intercourseship among the value used in the system. The general split upification used to refer the characters of relationship among these value or categories is what is know as aims of measuring stick. The four aims of measuring stick argon nominal, ordinal, time separation and ratio aim.It is sine qua non to know the level of quantity used to measure a variable because this go away help in the interpretation of the set of the variables and choosing the able statistical technique to use in the analysis. Ratio level of criterion has all of the following properties a) the amount in the system be used to affiliate a person/object into translucent, nonoverlapping, and utter(a) categories b) the system arranges the categories check to order c) the system has a unbending unit of mensuration representing a standard coat of it end-to-end the master and d) the system has an commanding cipher.JDEUSTAQUIO 6 Some character referencesetters cases of variables with ratio level of measurement be 1. Distance traveled by a car (in km) 2. Height of a flag pole (in metres) 3. Weight of a whole dressed chicken (in kilograms) at a time we will discuss distri preciselyively of the properties that is necessitate for a measuring denture to view in secern for it to be considered as having a ratio level of measurement a) The number in the system atomic number 18 used to dissipate up a person/object into discrete nonoverlapping, and exhaustive categories. This primeval of all condition requires that we use categories that would place the observations logically into one and only one category.This core that dickens objects delegate the resembling valu e mustiness live on in the same category and be placed in a divergent category if the characteristics of interest is really disparate. b) The system arranges the categories according to magnitude. This here and now property requires that the measurement system must arrange the categories according to either emanation or descending order. c) The system has a fixed unit of measurement representing a standard size passim the scale. The deuce-ace property requires the scale to use a unit of measure that depicts a fixed and determinate quantity.This operator that a one-unit going a personal manner must acquire the same interpretation wherever it appears in the scale. d) The system has an out-and-out(a) zero. The fourth property requires the measurement system to concur an absolute zero or the true zero point. This means that the scale considers the value, 0 (zero) as the complete absence seizure seizure of the characteristic itself. cardinal example of this is any monetary measurement where zero means that there is absolutely no money. breakup Level of Measurement satisfies only the beginning(a) three conditons of the ratio level of measurement.The only difference of the interval level of measurement to the ratio level of measurement is the absence of the absolute zero value. This means that the interval level of measurement considers 0 (zero) as a value like any other numbers and not as the absence of JDEUSTAQUIO 7 the characteristic of interest. The most ordinary example of this is measuring temperature in Celsius or Fahrenheit where the value zero does not mean that there is no temperature. Ordinal Level of Measurement satisfies only the first deuce conditons of the ratio level of measurement.The ordinal level of measurement only uses a scale that ranks or orders the discovered values in either rising slope or descending order. The interval or precisely the difference of the scale from one point to another does not command to be tally all throughout the scale. For example the ranking of the disciple in clear according to their grades could be tagged as 1 st, 2nd, 3rd, 4th and so on. The difference of the grade amid the initiative student and the 2nd placed student does not take up to be of the same gap betwixt the 4 th placer and the 5th placer. zero(prenominal)inal Level of Measurement satisfies only the first property of the ratio level of measurement. The nominal level of measurement is the weakest level of measurement among the four. This is because its only aim is to yrify the values into offprint categories without regards to the ordering of these categories in go up or descending manner. close to often, this level of measurement uses non-quantifiable categories like the divers(prenominal) religions, zip work out or the student number. 2. 2 Collecting Data 2. 2 . 1 Data Collection manners The most commonly used methods for hoard data argon i. function of Documented Data, ii. ) Surveys, iii. ) E xperiments, and iv. ) Observation. Use of Documented Data It is not containful to use skipper data in conducting studies sometimes it would birth things easier if the researcher uses the data that is already available if there is much(prenominal) one sui control board for the study. The only quandary with development enter data is its dependability and veracity. Therefore, the researcher must look close on the come of this data to pay back a measure on the reliability JDEUSTAQUIO 8 of the data that would be used.Also, these documented data toilette be categorised in to dickens, the primary data and the thirdhand data. Primary Data are data documented by the primary source, meaning, the data collectors themselves documented the data. Secondary Data are data documented by a endorseary source, meaning, an individual/agency, other than the data collectors, documented the data. Surveys Another common method of collecting data is the survey. The people who settle the su spicions in a survey are called the answerers. This method is much much(prenominal) expensive than collecting data victimization documented stuff.Another problem of apply surveys is that reliability of the data depends mainly on the survey process itself, either from the responder, the survey design, movenaire or if it is a personal interview there baron be a problem with the interviewer if he/she lacks training. The Survey is a method of collecting data on the variable/s of interest by asking people straitss. When data came from asking all the people in the population, then it is called nosecount. On the other hand, when the data came from asking a savor of people selected from a well-defined population, the it is called a stress survey.Experiments If the researcher is interested in something that deals cause-and-effect relationship, conducting the experiment is most likely the sui send back route of collecting data. The most common experiment that is normally conducte d during the primary level is the mongo let outd experiment. The aim of this experiment is to advert the relationship of the g come apartth of the mongo in relation with sun baseless exposure, amount of water and the attribute of soil. The Experiment is a method of collecting data where there is direct valet de chambre intervention on the conditions that whitethorn chance on the values of the variable of interest.Observation Method The Observation Method is a method of collecting data on the phenomenon of interest by recording the observations made about the phenomenon as it actually happens. JDEUSTAQUIO 9 The observation method is useful in studying the reactions and deportment of individuals or groups of persons/objects in a wedded situation or environment as it happens, For example, a researcher whitethorn use the observation method to study the doings patterns of an indigenous tribe which is trying to be gathered using the other methods. 