INTRODUCTION TO STATISTICS - KSU

CHAPTER 1Basic Concepts In Statisticsb. In a study of the evolving condition of the patients in ahospital, where there are many people of different types ofdiseases, but they all bind them recipe disease, so patients thatin the hospital make up a population.c. In a study to determine the technical condition of the aircraftof the Gulf Cooperation Council (GCC), where travel aircraft,military training aircraft, and Helicopters ., but they are allcharacterized by their ability to fly, so the aircraft in the GulfCooperation Council (GCC) are populationNote that a population can be a collection of any things, like set oftrees, people, animals or inanimate (books, cars, metal.). Thereforeit does not necessary deal with a people.Definition 1.1.5A sample is a subset of the population selected for study.Referring to the example of interest to know the average weight ofwomen that visited diet section, in this case the registered weightsof some women represent a sample.In practical life there are many ways to get a sample from the population under study, for example; face-to-face interview, online electronic questionnaires, paper questionnaires and using telephones.natiolupPo: : :Sample::: : :::::::: : : : : :: ::: : ::: :: : : : : : : : : : ::: : : : : : : :: : :: : : : : : : : :: :: : : :: : Figure 1.1 Population and SampleLet us discuss an example on determining the population and thesample for a studyExample 2If we take a class of the students in Stat 140 course at PY,then: all the students registered in the course represent thepopulation, and any class of them is represent a sample.4

BasicConceptsIn StatisticsEquationsand Inequalities1.2CHAPTERCHAPTER 11Variables and Types of DataBasic terms that will be used frequently in this section, and theyare very important tools in statistical problems, such terms are,an element, a variable and their types, a measurement, and adata set, Therefore to understand such terms, it is necessary toillustrate the following definitions.Definition 1.2.1An element (or member of a sample or population) is a specificsubject or object about which the information is collected.RemarkAny study is based on aproblem or phenomenon suchas heavy traffics, accidents,rating scales and grades orothers. The researcher shoulddefine the variables of interestbefore collecting data.Example 1 below discuss the definition of an element numericallyExample 1The following table gives the number of snake bites reportedin a hospital in 3 cities (A, B, C).CityNumber of Snake Bites10A17B11CEach one of the cities is a member, that is; city A is a member,city B is a member, and also city C is a member.Definition 1.2.2A variable is a characteristic under study that takes differentvalues for different elements.For example, if we collect information about income of households,then income is a variable .These households are expected to havedifferent incomes; also, some of them may have the same income.Note that a variable is often denoted by a capital letter like X, Y, Z, .and their values denoted by small letters for example x, y, z, .Definition 1.2.3The value of a variable for an element is called an observationor measurement.The following is an example to explain the difference in themeaning between variable and the measurement.5

CHAPTER 1Basic Concepts In StatisticsExample 2Referring to example 1, we see that the variable (let X forexample) is the number of snake bites and each one of thenumber of bites 10, 17, 11 represents an observation ormeasurement. Where we have X(A) 10 , X(B) 17 andX(C) 11.We know that the variable is a characteristic under study that takesdifferent values for different elements. In statistics, we have twotypes of variables according to their elements; first type is calledquantitative variable and the second one is called qualitativevariable.When a subject can be measured numerically such as (the price ofa shirt), then the subject in this case is quantitative variable. Thefollowing definition provides us with this concept.Definition 1.2.4Quantitative variable gives us numbers representing counts ormeasurements.When a subject cannot be measured numerically such as (eye color),then the subject in this case is qualitative variable. The followingdefinition provides us with this concept.Definition 1.2.5Qualitative variable (or categorical data) gives us names orlabels that are not numbers representing the observations.RemarkQuantitative variables give us quantitative data and inquires about thephrase “how much”, while the qualitative variables give us the qualitative data and inquires about thephrase “what or what is”.6The following examples illustrates the two type of variablesExample 3The following table shows some examples of the twotypes of variables

BasicConceptsIn StatisticsEquationsand InequalitiesCHAPTERCHAPTER 11Quantitative variablegives us quantitative dataThe age of people in years19, 2, 45, 23, 88, .Number of children in family5, 2, 4, 1, 14, .Qualitative variablegives us qualitative dataThe gender of OrganismsMale, Female .Results tossed a coin twiceHH, HT, TH, TT(H Head, T Tail)The heights of buildings in Eye color of peoplemetersBlack, Brown, Blue, Green, .15, 5.6, 12.7, 105, 27, .The weights of cars in tons Religious affiliationMuslim, Christian, Jew, .(ton 1000 Kg)2.35, 1.65, 2.05, 2.10, 1.30, .The speed of a car going on a The pressure in a boilerHigh, Moderate, Lowmain road in Km110, 105, 85, 120, 90, .Moreover, the variables measured in quantitative data dividedinto two main types, discrete and continuous. A variable thatassumes countable values is refer to discrete variable, otherwise thevariable is a continuous one. Accordingly, we provide the followingdefinitions.Definition 1.2.6Discrete variables assume values that can be counted.In following we illustrate some examples on a discrete variableExample 4-The number of children in a family, , where we have 1,2,3, .or k children.-The number of students in a classroom, where we have 21,25,32,18 and so on.-Number of accidents in a city, where we have 1,2,3,. or kaccidents.The other type of quantitative variable is the continuous variablewhich is assumed uncountable values, and offer us the followingdefinition.7

