Statistics Notes

Introduction to Statistics

• Studying statistics requires maintaining context to avoid losing sight of the bigger picture.

Everyday Use of Statistics

• Assessing student performance in exams.
• Comparing population growth rates between cities.
• Determining the safest means of transportation.
• Determining if a disease spread is an epidemic.
• Determining insurance premiums based on driving records.
• Predicting election results using opinion polls.
• Measuring customer satisfaction to improve product quality.
• Determining staffing needs for supplying goods and services.
• Identifying peak times for customer service demand.
• Finding the usual wait time to see a doctor.
• Identifying hotspots of criminal activities.
• Forecasting weather for the next few days.
• Tracking trends in stock market or revenue performances.
• Determining the number of operators needed at various times to handle calls.
• Analyzing player performance in hockey and basketball games.

Top 10 Uses of Statistics

1. Predictions: Statistics helps in making predictions about future events.
2. Forecasts: Used in weather forecasting and governmental planning.
3. Preparedness: Helpful in emergency preparedness.
4. Testing: Important for quality testing in various areas.
5. Political: Crucial in political campaigns.
6. Predicting: Plays a role in the medical field.
7. Financial: The financial market relies heavily on statistics.
8. Insurance: Used extensively in the insurance industry to determine premiums and assess risk.
9. Consumer: Widely used in consumer goods products.
10. Sports: Essential for making sports more effective.

Statistical Thinking

• Humans constantly observe and process information from their surroundings.
• Engage in statistical thinking knowingly or unknowingly in daily activities.
• Statistical awareness is an important part of daily lives.
• Observed events tend to follow patterns or regularities.
• Investigating these patterns helps in understanding their effects.
• Data collection is needed to investigate patterns through measurement, counting, or quantification.
• Measurements enable data analysis to identify connections, linkages, similarities, and differences.
• Understanding data allows for the application and generalization of knowledge to a wider context.
• Strive to avoid errors in generalization by collecting more data for higher confidence.
• Limited data reduces confidence in inferences, while large amounts of data increase confidence.
• 100% certainty in generalizing population characteristics is unattainable.
• Weigh chances or probabilities to assess the accuracy of generalizations, determining the degree of confidence.
• Statistical thinking summarizes experiences to understand essential features.
• Use summaries to estimate or predict future outcomes.

Statistical Thinking - Time Value

• Statistics helps explore the nature of variations over time.
• Describe patterns of change.
• Understand factors responsible for variations.
• Predict possible future outcomes.

Statistical Thinking - Key Processes

• Variation: Observe random variations of events.
• Pattern: Explore regularities in variations.
• Trend: Summarize variations to detect trends.
• Connect: Link variations to causes.
• Generalize: Draw reasonable conclusions and apply knowledge.
• Probability: Assign probability to measure confidence in conclusions.

What is Statistics?

• Statistics is the science of collecting, organizing, analyzing, presenting, and interpreting data to make informed decisions.
• It generates summary information and inferences about processes.
• Uses facts and figures to measure, assess, analyze, manage, and improve decisions.
• The goal is to rely on credible, data-driven evidence for effective decision-making, which is evidence-based management.
• Statistics:
  • Generates knowledge and useful intelligence.
  • Monitors business processes.
  • Manages situations or phenomena.
  • Evaluates and assesses options.
  • Mitigates risks or adverse effects.
  • Improves business operations.
  • Predicts outcomes and forecasts future trends.

Statistics as a Science

• Statistical analysis follows a scientific cycle.
  • Formulate a hypothesis.
  • Collect sample data.
  • Analyze the data.
  • Interpret the results to draw conclusions about the hypothesis.
• Statistics is a science because it provides proven methods for systematic examination.
• Based on mathematical theorems proven through logic.

Scientific Cycle

• Theory/Hypothesis: Explores the relationship between events or phenomena to understand their connections.
• Data Collection: Involves designing effective ways to obtain data from the field to represent the entire population.
• Data Analysis: Uses statistical methods to describe and analyze data, drawing conclusions about the validity of the hypothesis.

Why Study Statistics?

• To Conduct Research: Use statistical methods to publish scientific research, management reports, and academic journals, contributing to scientific knowledge.
• To Intelligently Read and Interpret Reports: Acquire numerical literacy skills to critically evaluate research articles, academic journals, and reports for sound decision-making.
• To Further Develop Critical and Analytical Thinking Skills: Enhance critical thinking and logical reasoning to understand issues, improving analytical reasoning.
• To Become an Informed Consumer of Information: Critically evaluate information to avoid being misled by misuse of statistics.

Data to Decision

• Statistics is about finding meaning behind numbers.
• Data: Isolated, raw, and unprocessed facts and figures without context (descriptive).
• Information: Organized, structured data providing meaning by exploring relationships (diagnostic).
• Knowledge: Learning derived from information through logical reasoning, revealing trends and patterns (predictive).
• Wisdom: Insights and actions obtained from knowledge to guide the best course of action for solving complex issues (prescriptive).
• A statistical process generally requires evidence to support an argument, obtained through data gathering and statistical analysis.
• Statistical analysis helps discover facts, information, knowledge, and critical insights for informed decisions.

