STAT 3206

“status” ; state

Statistics is derived from the Latin word ________ with the meaning ________.

Plural

In _______ sense, statistics is defined as any set of numerical data (e.g. vital statistics, monthly sales)

Singular

In ______ sense, statistics is defined as a branch of science that deals with the collection, presentation, analysis, and interpretation of data

Aids in decision making

Summarizes data for public use

Roles of Statistics

Descriptive

Inferential

2 Areas of Statistics

Descriptive Statistics

Area of Statistics that is concerned with describing a set of data **without drawing conclusions or inferences** from it.

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It includes collecting, presenting, and analyzing of data.

Inferential Statistics

Area of Statistics that utilizes sample data to **make inferences and draw conclusions** about a larger set of data.

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It includes interpreting, making inferences, hypothesis testing, determining relationships, and making predictions.

Data

facts or figures from which conclusions may be drawn

Data Set

collection of facts and figures or data

Elements/Units

entities on which data are collected

Variable

a characteristic or attribute of elements which can assume different values or labels under statistical study

Observation

set of measurements collected for a particular element

Qualitative Variable

Quantitative Variable

2 Types of Variables

Qualitative Variable

outcomes of the variables expressed **non- numerically or categorically**

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example: name, gender, eye color, religion, etc.

Quantitative Variable

outcomes are expressed **numerically** that are meaningful or indicate some sort of amount

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example: age, allowance, number of students, height, etc.

Quantitative Discrete Variable

Quantitative Continuous Variable

2 Kinds of Quantitative Variables

Quantitative Discrete Variable

It is a variable which can assume **finite**, or at most , countably infinite number of values.

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It is usually measured by counting. It answers the question “how many”.

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example: # of students, # of children

Quantitative Continuous Variable

It is a variable which can assume infinitely many values corresponding to a line interval

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It gives rise to measurement. It answers the question “how much”.

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example: weight, allowance, height

Nominal

Ordinal

Interval

Ratio

Scale/Levels of Measurement of Variables

Nominal

It is a classificatory scale. It is the **weakest** level of measurement where numbers or symbols are used simply for l**abeling or categorizing** subjects into different groups

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example: sex (male/female)

Ordinal

It is classificatory with ordering scale. It is numbers assigned to categories of any variable may be **ranked or ordered.**

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example: educational attainment (elementary/HS/college/MS/PhD)

Interval

It has the properties of the nominal and ordinal levels. The distances between any two numbers on the scale are of known sizes.

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It has arbitrary zero.

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example: temperature

arbitrary zero

zero does not mean nothing

Ratio

It is the highest level of measurement. It has the properties of the nominal, ordinal, and interval levels. It is anything that is countable or measurable.

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It has absolute zero or true zero.

absolute zero or true zero

zero means nothing

Primary Data

It is acquired directly from the **original** source of information. Data that are measured or gathered by the researcher themselves

Secondary Data

data taken from published or unpublished data which have been **previously gathered by others**

Subjective Data

It means “from someone’s point of view”. Data that is commonly about perceptions, beliefs, feelings, and opinions.

Objective Data

fact-based, measurable, countable, and observable data

Interview

Questionnaire

Experimental

Observation

Registration

5 Data Collection Methods

Interview

__**Data Collection Methods**__. There is a **person-to-person contact** or exchange of information between the interviewer and interviewee.

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It is more appropriate for obtaining **complex emotional-laden** topics probing sentiments underlying an expressed opinion. It provides consistent and more precise information since the interviewee may give clarifications.

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It is time consuming and has limited field of coverage

Questionnaire

__**Data Collection Methods**__. Data are collected by means of **written responses based on a list of questions** which are relevant to the problems of the study.

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It inexpensive and can cover a wide area in a shorter period of time

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It has high possibility of incomplete response or may not return the questionnaire, especially if it is mailed.

