An Overview of Basic Statistics

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AN OVERVIEW OF BASIC STATISTICS

Statistics being a branch of mathematics is often associated with anxiety or unease particularly among students who fear mathematics for one reason or another. Well, if you are one of those students then sit, relax, and read this chapter first as it will take you through a journey in which you will discover a world of fun, excitement, vision, and creativity. With minimum mathematical details, this chapter introduces key concepts and universal terms that are used among statisticians and briefly discusses common statistical tools, their underlying principles and their practical merits.

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Why should you learn statistics?In the general sense, statistics is the science of dealing with variability, uncertainty, and subjectivity to produce objective and quantitative information that can assist in making reliable decisions about numerous situations in life. Globally, statistics is a key tool in governments and organizations activities.

The reason we need statistics is that we are living in a world of numbers, or more precisely a world of data.

• Numbers are called data, only when they reflect information

• Data, on the other hand, are called statistics when they reflect specific or descriptive measures of the phenomenon or the event under study

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What is statistics?

Definition Source"The mathematics of the collection, organization, and interpretation of numerical data, especially the analysis of population characteristics by inference from sampling."

American Heritage Dictionary®

"A branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data."

The Merriam-Webster’s Collegiate Dictionary®

“The scientific application of mathematical principles to the collection, analysis, and presentation of numerical data.”

The American Statistical Association (ASA, http://www.amstat.org/)

Definitions of statistics

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http://www.amstat.org/

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What is statistics?

“The science and art of reading, describing, and manipulating data, which represents variables so that practical observations about a population can be made from a sample drawn from the population, and guidelines can be established to allow making precise and accurate conclusions about a certain process or system”

Statistics: between science and art

• Science stems from the use of mathematical concepts• Art stems from extrapolation, interpretation, and judgment

It is well-known, statistics does not provide causes and effects;it only yields analysis outcome based on the data used. It is then your job to provide causes and effects.

This is where the art of reading data and interpreting the results of the statistical analysis comes into play.

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What is statistics?


Numbers, data and statistics

100, 140, 213, 230, 180, 211, 120, 160, 200, 110, 260, 235, 280, 180, 300

NUMBERS

Human Weight (pound):

100, 140, 213, 230, 180, 211, 120, 160, 200, 110, 260, 235, 280, 180, 300

DATA

Statistics

Mean Value = 194.6 poundMinimum = 100 poundMaximum = 300 poundRange = 200 pounds

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What is statistics?


A Constant is a parameter that does not change:

- Your social security number will not change over time- Your birth date will not change over time

A Constant may also refer to a controlled variable:

*** If you can maintain your grade at an “A” from course to course, it becomes a controlled variable or a constant

A Variable is a parameter that is likely to change

- Grades of different students- Your income from one year to another - The heights of different students in your class

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What is statistics?


Population Sample

A population implies a totality or a complete collection of things (all students in a college, all people in a town, all machines in a factory, and so on)

A sample is a sub-collection of units or components selected from a population (few students from a college, few people from a town, and few machines from a factory, and so on) A census is a collection of data from every member of the population (all students’ grades, all people ages, and all machines’ efficiencies, and so on).

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Example: In a survey conducted in a community college of 5000 students, 800 students were selected randomly and asked if they would transfer to a four-year university. Five hundred and fifty of the students said yes. Identify the population and the sample. Describe the data set.

What is statistics?


Solution:

• The population consists of all students in the college (5000 students)

• The sample consists of all the students who were randomly selected (800 students)

• The actual data set consists of 550 yes’s and 250’s no’s. • 68.75% said “Yes” & 31.25% said “No”

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Working Problem 1.1:

Major television networks constantly monitor the popularity of their programs via asking

some specialized organizations such as Nielsen company to sample the preferences of

TV viewers.

(a) Suppose 1000 TV prime-time viewers selected randomly were asked if they watched a new talk-show, and 450 indicated they watched the show. Identify the population and the sample. Describe the data set.

(b) In another survey, suppose 1100 TV prime-time viewers selected randomly were asked if they watched the 2009 Super Ball on TV, and 999 indicated they watched the game. Identify the population and the sample. Describe the data set.

Cancelled

What is statistics?


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What is statistics?


Statistic and parameter

A statistic is any statistical measure of sample data

Examples: Sample Mean value and Sample Range

A parameter is any statistical measure of population data

It can either be calculated from an entire population data or estimated from sample data taken from a population

Examples: Population Mean value and Population Range

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Example: Decide whether the numerical values given below describe a sample statistic or a population parameter.