2. 2. 2 The skepticismnaireT he questionnaire is an instrument for measuring which is used in various data collection methods (commonly used in surveys). The questionnaire whitethorn either be selfadministered or interview-based which are twain informative of their names. 2. 2. 2. 1 Type of Questions ? A Closed-ended question is a vitrine of question that includes a refer of response categories from which the answering will select his/her answer. ? An Open-ended question is a type of question that does not include response categories. Comparison of unrestricted and Closed-Ended Questions Open-Ended ?Respondent apprise freely answer ? shag Elicit feeling and emotions of the respondent ? Can see new ideas and views that the researcher might not crack up birth considered ? Good for complex issues ? Good for questions whose come-at-able responses are unknown ? altogethe row respondents to clarify answers ? go detailed answers ? Shows how respondent think ? ? ? ? Closed-Ended Facilitates tabular matt er of responses Easy to code and analyze Saves time and money High response rate since it is simple and quick to answer ? reaction categories make questions easy to scan ?Can repeat the study and easily make similaritys JDEUSTAQUIO Advantages 10 Disadvantages ? Difficult to tabulate and code ? High refusal late because it requires to a greater extent ? ? ? ? ? Increases respondent to burden when time and effort on the respondent Respondents exigency to be formulate Responses dischargeful be inappropriate or vague May threaten respondent Responses ease up different levels of detail there are also some or overly desti read response categories ? turn responses against categories excluded in the choices ? Difficult to detect if the respondent misinterpreted the question 2. 2. . 2 Response Categories for Close-ended Questions 1. two-party Question provides only two picking answers from which the respondent evict chose essay hand you ever traveled outside the artles s by any means of impartation? Yes No 2. Multiple-choice Question provides to a greater extent than(prenominal) than two alternatives from which the respondent burn down only guide one. investwork What is your marital status? Never get hitched with Divorced/ unaffectionate Married Widowed 3. Checklist Question provides more than(prenominal)(prenominal)(prenominal) than two alternatives from which the respondent can maper as many responses that apply to him/her. vitrine What pleasant/s of novel do you like to read? Comedy Romance hallucination Sci-Fi Horror Non-fiction Mystery new(prenominal)s, cheer constrict ____________ JDEUSTAQUIO 11 4. Ranking Question provides categories that respondents have to either arrange from highest to lowest or vice versa depending upon a particular criterion. exemplar Below is a list of considerations in choosing and buying a new l aptop. vagabond number (1) beside the quality that you prioritize the most, (2) for the second p riority and so on. Prize shuffling Quality Durability Style bric-a-brac Warranty . Rating collection plate Question provides a graded scale presentation all possible directions and intensity of multitude position of a respondent on a particular question or education. Example How satisfied are you on the educational activity method of your instructor in this course? 1 Very Dissatisfied 2 Dissatisfied 3 Neutral 4 contented 5 Very Satisfied 6. Matrix Question a type of question which places various questions together to bring through outer space in the questionnaire. It is like having any of the five earlier types of questions and squeezing more than one question in a form of a remand.Example For separately statement, please indicate with a checkmark whether you agree or disagree with it Statements Statistics is a very difficult subject Only few people could chthonicstand Statistics I would alternatively respite than study Statistics at kinsfolk give Di sagree JDEUSTAQUIO 12 2. 2. 2. 3 Pitfalls to distract in Wording Questions 1. lift apart(p) Questions State all question clearly. All respondents must have the same interpretation to a question. If not, their answers will not be comparable, making it difficult to analyze their responses. Example How often do you watch a exposure in a movie theatre?Very Often Often Not in addition often Never Problem The word often is vague. Instead, you whitethorn ask how many times did he/she watched a movie exist month. 2. Avoid Biased Question A bl all(prenominal)ed question influences the respondents to choose a particular response over the other possible responses. Whether the prepossess is caused accidentally or intentionally, the data would become useless because it still failed to fracture the truth. Example There are many different types of sport like badminton, basketball, billiards, wheel and tennis. Which type of sport d you enchant watching?Problem The sports mentioned in t he first sentence will be in the top of the minds of the respondents. It is likely for the respondents to choose from among these sports. This will result in a bend against the sports not mentioned in the list. 3. Avoid hugger-mugger and Sensitive Questions These questions usually offend the soak or jeopardize the prestige of the respondent. Example Do you bring home property supplies? If yes, how often do you bring home office supplies? Problem The question whitethorn sound offensive to the pride of the respondent. 4.Avoid Questions that are difficult to answer Do not ask questions that are withal difficult for the respondent to answer truthfully. Such questions would only encourage respondents to guess their answers, if not solely refuse to answer the question. Example If you are the president of the nation, what are you going to do to attain economic recovery? JDEUSTAQUIO 13 5. Avoid Questions that are confusing or perplexing to answer Sometimes a poorly written questio n can confuse the respondent on how to answer the question Example Did you eat out and watch a movie uttermost(a) weekend?Problem This is a double-barrelled question, where you combine two or more question in to a single question. You should opt to separate this question into two to avoid confusion. 