Module 1

Module 1 Basic Concepts in Statistics

Introduction Everyday we often listen to our friends on such matters such as “How many units have you enrolled this semester” or “What is you average grade this semester” and so forth and so on. Or we listen to a barrage of statistic from our University Registrar telling us that “The total enrollment this semester is almost 6,500 with 3,500 males and 3,000 females. The total enrollment for the College of Arts and Sciences is 1600, with 879 number of students enrolled in the College of Agriculture and so forth and so on. We often listen to the different vital statistics of Miss CAS, Miss BA, Miss Agriculture, etc. during Miss CLSU as 34-24-34 or 36-25-35 and so on. Or you may ask the question of “What is the probability that you will graduate two years from now?” or “What is the probability that Sir will be absent next meeting?” However, statistics is more than these. Statistics enters into almost every phase of life in some way or another. A daily news broadcast may start with a weather forecast and end with an analysis of the stock market. In a newspaper at hand we see in the first five pages stories on an increase in the wholesale price of sugar, an increase in the number of crimes committed, new findings on mothers who smoke, the urgent need for laws, a school plan for evaluation of teachers, popularity of cell phones, a tuition fee increase and sex bias in the government working force. Each article reports some information, proposal or conclusion based on the organization and analysis of numerical data. Statistics in systematic and penetrating ways provides bases for investigations in many fields of knowledge, such as

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social, physical sciences, engineering, education, business, medicine and law. Information on a topic is acquired in the form of numbers; an analysis of these data is made in order to obtain a better understanding of the phenomenon of interest; and some conclusions may be drawn. Often generalizations are sought; their validity is assessed by further investigations. A definition of statistics is making sense out of figures. Statistics is the methodology which scientists and mathematicians have developed for interpreting and drawing conclusions from data. This chapter begins with the real definitions of statistics and the basic terminologies and concepts underlying the subject of statistics.

Objectives: At the end of this module, you should be able to: 1. Differentiate descriptive and inferential statistics. 2. Differentiate a continuous variable from a discrete variable and quantitative variable from qualitative variable. 3. Classify data according to level of measurement. 4. Employ summation notation and apply operations involving the summation. Definition of Statistics Statistics – is a branch of science which deals with the collection, organization, summary, presentation and analyses of quantitative data as well as drawing valid conclusions and making reasonable decisions on the bases of such analyses. The analysis of data collected in the course of study is among the most important activities performed by the researcher. Unfortunately, it is not given very much attention until the moment it is scheduled to begin. This module stresses that planning for data analysis begins when a study is just getting

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underway and it continues until all hypotheses or questions being considered have been satisfactorily resolved. Data analysis is the focus because: 1. The value of an entire study may depend upon the analyses one chooses to make or not to make. All the time spent obtaining permission to conduct a study and selecting samples and instruments may be wasted, in whole or in part, if careful attention is not given to how data will eventually be analyzed. 2. Planning data analysis procedures before data are collected insures that the right information will be collected and in a form suitable for later use. Learning too late that additional data would have made your study far more valuable is justification enough to plan ahead. It is not unusual for researchers to find that for every hour spent in planning the exact format in which data should be collected, as many as ten hours are saved in the analysis phase. 3. Knowledge of available procedures for data analysis should lead you to make more useful sets of findings and implications. This should cause others to take your work more seriously than they would if this section were given inadequate attention. 4. Data analysis is not that difficult anyway. There now exists a multitude of computer programs, many of which are designed in a “user friendly” format. All it need to take to do an analysis is the ability to push the button on the computer corresponding to the number of your selection. Data analysis can be simple as that and still be powerful enough to accomplish the purposes discussed above. With a bit more prodding, some computers can do much more to help us to generate study findings which will make our previous efforts even more worthwhile. DESCRIPTIVE AND INFERENTIAL STATISTICS