(a) A sample of community college professors in the U.S.A. revealed that the average starting salary of a college professor is $52,000 and the range of salaries is $62,000.

(b) In a college survey of all freshmen students, it was revealed that 85% of the students were fresh out of the high school and 15% were students who graduated from high schools more than 5 years ago


What is statistics?

Solution:

(a) The average is a sample statistic and the range is a sample statistic

(b) The proportion or percent of students is a parameter describing the population of freshmen students

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Determine whether the numerical values given below describe a sample statistic or a population parameter.

(a) Based on monitoring all units of a product prior to shipping, a company indicates that the percent of second-quality units is 2%

(b) New residents of an apartment complex were asked if they like the landscaping surrounding the complex. Eighty five percent indicated that they like it.

(c) The average salary of a group of 120 employees selected randomly from different divisions of a company was found to be $65,000


What is statistics?

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What is statistics?

What is the difference between precision and accuracy?

Accuracy means: all data meet a target value

Precision means: all data are very close in values

Accurate measure: A measure is said to be accurate when the measured values are very close to a target or an actual value.

Precise measure: A measure is said to be precise when the measured values are close to each other.

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Inaccurate/Precise Accurate/Imprecise

Inaccurate/Imprecise Accurate/Precise

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What is statistics?


Suppose a lab refrigerator holds a constant temperature of 38.0 F. A temperature sensor is tested 10 times in the refrigerator. The temperatures from the test yield the following values of temperatures:

38.3 37.8 36.0 38.3 38.2 37.6 38.2 38.4 37.9 38.3 39.0

Describe these values in terms of precision and accuracy and explain your answer.

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What is statistics?

• A process implies one or more of five basic elements: machine, material, methodology, people, and environment.• A system is an entity that has inputs and outputs

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Education Process

Environment(Classroom or

Laboratory)

Methodology(Face-to-Face,

Distance Learning)

Machine(Projectors & computers)

People(Teachers & Students)

Material(Books &

Notes)

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System(A)

(Weaving Machine)Yarns Fabric

Input Output

Examples of System Data:

Inputs: Outputs:- Fiber type - Fabric width- Yarn strength - Fabric thickness- Yarn diameter - Fabric weight

Output-Input Relationships

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Descriptive statistics: how to describe data

Problem: The four sets of data shown below represent student grades of four consecutive statistics quizzes given to a class of 10 students. Describe each set of data by closely observing it and writing your comments.

1 2 3 4 5 6 7 8 9 10Quiz 1 90 90 90 90 90 90 90 90 90 90Quiz 2 82 86 78 30 88 82 79 77 81 99Quiz 3 68 90 89 71 92 95 73 75 94 66Quiz 4 82 76 85 88 95 86 84 87 96 78

• The starting point in any statistical analysis is to read data using the language of statistics

• This language uses the so called ‘descriptive statistics’ to establish an organized and meaningful display of data with the goal being to reduce the flood of data presented down to few statistics that can fully describe the data

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Simple Numerical descriptive statistics

Grades 82 76 85 88 95 86 84 87 96 78

Using statistics, we can provide two types of description:

76 78 82 84 85 86 87 88 9596

(1) Data Center (e.g. Mean Value)

(2) Data Spread (Variability)(e.g. Range)

Example:

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Calculate the mean and the range for the following data set of people

income ($):

20,000, 22,000 ,28,000, 30,000, 27,000, 45,000, 50,000, 60,000

Simple Numerical descriptive statistics


Calculate the mean and the range for the following data set of people weight (pound)

125, 145, 160, 155, 110, 95, 175, 158


Calculate the mean and the range for the following data set of people temperature (Fo)

95.5, 98.0, 99.0, 96.5, 95.0, 97.0, 96.0, 98.0

Working Problem 1.7: Calculate the mean and range for the following data set of property taxes ($) 8100, 3500, 7000, 4200, 3000, 5000, 5100, 4000, 7500, 4800

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Histogram and frequency distribution

Observation # 1 2 3 4 5 6 7 8 9 10

Grades 75 65 90 75 90 75 65 90 75 100

A histogram or a frequency distribution is a simple x-y graph in which the horizontal x-axis represents the values (or classes of values) of the variable and the vertical y-axis represents the number of observations corresponding to each value (or the frequency).