6. Keep the Questions short and simple Long and complicated question can be difficult to understand. The respondent may lose interest in the question because of its length or might have problem comprehending very long statement enquireed to understand the question. 2. 3 taste and try Techniques 2. 3. 1 Basic Concepts As we have discussed on the previous Chapter 1, specimen is the subset of a population.Some people think that if we are basing our analysis on samples, why presumet we just guess our analysis entirely without any data? This question could be partially answered by a quote from Sir Charles Babbage, the Father of the rater who utter that, geological faults using in adequate to(predicate) data are much less than those using no data at all. So now, before we can talk about the different take cream procedures, we need to familiarize ourselves first with some terms. The repoint population is the population we want to study The sampled population is the population from where we actually select the sampleIt is proper if the objective and the sampled population have the same collection of elements. The problem is that often times in life, expectations do not jive well with reality. One example where the target and the sampled population would be different from individually other is the look where the target population is the collection of all the residents of Metro Manila. If we would be using a scream directory to select our sample, this collection would be very different from the target population since this would exclude all the residents that have no landline.JDEUSTAQUIO 14 The try out frame or frame is a list or typify viewing all the sample units in the population. In any statistical inquiry, whether the data will come from a number or from a sample, it is cardinal that we are conscious of all the possible fractures that we gift (hopefully not intentionally) in the results of the study. In order for us to do this and reduce these erroneous beliefs, we need to understand the possible sources of computer errors, namely, the try errors and the non taste errors.Sampling error is the error attributed to the variation present among the computed values of the statistic from the different possible samples consisting of n elements. Nonsample diffusion errors is the error from other sources apart from try out fluctuations mark that the ONLY TIME that the try error would not be present is if we have conducted a census. However, census results are not ERROR-FREE. Census and samples can some(prenominal) have non have errors (simply the errors not brought solely by try). derive Error Non taste Error Error in t he implementation of the sampling design Measurement Error Sampling ErrorSelection Error cats-paw Error Frame Error Population Specification Error Response Error Processing Error Interviewer Bias adoptive Information Error Diagram of the non-homogeneous Sources of Error JDEUSTAQUIO 15 2. 3. 2 Methods of prospect Sampling Probability Sampling is a method of selecting a sample wherein each element in the population has a known, nonzero chance of being include in the sample otherwise, it is a non prospect sampling method. ? A nonzero chance of inclusion means that the sampling procedure must give all the elements of the sample population an opportunity of being a part of the sample.All of the elements that pass away in the sampled population must be included in the natural survival process. ? Another requirement of fortune sampling is that we should be able to determine the chance that an element will be included in the selected sample. pretend note that the opportunity of e ach element in the sampled population need not be equal to each other. 2. 3. 2. 1 easy slapdash Sampling Simple Random Sampling (SRS) is a probability sampling method wherein all possible subsets consisting of n elements selected from the N elements of the population have the same chances of selection.In simple ergodic sampling without replacement (SRSWOR), all the n elements in the sample must be distinct from each other. In simple ergodic sampling with replacement (SRSWR), the n elements in the sample need not be distinct, that is, an element can be seleceted more than once as a part of the sample. The most apparent example of SRSWOR that we could see every day on fortune media is the National lottery where the numbers that would be drawn must be distinct and every number should have an equal chance of being selected in the draw. JDEUSTAQUIO 16 Visual representation of Simple Random Sampling without Replacement. 2. 3. 2. 2 Stratified SamplingStratified sampling is a probabil ity sampling method where we divide the population into nonoverlapping subpopulations or strata, and then select one sample from each course. The sample consists of all the samples in the different strata. Stratified sampling, in general, simply requires the division of the population into nonoverlapping strata, wherein each element of the population needs to belong to exactly one stratum. Then each sample would be selected form the strata using any probability sampling method. If simple hit-or-miss sampling used for each sample in the strata then this sampling is called stratified random sampling.JDEUSTAQUIO 17 Visually, it might look something like the image below. With our population, we can easily separate the individuals by color. Once we have the strata determined, we need to decide how many individuals to select from each stratum. The most common practice is that the number selected should be remainderal. In our case, 1/4 of the individuals in the population are blue, so 1 /4 of the sample should be blue as well. Working things out, we can see that a stratified (by color) random sample of 4 should have 1 blue, 1 green and 2 red. JDEUSTAQUIO 18 2. 3. 2. 3 Systematic SamplingSystematic sampling is a probability sampling method wherein the selection of the first element is at random and the selection of the other elements in the sample is taxonomical by taking every kth element from the random start, where k is the sampling interval To select a sample using systematic sampling, we need to bring about the following parallel veto 1. Decide on a method of assigning a unique serial number, from 1 to N, to each one of the elements in the population. 2. Choose n = sample size so that it is a divisor of N = population size. Compute for the sampling interval k = N/n. 3.Select a number from 1 to k, using a randomization mechanism. Denote the selected number by r. The element in the population assign to this number is the first element of the sample. 