Experimental

__**Data Collection Methods**__. It is used when the objective is to determine the **cause-and-effect relationship** of certain phenomena under controlled conditions

Observation

__**Data Collection Methods**__. The researcher **observes** the behavior of persons and their outcomes. The potential bias caused by the interviewing process is reduced and eliminated in this method

Registration

__**Data Collection Methods**__. This method of collecting data is **enforced by** **certain laws** such as registration of births, deaths, licenses, etc.i Information are kept systematized and made available to all because of the requirement of the law.

Population

entire group of observations or elements where inferences and conclusions are made

Parameter

a numerical characteristic of the population

Sample

subset of the entire group of observations or elements where data is collected

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representative of the population

Statistic

a numerical characteristic of the sample

Census/Complete Enumeration

Sampling/Survey Sampling

General Classification of Collecting Data

Census/Complete Enumeration

process of gathering information from every unit or all the units of the population

Sampling/Survey Sampling

process of obtaining a part or subset of the population

less cost

greater accuracy

greater speed

greater scope

Why do we sample?

Probability Sampling

Nonprobability Sampling

Types of Sampling Methods

Probability Sampling

__**Types of Sampling Methods.**__ Each unit in the population has a known, non-zero probability of selection, and **have equal chances** of being selected as a sample.

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It uses some **chance mechanism**

Nonprobability Sampling

__**Types of Sampling Methods.**__ The elements in the population **do not have equal chances** of being selected as a sample.

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Elements of the population are taken depending to a large extent on the **personal feelings or purpose** of the researcher and **without regard for some chance mechanism** for choosing an element

sampling frame,

listing of all individual units in the population which is required in the execution of probability sampling methods

Simple Random Sampling (SRS)

Systematic Sampling

Stratified Sampling

Cluster Sampling

4 Types of Probability Sampling Methods

Simple Random Sampling (SRS)

Method of selecting ***n*** units out of ***N*** units in the population where all elements in the population have an **equal chance** of being included in the sample.

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This sampling method is suitable when the **population** being studied is **homogeneous or have the same characteristics.**

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ex. draw lots, random number generator

SRS with Replacement (SRSWR)

SRS without Replacement (SRSWOR)

2 Types of Simple Random Sampling (SRS)

SRS with Replacement (SRSWR)

Type of SRS where a chosen element is **always replaced** before the next selection is made.

SRS without Replacement (SRSWOR)

Type of SRS where a chosen element is **not replaced** before the next selection is made

Systematic Sampling

Type of probability sampling method which is a method of selecting a sample by taking **every** **kth** **unit** from an ordered population, where the first unit being selected at random

sampling interval

In systematic sampling, what is k?

Stratified Sampling

It is done if the population is **heterogeneous** and can be subdivided into non-overlapping **homogeneous subpopulation** called ***strata***.

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Samples are then randomly selected from all the strata using SRS or systematic sampling

Cluster Sampling

A method of sampling where a sample of distinct groups, or ***clusters***, of elements is **randomly selected** and then a census or all elements in the selected clusters is taken.

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Clusters are non-overlapping subpopulations which together comprise the entire population, and is preferably formed with heterogeneous.

Purposive Sampling

Convenience Sampling

Quota Sampling

Snowball Sampling

4 Types of Non-probability Sampling

Summation Symbol

Upper Limit

Index of Summation

Lower Limit

Summand

Parts of Summation Notation (from upper left, upper right, lower left, …)

Rules on Summation

(ewn paano ipapasok huhu)

measure of central tendency

a value at the center or middle of a data set, that is, the value where the data tend to cluster

Mean

Measure of Central Tendency that is **average** value.

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It is susceptible to extreme values, single value, and continuous data. It works well with many statistcial methods.

Median

Measure of Central Tendency that is the **middle value** **of an ordered data**.

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It is not susceptible to extreme values, single value, and continuous data. It is often a good choice if there are some extreme observations.