65 70 75 80 85 90 95 1000

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nWeigh

t nWeigh

t n Weight n Weight n Weight n Weight1 146 11 145 21 144 31 153 41 127 51 1462 145 12 157 22 267 32 162 42 145 52 1593 147 13 148 23 151 33 144 43 137 53 1574 120 14 155 24 143 34 160 44 160 54 1445 187 15 158 25 161 35 110 45 141 55 1596 157 16 195 26 148 36 142 46 154 56 1627 143 17 142 27 240 37 155 47 152 57 1578 117 18 154 28 128 38 145 48 149 58 1499 170 19 160 29 136 39 150 49 125 59 28310 138 20 160 30 110 40 136 50 139 60 154

Typical Example of a Histogram: Data of human weight (lb) of a random sample of 60 people

The Bulk of the Data

Extreme Values: Outliers

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Key Points about Descriptive Statistics:

• Any statistical analysis should begin by performing descriptive statistics

• Descriptive statistics represent a powerful tool of data description particularly when a large amount of data must be analyzed

•The key elements of descriptive statistics are the measures of central tendency and the measures of variability

• Using descriptive statistics, data abnormality or data errors can be detected even for a large amount of data

• Frequency distributions and histograms represent graphical approaches of data description

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What is probability?

Most people use terms such as chance, likelihood, or probability to reflect the level of uncertainty about some issues or events. Some of the common examples of using these terms are as follows:

• As you watch the news every day, you hear forecasters saying that there is a 70% chance of rain tomorrow.

• As you plan to enter a new business, an expert in the field tells you that the probability of making a profit in this business is only 0.4, or there is a 40% chance that you will make a profit.

• As you take a new course, you may be wondering about the likelihood of passing or failing the course. • Your friend is undergoing a surgery and the physician is telling him that his chance of surviving the surgery is 95%.

• You hear it on health news all the time that a smoker has a greater chance of getting lung cancer than a nonsmoker. These are all expressions of probability that we often hear or read about and they can affect our planning or intention to do or not to do things in life.

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What is a probability Value?

The general definition of probability is that it is a value between zero and one, which reveals the relative possibility an event will occur.

• A probability of zero or close to zero implies that an event is very improbable to occur

• A probability of one or close to one gives us higher assurance that an event will occur.

• Between these two extremes, different values of probability will be expressed as a decimal such as 0.33, 0.7, or 0.50, as a fraction such as 1/3, 7/10, or 1/2, or as a percent such as 33.33%, 70%, or 50%.

Classically, probability is defined by the ratio of the number of particular target outcomes of an event to the number of all possible equally likely and mutually exclusive outcomes. For example, we know that in tossing a coin, the chance of head being the outcome is 50% or, in the context of probability, we say that the probability of head occurrence is 0.5. This value is a direct calculation from the classic definition of probability where the total number of possible outcomes in tossing a coin is 2 (head or tail), and the chance of head being the outcome is 1/2.

H T

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Sampling and sampling techniques

Sampling:

•Selecting a number of representative samples (say, k samples, each of size n) from a population using a technique suitable for the population under consideration and the purpose of testing

•Testing the samples and listing sample data

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Why Sampling?

• Population can be too immense to test

• Test can be destructive

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What are the different types of sampling?

Random Sampling

Population Sample

Stratified Sampling

I II

III IV

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What are the different sources of variability?

INHERENT VARIABILITY

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Working Problem 1.9: Identify the following sources of variability as inherent or induced:

(a) If one picks any boll of cotton from the field, one will not find two cotton fibers in this boll that are similar in length, diameter, or maturity.

Inherent ( ) Induced ( )

b. You open a box of water bottles and you find that some bottles are completely filled and some are half-empty


c. At the workplace, you find some people performing better than other people of the same experience and background


d. In a sample of natural soil aggregates taken from a certain area you find no two soil particles that are alike in size or texture.


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What are the different types of variables?

Classification by data nature or measurement levels:

•Continuous variables

•Discrete variables •Special variables

Classification by Information Nature:

Quantitative Variables

Qualitative Variables

Two Ways to Classify variables:

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Working Problem 1.10: Decide whether the following variables are discrete or continuous. Explain your reasoning.

1. The weights of football players in a team:

Discrete ( ) Continuous ( )

2. The number of defects in a shipment of cellular phones:


3. The number of passing students in a course:


4. Student grades in a course:


5. The speed at which different cars run on a highway:


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What are the levels of measurements?