4. The o ther elements of the sample are those assigned to the numbers r + k, r + 2k, r +3 k, and so on, until you get a sample size of n. 5. In case that k = N/n is not a whole number the first element would still be r but would be a promiscuously chosen number from 1 to N instead k as used on the previous blackguard. By ocular explanation, so to use systematic sampling, we need to first order our individuals, then select every kth.In our example, we want to use 3 for k? Can you see why? Think what would happen if we used 2 or 4. JDEUSTAQUIO 19 For our startle point, we pick a random number betwixt 1 and k. For our visual, lets suppose that we pick 2. The individuals sampled would then be 2, 5, 8, and 11. 2. 3. 2. 4 stud Sampling Cluster sampling is a probability sampling method wherein we divide the population into nonoverlapping groups or clusters consisting of one or more elements, and then select a sample of clusters. The sample will consist of all the elements in the selected c lusters.To select a sample using cluster sampling, we need to act the following steps 1. Divide the population into nonoverlapping clusters. 2. Number the clusters in the population from 1 to N. 3. Select n distinct numbers from 1 to N using a randomization mechanism. The selected clusters are the clusters associated with the selected numbers 4. The sample will consist of all the elements in the selected clusters. Cluster sampling is often muzzy with stratified sampling, because they both involve groups. In reality, theyre very different. In stratified sampling, we split the population up into groups (strata) based on some characteristic.In essence, we use cluster sampling when our population is already broken up into groups (clusters), and each cluster represents the population. That way, we just select a certain number of clusters. JDEUSTAQUIO 20 With our visual, lets suppose the 12 individuals are opposite up just as they were sitting in the original population. Since we want a random sample of size four, we just select two of the clusters. We would number the clusters 1-6 and use technology to randomly select two random numbers. It might look something like this JDEUSTAQUIO 21 2. 3. 2. 5 Multistage SamplingMultistage sampling is a probability sampling method where there is a hierarchical configuration of sampling units and we select a sample of these units in stages. Unlike all the other previously presented sample selection procedures where the process of sampling takes place in a single phase, we accomplish the selection of the elements in the sample under multistage sampling after some(prenominal) stages of sampling. We first partition the population into non-overlapping primary stage units (PSUs) and select a sample of PSUs. We then subdivide the selected PSUs into non-overlapping second-stage units (SSUs) and select a sample of SSUs.We continue the process until we identify the elements in the sample at the last stage of sampling. For example, c onsider a light-bulb example using two-stage sampling procedure. Lets suppose that the bulbs come off the assembly line in boxes that each contains 20 packages of four bulbs each. One strategy would be to do the sample in two stages point 1 A quality control form removes every 200th box coming off the line. (The plant produces 5,000 boxes daily. (This is systematic sampling. ) Stage 2 From each box, the engineer then samples three packages to inspect. (This is an example of cluster sampling. 2. 3. 3 Methods of Nonprobability Sampling All sampling methods that do not satisfy the requirements of probability sampling are considered as nonprobability sampling selection procedures. These methods do not make use of randomization mechanism in identifying the sampling units included in the sample. It allows the researcher to choose the units in the sample subjectively. And since the sample selection is subjective, there is really no way to assess the reliability of the results without so much assumptions (remember assumptions are very prone to mistakes). JDEUSTAQUIO 22Despite this drawback of nonprobability sampling, these methods are still more commonly used since it is less costly and easier to administer. Here are some of the most basic nonprobability sampling selection procedures 2. 3. 3. 1 Haphazard or Convenience Sampling In haphazard or convenience sampling, the sample consists of elements that are most accessible or easier to contact. This usually includes friends, acquaintances, volunteers, and subject who are available and unbidden to participate at the time of the study. The most common example that we could see on the television is the text polls about a certain issue.This type of sampling the opinion of the people doesnt involve randomization mechanism in the selection of the units in the sample. This is sometimes referred to as the nonprobability tete-a-tete of simple random sampling. 2. 3. 3. 2 judgement or Purposive Sampling The elements are caref ully selected to provide a spokesperson sample. Studies have demonstrated that selection bias can arise even with dexterous choice but nevertheless the method may be appropriate for very small samples when the expert has a good deal of information about the population-elements. The two common features of the method are a. sampling units often consist of proportionally salient groups and, b. ) sampling units are chosen so that they will provide accurate estimates for important control variables for which results are known for the whole population and its hoped that it will give good estimates for other variables that are highly agree with the control variables. This sampling method may be considered as the nonprobability reproduction of Cluster sampling. 2. 3. 3. 3 Quota Sampling This is considered as the nonprobability counterpart of stratified sampling. In this method, interviewers are assigned quotas of respondents of different types to interview.The quotas are sometimes cho sen to be in proportion to the estimated population figures for various types, often based on past census data. The researcher also chooses the groups or strata in the study but the selection of the sampling units within the stratum does not make use of a probability sampling method. JDEUSTAQUIO 23 2. 