CHAPTER 1Basic Concepts In StatisticsDefinition1.2.7Continuous variables assume all values between any two specificvalues, i.e. they take all values in an interval. They often includefractions and decimals.In the following we illustrate some examples on a continuousvariableExample 5- Temperature: For example the temperature in Riyadh cityin last summer was between 15 and 56, i.e. the temperaturet ! [15, 56].- Age: For example the age of a horse is between 0 (Stillborn)and 62 years (Said the oldest horse was 62 years, but the middleage of a horse is 30 years ), i.e. the age of a horse x ! [0, 62]- Height: For example the height of a student in a Countryis between 110 cm (person elf) and 226 cm (person giant), i.e.the height of a student x ! [110, 226]Variables classified according to how they are categorized ormeasured. For example, the data could be organized into specificcategories, such as major field (mathematics, computers, etc.),nationality or religious affiliation. On the other hand, can the datavalues could be ranked, such as grade A, B, C, D, F h or ratingscale (poor, good, excellent), or they can be classified accordingto the values obtained from measurement, such as temperature,heights or IQ scores. Therefore we need to distinguish betweenthem through the measurement scale used. There are four levelsof measurement scales; nominal, ordinal, interval, and the ratiolevel of measurement, the difference between these four levels isexplained in the following definitions.Definition1.2.8The nominal level of measurement classifies data into mutuallyexclusive (disjoint) categories in which no order or ranking canbe imposed on the data.The following examples include nominal level of measurements indifferent cases.8

BasicConceptsIn StatisticsEquationsand InequalitiesCHAPTERCHAPTER 11Example 6- Gender: Male, Female.- Eye color: Black, Brown, Blue, Green, .- Religious affiliation: Muslim, Christian, Jew, .- Nationality: Saudi, Syrian, Jordanian, Egyptian, Pakistani, .- Scientific major field: statistics, mathematics, computers,Geography, .When the classification takes ranks into consideration, the ordinallevel of measurement is preferred to be used. The following definition provided us this concept.Definition1.2.9The ordinal level of measurement classifies data into categoriesthat can be ordered, however precise differences between theranks do not exist.The following examples include some ordinal level of measurements.Example 7Grade A, B, C, D, F h: Grading technique is the most commonexample on ordinal level. For example we find that the systemof appreciation in Saudi universities are (in descending order)A , A, B , B, C , C, D , D, F.- Rating scale (bad, good, excellent and so on .): To test thequality of the canned product, we find that the state of thetested object either excellent or good or bad.- Ranking of football players: A football player can be rankedin first grade, second grade, third grade, .- Ranks of university faculty members: Academic ranks usuallyclassified as professor, associate professor, assistant professor ,and instructor.The third level of measurement is called interval level. The following definition provided us this concept.9

CHAPTER 1Basic Concepts In StatisticsDefinition1.2.10The interval level of measurement orders data with precisedifferences between units of measure. (in this case there is nomeaningful zero). On the other hand, the resulting measurementvalues belong to an interval of the real numbers.Example 8- IELTS: An International English Language Testing System.ILETS is an international system to test the English languagein order to study and work. The degree x to which the grantwill be between zero and 9, i.e. x ! [0, 9]-TOEFL: Test of English as a Foreign Language. TOEFL isa standardized test of English language proficiency for nonnative English language speakers wishing to enroll in someuniversities in the world.- SAT score and IQ test: The SAT is a standardized test widelyused for college admissions in some universities in the world.It is a good predictor of a student’s performance in the firstyear of college. Degree to which the grant will be betweenzero and 2400, i.e. x ! [0, 2400]- Temperature: When the degrees of temperatures aremeasured in Celsius or Fahrenheit, then the values that weobtain from absolute zero (-273.15 but without this degree)extends to millions as is the case in the sun and stars.Note that if we compare between two temperature degrees, like30c C and 60c C we can’t say that 60c C is as high as twice the degree30c C ; but we can say there is a 30c C difference between them. Inthe sense that we can not be compared to some of the quantitiesto others in this case. On the other hand, if the temperature ofsomething equals to zero, that does not mean it does not have atemperature.That mean in the Celsius temperature the zero means there isa temperature and it is very cold that is the zero does not meannothingness. The data at this level do not have a natural zerostarting point. The measurements that rely (or that adopt) zeroas starting point called ratio level and offered us the followingdefinition10