Nominal Measures

The Pyramid of Data Levels of Measurements

Qualitative, no magnitude, no order(e.g. colors, genders, country name, person name)

Ordinal Measures

Zero has no numerical meaning, and meaningful order (e.g. ranking products as 1= superior, 2= very good, 3= good, 4= fair, 5= poor)

Interval Measures

Zero has no numerical meaning, meaningful order but constant incremental difference (e.g. temperature)

RatioMeasures

Quantitative, meaningful magnitude, zero has numerical meaning, ratio between numbers is meaningful, and meaningful order (e.g. weight, and length)

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In order to avoid any confusion about what level of measurement a variable belongs to, we should begin by addressing the following key questions (Michael Sullivan III, 2010, Statistics-Informed Decisions Using Data, Third Edition, Prentice Hall, Pearson Education, Inc.):

• Does the variable simply categorize each individual? If so, the variable is nominal (e.g. gender)

• Does the variable categorize and allow ranking of each value of the variable? If so, the variable is ordinal (letter grade in your calculus class)

• Do differences in values of the variable have meaning, but a value of zero does not mean the absence of the quantity? If so, the variable is interval (e.g. temperature).

• Do ratios of values of the variable have meaning and there is a natural zero starting point? If so, the variable is ratio (e.g. human weight and number of hours you study every week)

Key tips to figure out the level of measurement:

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Example: What is the level of measurement for each of the following variables?

a. Your score in the first Math quiz

b. People income in a given business

c. Classification of students by gender

d. Ranking of education by K-12, community college, and four-year University

e. The number of hours you spend watching TV every week

What are the levels of measurements?

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Working Problem 1.11: What is the level of measurement reflected by the following data?

(a) The ages of a sample of students entering first year of college:

18, 21, 19, 17, 16, 22, 21, 22, 23, 18, 18, 17, 19

(b) In a survey of 500 luxury-house owners (above $2million price), 200 were from California, 150 from New York, and 150 from Florida.

Working Problem 1.12: What is the level of measurement for each of the following variables?

a. Student scores in the first stat testb. Waiting time (minutes) waiting for a school busc. Classification of employee by gender d. A ranking of students as freshman, sophomore, junior, and seniore. The weight of football players

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What is a Normal Distribution?

The normal distribution is the key to numerous Statistical Concepts

A normal data set of a variable will typically have the following basic characteristics:

• Relatively few components or elements will exhibit low values of the variable

• Relatively few components or elements will exhibit high values of the variable

•The majority of values will be in the middle

Freq

uenc

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Characteristic value 10 20 30 40 50 60 70 80

Examples:• People Income• Student Grades• People Weight

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Center Values

High Values

Low Values

Freq

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Characteristic value

Measures of Central Tendency

Measures of Data Dispersion

MeanMode

Median

General Features of a Normal Distribution

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Comparison of the Distributions of Area of Ceramic Tiles of Two Different Types

Type A

Type B

Freq

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AreaMeanMode

Median

High variabilityLow variability

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Example: The Areas of Two Different Types of Ceramic Tiles

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Characteristic value

Freq

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Distributions associated with Different Categories of Variables

Larger-the-Better Variables- Students grades- Machine efficiency- People income

Nominal-the-Best Variables- Thickness of wood board- Area of ceramic tiles- Shoe size

Smaller-the-Better Variables- Number of defects- Number of failing students- Number of car accidents

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Classifying Variables by the Nature of Values:

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Working Problem 1.13:The frequency distributions below illustrate the grades of three quizzes for students taking a biology class. The final frequency distribution combines all the grades of the three quizzes. Describe the students’ performances in each quiz and the overall performance of the class.

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Working Problem 1.14:The frequency distributions shown below are the same as those in Working Problem 1.13 but superimposed. Describe the progressive students’ performances in each quiz and the overall performance of the class.

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The frequency distribution shown below represents the average grades of the three quizzes in Working Problems 1.13 or 1.14. Describe this distribution.

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Frequency Distribution of Average Grades of the Three Quizzes

Grade Averages

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Inferential statistics: how do you estimate population parameters from sample statistics?

Population

Sample

Population Parameters

Population mean (m)Population ModePopulation RangePopulation Standard Deviation (s)

Inferential Statistics

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Descriptive Statistics: The analysis of determining sample statistics (e.g. mean and range)Inferential Statistics: The analysis of estimating population parameters from sample statistics

Descriptive Statistics

Sample Statistics

Sample mean (X-bar)Sample ModeSample RangeSample Standard Deviation (s)

An Overview of Basic Statistics

Education

Transcript of An Overview of Basic Statistics