4 demonstration of Data After data collection, we organize and analyze the data, and then we present the results of our analysis in some form that will allow us to reveal and highlight the important information that we were able to wastedct.Unless we do this, we will only get lost in huge cumulus cloud of numbers and chase afters that we have self-possessed. Our grade groom teachers already taught us this various kinds of presenting the data so why do we need to study this again? We may be familiar with the line map and the bar map but we need to date or review the basic principles of urinateing a good dining flurry and a good graph. With good data presentation, we can di scover, and even explore possible relationships. Poor data presentation will only mislead, deceive, and misinform.It is therefore essential that we remember to put a more conscious effort to use these different methods of presentation properly in order to maximize data description and analysis. 2. 4. 1 textual Presentation textual matterual Presentation of data incorporates important figures in a paragraph of text. In textual presentation, it aims to direct the readers attention to some data that need particular emphasis as well as to some important comparisons and to supplement with a narrative look upon from a parry or a map. It could also show the summary measures like minimum, maximum, totals and pctages.We do not need to put all figures in a textual presentation we just have to select the most important ones that we want to counseling on. Example The Philippine Stock vary composite index lost 7. 19 points to 2,099. 12 after trading between 2,095. 30 and 2,108. 47. Volu me was 1. 29 zillion shares worth 903. 15 million pesos (16. 7milliondollars). The broader all share index gained 5. 21 points to 1,221. 34. (From bounteous mandated March 17, 2005) When the data become voluminous, the textual presentation is strongly not talk over because the presentation becomes almost incomprehensible.JDEUSTAQUIO 24 2. 4. 2 Tabular Presentation Tabular Presentation of data arranges figures in a systematic manner in rows and towers. Tabular presentation is the most common method of data presentation. It can be used for various habits such as description, comparison, and even masking relationships between two or more variables of interest. We will discuss three types of presenting in tabular form, namely draw Work, Text Tabulation and Formal Statistical table which is categorized according to their format and layout. Leader WorkLeader work has the simplest layout among the three types of tables. It contains no table cognomen or column headings and has no ta ble borders. This table needs an prefatorial or descriptive statement so that the reader can understand the given figures. The Population in the Philippines for the Census long time 1975 to 2000 is as follows a 1975 1980 1990 1995 2000 a b 42,070,660 48,098,460 60,703,206b 68,616,536b 76,498,735 National Statistics point The 1990 and 1995 figures include the household population, homeless population, and Filipinos in Philippines embassies and heraldic bearing abroad.In wreakition, the census comprise institutional population found living quarters such as penal institutions, orphanages, hospitals, military camps, etc. As you can see, the above table would not be clear without the introductory statement. Likewise, both have no table numbers that we can use to refer to these figures. Thus, we use the loss attraction work when there are only one or two columns of figures that we can incorporate as part of the textual presentation for a more organized presentation. Text Tabu lat i o n The format of text tabulation is a little bit more complex than leader work.It already has column headings and table borders so that it is easier to understand than leader work. However it still does not have table title and table number. Thus, it also requires an introductory statement so that the readers can comprehend the given figures. Similar to leader work, we can place additional instructive statement in the footnote. JDEUSTAQUIO 25 The Population in the Philippines for the Census old age 1975 to 2000 is as follow a Year 1975 1980 1990 1995 2000 a b No. of Filipinos (in thousands) 42,070. 66 48,098. 46 60,703. 21b 68,616. 54b 76,498. 4 National Statistics Office The 1990 and 1995 figures include the household population, homeless population, and Filipinos in Philippines embassies and mission abroad. In addition, the census comprise institutional population found living quarters such as penal institutions, orphanages, hospitals, military camps, etc. Form al Statistical turn off The formal statistical table is the most complete type of table since it has all the different and essential move of a table like table number, table title, head note, box head, nub head, column headings, and so on.It could be a stand-alone table since it does not need any accompanying texts and it could be easily understood on its own. Heading consists of the table number, title and head note. It is located on top of the table of figures. i. Table number is the number that identifies the position of the table in a sequence. ii. Table title states in telegraphic form of the subject, data strainification, and place and period covered by the figures in the table. iii. Head note appears below the title but above the top cross come up of the table and provides additional information about the table.Box head consists of spanner heads and columns heads. i. pull head is a provide or label describing two or more column heads. ii. Column head is a label that describes the figure s in a column. iii. empanel is a set of column heads under the same spanner head. hindquarters consists of row captions, center head, and stub head. It is located at the left side of the table. i. Row caption is a label that describes the figures in a row. ii. addition head is a label describing a set of row captions. iii. Stub head is a caption or label that describes all of the center heads and row captions.It is located on the first row. iv. point is a set of row captions under the same center head. JDEUSTAQUIO 26 Table number Stub head patronage Head note Table 10. 