BasicConceptsIn StatisticsEquationsand InequalitiesDefinitionCHAPTERCHAPTER 111.2.11The ratio level of measurement is the interval level withadditional property that there is also a natural zero startingpoint. In this type of measurement zero means nothingness.Another difference lies in that we can attribute some of thequantities to others.The following examples include some of ratio level measurement,we note what is the meaning of something that takes a temperatureequal to zeroExample 9- Distance: The distance between two cities X and Y, wherewe find that the measurement is an interval level, but becauseof that we can say that the distance between the two citiesX and Y is equal twice the distance between the cities X andZ, they become standard ratio scale. Note that here zero hasa meaning, because if the distance is equal to zero, it meansthat the city (position) itself. Here we note that the conceptsof length and height are a special case of the distance concept.- Age: The ages of people fall under this category ofmeasurements, because the zero here means that the personwas born dead and that old equals zero.- Time: The time required to get from home to work is ameasurement of type ratio level and zero here means that ithas not yet kicks off. Note that here we can attribute someof the time to others, if we say the time required to get fromhome to work is equal twice the time needed to get home fromwork.- Salary: The value of salary for someone is a measurement oftype ratio level, where we can attribute values of wages to eachother, as if to say that the person X receives a salary twice thesalary of the person Y. And zero here means that the persondid not receive a salary.- Weights: If we take weights of fruit boxes, then note thatthe values are of measurements of type ratio level, due to thepossibility of the weights attributed to each other, where wecan say that an apple box weight is equal twice the weight ofthe orange box, and the zero here it is linked to accuratelydevice which measures the weight. And so that there is nothing11

CHAPTER 1Basic Concepts In Statisticson the earth has no weight because of Earth’s gravity. Forexample, if we have the weight of a box containing apple andusing the balance of accuracy 200 grams, we will not get tozero absolute (it is enough having one apple in the box so thatdoes not refer to zero), while if we weigh it in another balanceaccuracy tons, the balance will refers to the value zero even ifthe box full of apples.Here there is a meaningful zero.The graph below summarize the classification of ntinuousFigure 1.2: Classification of variables1.3Sampling TechniquesIts known that in some cases, it’s hard to study a large populationin order to make conclusions about certain phenomena, for exampleif we interested in studying the obesity in kingdom of SaudiArabia, imagine that the researcher has a limited period of timesay three months, its ‘ impossible’ to survey all citizens in KSA tomake conclusions about such phenomenan during the determinedperiod of time, so that sampling methodology is the best solutionin order to perform the study and get representative results duringshorter period and also it saves efforts and money.Sampling methodology as illustrated in section (1.1) suggestsselecting a portion of elements of population under study in orderto make statistical analyses to make decisions about phenomenanunder study. Therefore, sampling is not include the selectionof elements arbitrary. So that, there are several techniques ofsampling were established according to the type of analysis used12

BasicConceptsIn StatisticsEquationsand InequalitiesCHAPTERCHAPTER 11Some of these techniques are the simple random sampling method,the systematic method, the stratified method and the clusteredsampling method. Differences between such methods refer tomany circumstances such as population size, degree of accuracydetermined by the researcher, type of elements of population andthe number of categories in the population under study.Now, let us discuss the mentioned methods in details starting withthe simple random sampling method.A Simple Random Sampling MethodIt’s the simplest method for sampling and it is applicable whenthe population is slightly small. In order to get a sample of thistype the elements of population should be to achieve the followingconditions:1. All elements of population have the same chance of choice,2. All elements of population are independent.After verification of the fulfillment of these conditions the elementsof population take serial numbers and then use one of the methodsthat used in randomization order to pull the required elementsof the sample, and in this regard can be used a table of randomnumbers (See table 1, 98064688Table 1: Random Numbers13

CHAPTER 1Basic Concepts In StatisticsThree steps to use the random number table such steps are:1. Close your eyes2. Point your finger anywhere in the random numbers table.3. Open your eyes and begin reading the digits beginning whereyour finger touches the table)Example 1To select a random sample consisting of 10 elements out of90 elements, it is necessary to number each element from01, 02, 03, g to 90 . Then select a starting number by closingyour eyes and placing your finger on a number in the table 1.Suppose, in this case your finger is landed on the number 19in the forth column. Then proceed downward until you haveselected 10 different numbers between 01 and 90 . When youreach the bottom of the column, go to the next column. If youselect a number greater than 90 or the number 00 or duplicatenumber, just omit it. In our example, we will use the elementsnumbered: 19, 69, 17, 07, 31, 27, 75, 42, 67, and 55.Even that simple sample method is easy to perform, but it has somedisadvantages that make it not the best choice to use, especiallywhen we are talking about large populations, it costs more moneyand much time, many samples can be selected using this method, butthey might give same results. So that, statisticians and researchersdeveloped other alternatives to be used to get more convenient.B Systematic Sampling MethodSuppose we want to take a sample with size n using this method,we are including the following:1- We giving the elements of the population serial numbers from1 up to N2- Determining an interval (called the withdrawal period). Thisinterval can be computed their width by dividing the size of thepopulation that we are interested by the required sample size.k Nn3- Then we randomly select number located between 1 and k(Let s, for example), so the element that holds this number sis the start element in the sample.4- Take elements from population that bear numbers s t k with1 t n 1 . So we get the required sample.14

BasicConceptsIn StatisticsEquationsand InequalitiesCHAPTERCHAPTER 11To understand this, assume example 2Examp

Statistics is a branch of science dealing with collecting, organizing, summarizing, analysing and making decisions from data. Definition 1.1.1 Statistics is divided into two main areas, which are descriptive and inferential statistics. A Descriptive Statistics