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The emphasis on the decision-making aspects of statistics is a recent one. In its early years, the study of statistics largely consisted of methodology for summarizing or describing numerical data. This area of study has become known as descriptive statistics because it is concerned largely with summary calculations and graphical displays. These methods are in contrast to the modern statistical approach in which generalizations are made about the whole by investigating a portion. Thus, the average income of all families in the Philippines can be estimated from figures obtained from a few hundred families. Such a prediction or estimate is an example of inference. The study of how inferences are made from numerical data is called inferential statistics. VARIABLES Variables are the factors that we focus on in a given study. They are the characteristics of interest of the study which are inherent of the object or person. Example of such variables are sex of the grade I pupils, number of children in the family, age of father, family income, color of the eye of the person, nationality, attitude of farmers, behavior of kindergarten pupils, etc… Kinds of Variables: 1. Continuous variable – takes any value within a specified range of values. It usually gives rise to measurement. Example: height, weight, volume, age 2. Discrete variable – takes integral values. It usually gives rise to counting numbers. Example: number of children, number of road accidents Types of Variables: 1. Quantitative variables – those variables which are expressed numerically. Example: height, weight, number of children in the family

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2. Qualitative variables – those variables which are expressed categorically. Example: color of the eye, sex, military rank Levels of Measurements 1. Nominal: Nominal data consist of numbers which indicate categories for purely classification or identification purposes. With nominal data, the numbers themselves have no mathematical value assigned to them. The number on the back of a basketball player, for example, is at a nominal level. It makes no sense to add the number of the center (12) to the number of the guard (33). In effect the numbers are really names. Mathematically, all one may do with nominal data is to count how many are in each category. Another example is sex where we assigned 1 for male and 2 for female. 2. Ordinal: Ordinal data possess a rank order characteristics, but they do not provide information about the distance between each rank. Thus if I know that Marvin is ranked best in Mathematics, Melvin is next and MJ is third best, they may be assigned values of “1”, “2”, and “3”, respectively. One does not know, however, how much better in mathematics Marvin is as compared to Melvin, or Melvin compared to MJ. With ordinal data the intervals between the ranks are not equal. Likewise, the only mathematical symbol that can be used in an ordinal data is greater than (>) or less than (<). 3. Interval: Interval data possess equal intervals providing information about how much better one value is compared with another. Usually, we assume that our mental ability test, achievement test and attitudinal test scores are examples of interval data. Further, interval data have no absolute zero, that is, zero is just an arbitrary value. If the temperature reading is 0oC, it does not mean the absence of temperature but rather the temperature reaches the freezing point. Or if the IQ score is zero, it does not mean the absence of knowledge but rather the individual belongs to the low (or very low) performer category. Furthermore, interval level can differentiate between any two classes in terms of degrees of differences. Aside from the mathematical symbol > and <, addition and subtraction have meanings.

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4. Ratio: Ratio data possess the characteristics of interval measures and they have a true zero or absolute zero which indicates the total absence of the property being measured. For example, measures of height, weight and age are typical ratio scales since all of them have a zero value. All mathematical procedures are appropriate with ratio scales.

SAQ1 1. Categorize each of the following as either nominal, ordinal, interval or ratio measurement: a. first, second and third place in a singing contest b. metric measurement of distance c. house numbers d. cell phone numbers e. number of live births in December, 2000 Notations and Symbols f. attitude towards impeachment: 1=high 2=moderate 3=low 2. Categorize each of the following variables as either discrete or continuous. a. number of students who score 80 and above in the NSAT exam b. distance of the school to your house c. number of chairs in the auditorium d. number of faculty members in you school e. floor area of our classroom (in sq. ft.) f. scores of students in an examination

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Notations and Symbols In the study of statistics, we cannot avoid the use of the different notations and symbols. If our variable of interest is age, then we let the symbol X stand for the variable age. Similarly, if the ages of the 4 students are 15, 18, 19, and 15 then we can write the following as: X1 = 15, X2 = 18, X3 = 19 and X4 = 15. Generally X1,X2, X3, and X4 can be written down as Xi (read as X sub i) where i is known as the index which locates the value of the variable in the set. Note that X1 and X4 have the same value, however, the value of X1 refers to student number 1 while X 4 refers to student number 4. Formally, we write the symbol for the variable and the value of the variable (known as the variate) as follows: X = (X1,X2, …, Xn ) Frequently, it is necessary to work with sums of numerical values. Using the Greek letter  (capital sigma) to indicate “summation of”, we can write the sum of the 4 ages as 4