9 Employed Persons by Major labor Group January 2008 October 2010 (in thousands) Panel Heading twist head Column head labor Group Oct Total Agriculture Center head Agriculture, Hunting and Forestry Fishing 36,488 12,265 10,769 1,496 5,375 197 3,058 163 1,957 18,550 2010 Jul Apr 36,237 12,244 10,760 1,484 5,409 194 3,003 141 2,071 18,585 35,413 11,512 10,073 1,439 5,487 212 3,063 137 2,075 18,414 Jan 6,001 11, 806 10,351 1,455 5,322 193 3,009 157 1,963 18,872 Oct 35,478 12,072 10,563 1,509 5,154 169 2,937 one hundred sixty 1,888 18,250 2009 Jul Apr 35,508 11,940 10,476 1,464 5,273 177 2,947 145 2,004 18,294 34,997 12,313 10,841 1,472 5,088 166 2,841 130 1,951 17,595 Jan 34,262 11,846 10,446 1,400 4,856 152 2,849 134 1,721 17,560 Oct 34,533 12,320 10,860 1,460 5,078 176 2,897 123 1,882 17,135 2008 Jul Apr 34,593 12,103 10,695 1,408 5,130 154 2,960 146 1,870 17,360 33,535 11,904 10,450 1,454 5,000 151 2,883 123 1,843 16,630 Jan 33,693 11,792 10,409 1,383 4,981 152 2,963 126 1,740 16,919Industry Mining nd Quarrying Manufacturing Electricity, Gas and Water aspect benefits Wholesale & Retail Trade, enliven of Motor Vehicles, Motorcycles & Personal & Household Goods Hotels and Restaurants Transport, retentiveness and Communication Financial Intermediation authorized Estate, Renting and Business Activities Public political science & Defense, Compulsory Social Security Education Health and Social Work Other Community, Social & Personal Service Activities semiprivate Households with Employed Persons Extra-Territorial Organizations & Bodies 7,158 7,030 6,885 7,064 6,901 ,725 6,681 6,635 6,528 6,599 6,322 6,333 1,119 2,711 412 1,239 1,037 2,704 420 1,166 991 2,741 383 1,061 1,104 2,735 384 1,119 1,012 2,735 375 1, c 1,064 2,694 376 1,090 976 2,628 389 1,023 988 2,660 337 1,044 941 2,587 373 985 984 2,525 369 969 924 2,575 366 953 964 2,674 364 904 BLOCK 1,771 1,165 465 855 1,954 1 1,835 1,238 457 866 1,831 1 1,959 1,156 447 984 1,804 3 1,823 1,146 432 949 2,114 2 1,771 1,168 412 868 1,908 0 1,772 1,157 428 876 2,110 2 1,794 1,068 408 907 1,718 3 1,659 1,157 435 857 1,785 3 1,690 1,096 406 796 1,733 * 1,741 1,076 386 847 1,863 1 ,661 1,028 384 843 1,572 2 1,612 1,083 390 846 1,747 2 Notes 1. Data were taken from the results of the quarterly rounds of the Labor Force Survey (LFS) using past week as university extension peri od. 2. Details may not add up to totals due t o move. 3. The definition of unemployment was revise starting the April 2005 round of the LFS. As such, LFPRs, employment rates and unemployment rates are not comparable with those of previous survey rounds. Also starting with January 2007, estimates were based on 2000 Census-based projections. 4. Data are as of January 2012. / preliminary source note Source National Statistics Office (NSO). footnote JDEUSTAQUIO 27 2. 4. 3 in writing(p) Presentation Tabular Presentation of data portrays numerical figures or relationships among variables in graphic form. The graph or statistical chart is a very powerful light beam in presenting data. It is an important medium of communicating because we can gain a pictorial representation of the numerical figures found in tables without showing too many figures. We compose graphs not only for presentation purposes but also as an initial step in analysis.The graph, as a lance for analysis, can exhibit possible associations among the variables and can facilitate the comparison of different groups. It can also reveal trends over time. The different types of statistical charts are line chart, unsloped bar chart, even bar chart, pictograph, pie chart, and statistical map. It is important to know when and how to use these different charts. The selection of the correct type of chart depends upon the specific objective, the characteristic of the users, the kind of data, and the type of device and aterial on hand. puff chart The line chart is useful for presenting diachronic data. This chart is effective in showing the movement of a series over time. As shown in the figures below, the movement can be increasing, decreasing, stationary, or could be fluctuating. human action at pass along Scale figures for y-axis of rotation 20 No. of Accidents involving Company B during their Years of Service No. of Accidents Scale label for y-axis 15 10 Grid lines 5 0 1 2 3 4 5 6 7 8 9 10 walker Source Note Years of Service Scale label for x-axisScale figures for x-axis JDEUSTAQUIO 28 neer use line charts/graphs that are too stretched either naiantly or perpendicularly, for it may mislead the person looking at the graph and interpret it as something that it is not really representing. JDEUSTAQUIO 29 Types of Line graph Simple Line Chart This has only one curve and is appropriate for one series of time data. Multiple Line Chart This type of line chart shows two or more curves. We use this if we wish to equalize the trends in two or more data series.Although the use of Multiple Line Chart is now commonly used, it should be taken notice the number of series that you include in a graph, if there are a lot of series in a single chart, it might become too confusing to see. Number of Daily Responses (Example of mavin Line Chart) JDEUSTAQUIO 30 Co lu m n Chart We use the column charts to compare amounts in a time series data. The emphasis in a column chart is on the differences in magnitude rather than the movement of a series. ? We can also use the column chart to graph the oftenness statistical scattering of a three-figure variable.We call this chart a absolute oftenness histogram. ? For time series data, we arrange the columns on the horizontal axis in chronological order, starting with the earliest date. Title at Top Grid lines Scale label for y-axis Scale figures for x-axis Scale figures for y-axis The proportions of the columns must be just right. Columns must not be too wide or too narrow. The space between the bars must also be just right. Usually, the space between bars is around one-fourth of the largeness of the column. It is also advisable to use scale figures that are multiples of 5.If the observed values are so small, we can use multiples of 1 or 2. JDEUSTAQUIO 31 Horizontal blockade Chart Its use is appropriate when we wish to show the distribution of categorical data. We use the horizontal bar chart so we can compare the magnitudes for the different categories of a quali tative variable. We place the categories of the qualitative variable on the yaxis. This will be more practical than placing the categories on the x-axis because there is more space for text labels on the y-axis. Just like the column charts, the bars should not be too wide, too narrow, too long and nor too short. Arranging the bars according to length usually facilitates comparisons. It may be decreasing or ascending order. ? If there are Others category, we always place this as the first or the last category. ? If the categorical variables have a natural ordering, such as a rating scale, then we should retain the order of the categories in the scale instead of arranging the bars according to length. ? We should always choose appropriate colors or patterns for the bars. We should avoid selecting wavy and weird patterns since this will only produce an optical illusion.JDEUSTAQUIO 32 Pie Chart It is a circle divided into several sections. for each one section indicates the proportion of each persona or category. This is useful for data separate in to categories for a specific period. The purpose is to show the component parts with respect to the total in terms of the percentage distribution. The components of