Mode

Measure of Central Tendency that is the **most frequent value**. It locates the point where the observation values occur with the greatest density

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It can be a single value, multiple values, may not exist. It can be categorical or continuous data. It is appropriate for data at **nominal and ordinal level.**

Measure of variability or dispersion

It indicates the extent to which observations in a data set are **scattered about an average**. It is also used as a **measure of reliability** **of the average value**

True

__**True or False**__. The higher the measure of variability, the more dispersed the data is

Range

Variance

Standard Deviation

Standard Error of the Mean

Coefficient of Variation

Measures of Variability

Range

A measure of variability which is the difference between the **highest value and the lowest value** in the data set.

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It uses extreme values; an outlier can greatly alter its value. It fails to communicate any information about the clustering.

Variance

A measure of variability which refers to the **mean of the squared deviations** **of the observation from the mean**. It is not a measure of absolute dispersion. It can only take the values from 0 to +∞

Standard Deviation

A measure of variability which refers to the **positive square root of variance**. It is the **measure of absolute dispersion**.

Standard Error

A measure of variability which refers to the **standard deviation of the sampling distribution of the mean**. It provides a tolerance of an estimate of the mean which is calculated from a sample

Coefficient of Variation

A measure of variability which refers to the **ratio of the standard deviation and the mean** and is expressed in percentage.

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It is Unitless. It is used to **compare the variability** of two or more data sets

Measures of Skewness

It tells us the distribution of data. It can be revealted through a comparison of the mean, median and mode.

skewed

A distribution of data is _____ if it is **not symmetric** and extends more to one side than the other

symmetric

A distribution of data is ______ if the left half of its histogram is roughly a **mirror image** **of its right half**

Positive Skew

__**Measures of Skewness.**__ It is ‘skewed to the right’. It has more concentration of values below the mean.

Negative Skew

__**Measures of Skewness.**__ It is skewed to the left. It has more concentration of values abovethe mean.

Symmetrical Distribution

__**Measures of Skewness.**__ It is ‘normally distributed’. It has approximately same values for the three central tendencies

Percentile

Quartile

Measure of Location

Quartile

It divides a set of data into **four groups** with about 25% of the values in each group.

Boxplot

Graphical Presentation that gives information about the **distribution and spread of data.** It shows information on the minimum and maximum value, Q1,Q2, and the median

Histogram

Graphical Presentation in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis.

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The class frequencies are represented by the heights of the bars, and the bars are drawn adjacent to each other.

Scatterplot

Graphical Presentation that is used to evaluate the relationship between two different continuous variables.

Probability

It is a quantitative measure of uncertainty. It is a number that expresses the strength of our belief in the **occurrence of an uncertain event**

Random experiment

It is any process that allows researchers to obtain observations. It is any process that can be repeated under basically same conditions and yields well defined outcomes.

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ex. toss coin

Sample space (S)

It is the set of all possible outcomes of a random experiment.

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ex. {head, tail}

sample points

Elements of the sample space.

n(S)

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ex. toss coin, n(S) = 2

number of sample points is denoted by ____

Event

A subset of the sample space

Classical Approach

Relative Frequency Approach

Subjective Probability

Three Types of Probability

Classical Approach

Based on the idea that **certain occurrences are equally likely**, that is, we assume that in a given experiment, **all the sample points in the sample space have equal chances** of occurring

priori probability

Classical Approach is also called as ______. We can state the answer in advance without performing the experiment.

Relative Frequency Approach

An **experiment is conducted or observed** in large number of times that an event actually occurs, that is, probabilities are determined based on **experimental approach**

posteriori probability or empirical method

Relative Frequency Approach is also called as ______

Subjective Approach

It is based on the beliefs of the person making the probability assessment.

Law of Large Numbers

This law states that “**as a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability**”

random variable

a variable that has a single numerical value (determined by chance) for each outcome of a random experiment

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It is denoted by X, Y, Z.

Discrete Random Variable

Continuous Random Variable

2 Types of Random Variable

Discrete Random Variable

__**Types of Random Variable**__. It has either a **finite** number of values or a countable number of values.

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ex. coin (h or t)

Continuous Random Variable

__**Types of Random Variable**__. It has **infinitely** many values which can be associated with measurements on a continuous scale

Probability distribution

the listing of all possible value that a random variable can take on together with their corresponding probabilities.