X i i 1

where we read “summation of Xi, i going from 1 to 4”. The numbers 1 to 4 are called the lower and upper limits of summation. Hence 4

 X i= i 1

X1 + X2 + X3 + X4

= 15 + 18 + 19 + 15 = 67 n

In general, the symbol

 i 1

means that we replace i whenever it

appears after the summation symbol by 1, then by 2, and so on up to n and then add up the terms. The subscript may be changed to any letter, although i is seen to be written in most textbooks. When we are summing over all the values, instead of using we use  to mean that the sum is taken from the first observation to the nth observation. PROPERTIES OF SUMMATION SIGN

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1. The sum of n number of constant k is  k = nk 2. The sum of the constant k times the variable Xi is equal to the constant k times the sum of the variable Xi, that is,  k X i = k  Xi 3. The summation of the sum or difference of 2 or more variables is equal to the sum or difference of terms taken separately, that is  (Xi ± Yi) =  Xi ±  Yi 4. The sum of the squares of variables is obtained by first taking the square of all the observations and then get the sum, that is,  Xi2 = X12 + X22 + X32 + … + Xn2 5. The square of the sum of the variable is obtained by first taking the sum of all the observations and then take the square, that is ( Xi )2 = (X1 + X2 + X3 + … + Xn )2 6. The sum of the product of 2 variables X and Y is obtained by first taking the product of the 2 variables then take the sum, that is  (Xi Yi) = (X1Y1) + (X2Y2) + (X3Y3) + … + (XnYn) 7. The sum of the product of 2 variables X and Y is obtained by first taking the sum of X and the sum of Y separately and then take the product, that is ( Xi )( Yi )= (X1 + X2 + X3 + … + Xn)(Y1 + Y2 + Y3 + … + Yn)

Activity

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A. Write out in full the sums represented by the following expressions: 8

1.

 4Wi

4.

 X

2.

i 4 6

j 1

j

 Yj 

9

5

5

i4

i 1

i 1

  Xi / Yi  3.  X i  Yi  4

2

4

5

i2

i 3

5.   X i  3   Yi  2 

B. Write each of the following expressions in summation notation with appropriate limits. 1.  a1  b1    a 2  b2    a 3  b3    a 4  b4  3 2.  R1  R2  R3  R4  R5  R6  R7  R8  3 4 3.  X 1  X 2  X 3  X 4   Y1  Y2  Y3  Y4 

2 2 2 2 2 4. W1  W2  W3  W4  W5

5.

X1  X2  X3  X4  X5  X6  X7  X8 Y1  Y 2  Y 3  Y 4  Y 5  Y 6  Y 7  Y 8

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C. For data set below, compute the following:

a.

X

b.

i

1 f.  X i n

j.

X Y

i i

Y

i

1 g.  Yi n



c.

X

h.

X

2 i

2 i

d.

Y

i

 X   i

n

2

i.

2

e.

Y

i

2

X Y i

Y  

i

2

i

n

 X Y i

i

n

Data on the frequency of watching TV (X) and frequency of reading books (Y) of n = 10 pupils per week. Pupi 1 2 3 4 5 6 7 8 9 10 l X Y

5 2

3 4

4 3

2 5

1 6

0 7

6 1

5 2

D. Given a set of values X 1 , X 2 ,..., X N and Y1 , Y2 ,..., YN . If A 

X

i

N

and B 

Y

i

N

Prove the following: a.

 X

b.

 X

c.

 X

d.

 X

e.

 X

i

 A  0

 A   X i  NA2 2

i

i

 A X i  B  

i

 X Y  B i

i

 1   X i  2 X i  N 2

i

2

2

 A X i  B  

 X Y  NAB i i

2 5

3 4