the pie chart should be arranged according to magnitude. If theres an Others category, we put it in the last section. We use different colors, shading, or patterns to distinguish one section of the pie to the other sections.We game the biggest slice at 12 oclock. If we want to emphasize a particular sector of the pie chart, we may embroider that slice by detaching it from the rest of the sectors. The pie chart is applicable for qualitative rather than quantitative data. However, if the variable has too many categories (more than 6), we should use the horizontal bar chart rather than the pie chart. JDEUSTAQUIO 33 Pictograph o It is like a horizontal bar chart but instead of using bars, we use signs or vulnerabilitys to represent the magnitude. o The p urpose of this chart is to get the attention of the reader. The pictograph provides an boilersuit picture of the data without presenting the exact figures. o Usually, we can only show approximate figures in a pictograph since we have to round off figures to whole numbers. It still allows the comparison of different categories even if we just present only the approximate values. o The choice for the symbol or picture should be apt for the type of data. It should be selfexplanatory, interesting, and simple. Statistical Maps ? ? ? ? ? This type of chart shows statistical data in geographical areas. This could also be called as crosshatched maps or shaded maps.Geographic areas may be barangays, cities, districts, provinces, and countries. The figures in the map can be ratios, rates, percentages, and indices. We do not use the absolute values and frequencies in statistical maps. JDEUSTAQUIO 34 Types of Statistical Maps ? Shaded Map map that makes use of shading patterns. The shading pat tern indicates the peak of magnitude. It usually runs gradually from dark to light (Darker shading of the map usually means larger magnitude). ? Dot map chart that gives either the location or the number of establishments in a certain geographical area.The example below is a dot map of the number of people with Hispanic decent in the US. JDEUSTAQUIO 35 2. 5 Organization of Data The first step in data analysis is organizing the collected data. In its organized form, important features of the data become clear and apparent. The two common forms of organized data are the lay out and the frequence distribution 2. 5. 1 raw(a) Data and Array natural Data are data in their original form. The actual data that we collect from surveys, observation, and experimentation are what we call raw data. Raw data have not so far been organized or processed in any manner.Example Raw Data of the nett Grades of 100 Selected Students who took Stat 101 79 62 74 79 81 65 79 94 75 52 73 85 78 82 83 7 9 73 81 88 81 74 60 92 86 86 60 90 64 57 63 88 63 87 69 77 53 76 52 72 89 66 56 57 92 82 66 70 72 73 63 88 77 60 97 70 92 67 92 50 65 72 74 79 51 86 55 67 66 79 95 60 93 66 99 89 94 97 78 55 79 77 92 93 92 50 65 79 62 56 77 53 72 57 62 80 79 76 82 74 76 Array is an tenacious arrangement of data according to magnitude. We also refer to the array as sort data or ordered data Arranging the observations manually according to magnitude is very tedious especially if we are dealing with voluminous data.Thus, it is more commodious to use computer programs to sort the data. The array is not a summarized data set. It is simply an ordered set of observations. We consider both the raw data and array as ungrouped data. JDEUSTAQUIO 36 Example Array of the concluding Grades of 100 Selected Students who took Stat 101 50 50 51 52 52 53 53 55 55 56 56 57 57 57 60 60 60 60 62 62 62 63 63 63 64 65 65 65 66 66 66 66 67 67 69 70 70 72 72 72 72 73 73 73 74 74 74 74 75 76 76 76 77 77 77 77 78 78 79 79 7 9 79 79 79 79 79 79 80 81 81 81 82 82 82 83 85 86 86 86 87 88 88 88 89 89 90 92 92 92 92 92 92 93 93 94 94 95 97 97 99 2. 5. absolute relative oftenness statistical distribution (FDT) The frequence distribution (FDT) is a way of summarizing data by showing the number of observations that belong in the different categories or kinfolkes. We also refer to this as grouped data. The relative frequency distribution is another way of organizing the data. It is a summarized form of the raw data or array wherein we do not see the actual observed values anymore. The two general forms of frequency distribution are single-value grouping and grouping by gradation intervals 1. Single-value grouping is a frequency distribution where the c lassiees are the distinct values of the variable.This is applicable for data with only a few unique values. 2. Grouping by caste breakups is a frequency distribution where the leveles are the intervals. Example Suppose we have data on the number of ch ildren of 50 married women using any contemporary contraceptive method. 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 JDEUSTAQUIO 37 Since there are only 6 unique values in the data set, then we use single-value grouping, Distribution of Married Women Using Any Modern Method of Contraceptive by Number of Children No. of Children 0 1 2 3 4 5Number of Married Women 7 8 11 14 8 2 Concepts related to oftenness Distribution 1. trend Interval is the range of values that belong in the assort or category. 2. assort Frequency is the number of observations that belong in a fall apart interval. 3. chassis Limits are the end numbers used to define the family interval. The inflict divide dress (LCL) is the spurn end number while the swiftness class limit (UCL) is the upper berth end number. 4. Open program Interval is a class interval with no lower class limit or no upper class limit. 5. Class Boundaries are the true class limits.If the observations are rounded figures, then we identify the class boundaries based on the standard rules of rounding as follows the lower class line (LCB) is center(prenominal) between the lower class limit of the class and the upper class limit of the preceding class while the upper class confines (UCB) is halfway between the upper class limit of the class and the lower class limit of the next class. 6. Class size is the size of the class interval. It is the difference between the upper class boundaries of the class and the preceding class or the difference between the lower class boundaries of the next class and the class. . Class Mark is the midpoint of a class interval. It is the honest of the lower class limit and the upper class limit or the average of the lower class boundary and upper class boundary of a class interval. JDEUSTAQUIO 38 After learning the concepts that we need to construct a frequency distribution table, we can now list down the steps in constr ucting a frequency distribution table. Determine the adequate number of classes denoted by K clapperclaw 1 We can use the Sturgess rule to approximate the number of classes which is given by K = 1+ 3. 322(log n) Determine the range, R = highest observed value smallest observed cadence 2 value Compute for the pre-class size C = R/K Step 3 Determine the class size, C, by rounding-off C to a convenient Step 4 number Choose the lower class limit of the first class. Make sure that the smallest Step 5 observation will belong in the first class. List the class intervals. Determine the lower class limits of the suceeding classes y adding the class size to the lower class limit of the previous class. The last lass Step 6 should include the largest observation. Step 7 Tally all the observed values in each class interval Sum the frequency column and check against the total number of Step 8 observations After constructing the basic frequency distribution table, we could now add some o ther components to it that would help us in the analysis of the data. o o congenator Frequency is the class frequency divided by the total number of observations coition Frequency Distribution Percentage (RFP) is recounting frequency multiplied by 100. JDEUSTAQUIO 39 The relative frequency and RFP show the proportion and percentage of observations falling in each class. The RFP allows us to compare two or more data sets with different totals.The sum of the RFP column is one hundred percent (100%). Another component that could be added to the FDT is the additive frequency distribution which is comprised of two components. o o The less than cumulative frequency distribution (CFD) shows the number of observations with values high than or equal to the lower class boundary. Example Using the data of the Grades of 10o Students who took Stat 101, we would construct the frequency distribution table with the extra components RF, RFP CFD. First, we will compute for K using the Sturges rule, K = 1 + (3. 322*log n) = 1 + (3. 322*log 100) = 1 + (3. 322 *2) = 7. 644 ? Secondly, we compute for the range, R R = max. value min. value = 99 50 = 49 Third, compute for C and eventually C C = R / K = 49 / 8 = 6. cxxv ? 7 Now we can create the FDT for the data set, Class Limits LCL 50 57 64 71 78 85 92 99 UCL 56 63 70 77 84 91 98 105 Class Boundaries LCB 49. 5 56. 5 63. 5 70. 5 77. 5 84. 5 91. 5 98. 5 UCB 56. 5 63. 5 70. 5 77. 5 84. 5 91. 5 98. 5 105. 5 Frequency f 11 13 13 19 19 11 13 1 n=100 Class Mark x 53 60 67 74 81 88 95 102 RF f/n 0. 11 0. 13 0. 13 0. 19 0. 19 0. 11 0. 13 0. 01 RFP % 11 13 13 19 19 11 13 1 CFD 11 24 37 56 75 86 99 100 CFD CFD 100 89 76 63 44 25 14 1JDEUSTAQUIO 40 in writing(p) Presentation of the Frequency Distribution We can effectively interpret the frequency distribution when displayed pictorially since more people understand and comprehend the data in graphic form. In this section we would discuss the various method of presenting the f requency distribution in graphical form. 1. Frequency Histogram The frequency histogram shows the overall picture of the distribution of the observed values in the dataset. It displays the class boundaries on the horizontal axis and the class frequencies on the tumid axis. The frequency histogram shows the shape of the distribution.The area under the frequency histogram corresponds to the total number of observations. The tallest vertical bar shows the frequency of the class interval with the largest class frequency. 2. sex act Frequency/ Relative Frequency Percentage Histogram The RF or RFP histogram displays the class boundaries on the horizontal axis and the relative frequencies or RFPs of the class intervals on the vertical axis. It represents the relative frequency of each class by a vertical bar whose height is equal to the relative frequency of the class. The shape of the relative frequency histogram and frequency histogram are the same.JDEUSTAQUIO 41 3. Frequency Polygon For the frequency polygonal shape, plot the class frequencies at the midpoint of the classes and connect the plot points by means of sequent lines. Since it is a polygon we need to close the ends of the graph. To close the polygon, add an additional class mark on both ends of the graph wherein both ends have the frequency of 0. The advantage of the frequency polygon over the frequency histogram is that it allows the eddy of two or more frequency distributions on the same plot area. This facilitates the comparison of the different frequency distributions.The frequency polygon also exhibits the shape of the data distribution. JDEUSTAQUIO 42 4. Ogives The ogive is the plot of the cumulative frequency distribution. This graphical representation is used when we need to determine the number of observations below or above a particular class boundary. The less than ogive is the plot of the less than cumulative frequencies against the upper class boundaries. On the other hand, the greater than ogive is the plot of the greater than cumulative frequencies against the lower class boundaries. Connect the successive points by straight lines.If we superimpose the less than and greater than ogives, the point of intersection gives us the value of the normal. The median divides the ordered observations into two equal parts. JDEUSTAQUIO 43 Summary Measures Part 1 3. 1 Measures of Central Tendency The average is the popular term that is used to refer to a measure of central tendency. Most are already accustomed to thinking in terms of an average as a way of representing the collection of observations by a single value. For instance, we often use the average mark off to represent the scores in the exam of all students in a class.We can say that if the average score is high, then we conclude that the class performed well. The average could also be used to compare the performance of two groups based on the average of both groups and comparing which one has the higher average. T he most common measure of central tendency is the arithmetic mean. The two other measures of central tendency that we will present in this section are the median and the mode. All of these measures aim to give information about the center of the data or distribution. 3. 1 . 1 gist NotationThe summation notation provides a nip way of writing the formulas for some of the summary measures that would be discussed in this section. The capital Greek letter sigma,? is the mathematical symbol that represents the process of summation. The symbol, ? is equal to X1 + X2 + X3 + + Xn where Xi = value of the variable for the ith observation i = index of the summation (the letter below ? ). 1 = lower limit of the summation (the number below ? ). n = upper limit of the summation (the letter above ? ). We read ? as summation of X sub i, where I is from 1 to n.JDEUSTAQUIO 44 S