Data to Decisions Breakdown

1. Data:
   • Signal, know-nothing.
   • What? Reveals relationships.
2. Information:
   • Useful, organized, structured, given context.
   • Why? Reveals patterns.
3. Knowledge:
   • Contextual learning, given meaning.
   • What is Best? Reveals principles.
4. Wisdom:
   • Understanding, given insight, actionable.
   • What Action? Reveals direction.
5. Decisions:
   *Change, movement, given purpose

Applied Statistical Research

1. Exploratory Research:
   • Used to find out if a particular issue is becoming a growing concern in a community.
   • Uses facts and figures to identify and explore a problem.
   • Examples:
     • Is teenage pregnancy a growing societal problem?
     • Is school bullying a serious social problem?
     • Is mental health a serious issue facing the youth?
     • Whether a neighborhood is becoming a crime hotspot?
     • Whether the police are racial profiling?
2. Descriptive Research:
   • Used to describe a prevailing issue of interest.
   • Provides a method of measuring and describing phenomena.
   • Examples:
     • Monitor the teenage pregnancy rate for the past 5 years.
     • How many students have become victims of school bullying in the past 5 years?
     • What is the statistical trend in mental health cases among the youth population?
     • Track the crime rates of the neighborhood for the past 5 years.
     • What is the racial breakdown of police stops?
3. Explanatory Research:
   • Used to identify causes and effects of phenomena, to explain and predict how one phenomenon will change in response to a change in the other phenomenon. *Examples:
     • What factors are related to teenage pregnancy?
     • Whether poor parenting leads to school bullying?
     • What factors are contributing to rising mental health among the youth?
     • Whether gang activities are contributing more to crime in the neighborhood?
     • Whether lack of diversity is affecting police relationship with the community?
4. Evaluation Research:
   • Statistics is used to assess the effectiveness of policies and program outcomes against a baseline or comparing post outcome against prior outcome.
   • Examples:
     • Whether after-school programs reduce teenage delinquency?
     • Whether cameras reduced red light traffic violation?
     • How effective are early intervention programs for the youth population?
     • Whether police on school campus reduced school violence?
     • Whether community policing has improved police relationship with the community

Sampling and Inference

• Population: Complete collection of all objects or persons of interest; a measurement of any population characteristics is known as a parameter.
• Sample: A subset of data drawn from a population for study; a measurement of any sample characteristics is known as a statistic.
• Sampling: Process of collecting data from the field to represent an entire population.
  • Needed because investigating every member is impractical, expensive, and time-consuming.
• Inference: Process of making decisions or conclusions about the population based on sample evidence.

Types of Statistics: Descriptive vs. Inferential

• Descriptive Statistics:
  • Used to describe, organize, and summarize information about an entire population.
  • Example: 90% satisfaction of all customers.
• Inferential Statistics:
  • Used to generalize about a population based on a sample of data.
  • Example: 90% satisfaction of a sample of 50 customers --> 90% satisfaction of all customers.

Descriptive Statistics

• Methods used to summarize, describe, and present data in a meaningful way.
• Includes tabular, graphical, and numerical descriptions.
• Organize data using tables (e.g., frequency table), graphs (pie chart and bar graph), and calculate quantities (e.g., mean, median, and mode).
• Effective way of describing the characteristics of a sample or observed data.
• Does not allow making conclusions beyond the analyzed data; uses a deductive (top-down) approach.
• Statistical information in newspapers, magazines, company reports are summarized and presented in this format.

Inferential Statistics

• Since studying an entire population is often impossible, data is collected from a sample.
• Sampling involves collecting sample data from the field to represent a population, which is often large, expensive, and time-consuming to study.
• After analyzing sample data, results are inferred back to the population for estimates, predictions, or conclusions about population characteristics.
• Statistical inference is the process of making decisions or conclusions about the population based on sample evidence.
• Researchers generalize from the sample to make conclusions beyond the sample data.
• Examples include hypothesis testing, probability, interval estimation, and regression; uses an inductive (bottom-up) approach.

Example of Inferential Statistics

• Estimating the average income of residents in Toronto.
• Studying the income of the entire Toronto population (
  $2.5$ million) is time-consuming and expensive.
• A survey of 200 residents shows an average income of $\$43,000$.
• Inference: The entire population in Toronto has an average income of $\$43,000$.
• Population: All elements of interest in a study (parameter).
• Sample: A subset of the population (statistic).

Descriptive vs. Inferential Statistics - Comparison Chart

Statistical Reasoning

Inductive Reasoning

• Infer the characteristics of the population based on what you know about the sample (inferential statistics).
• Moves from precise observation to a generalization.
• Begin with research questions and collection of empirical data, which are used to generate hypotheses and theory.
• Conclusions are probabilistic (weak or strong).
• You need evidence instead of true fact.
• Example: The quiz is easy therefore the final exam will be easy.
• Sample è Population

Deductive Reasoning

• Deduce the characteristics of the sample based on existing theory about the population (descriptive statistics).
• Moves from generalized statement to an effective conclusion.
• Begin with a theory-driven hypothesis, which guide data collection and analysis.
• Conclusions are sure (valid or invalid).
• You use true facts to assume a valid conclusion.
• Example: All students in the class play guitar. Jane is a student in the class so Jane plays guitar.
• Sample ç Population

Statistical Variables

• Independent Variable:
  • The cause or the explanatory variable.
  • Explains the existence of the dependent variable.
  • Influences the dependent variable.
  • Represented by “X” on a chart.
• Dependent Variable:
  • The effect or the response variable.
  • Responds to the effect of the independent variable.
  • Influenced by the independent variable.
  • Represented by “Y” on a chart.
• Example: In a relationship between income and expense, your level of expense depends on your level of income; expense is a dependent variable, and income is an independent variable.
• The independent variable is controlled or manipulated by the researcher, while the dependent variable changes in response to the independent variable.
• A statistical value can be a constant or variable; a constant is fixed, while a variable changes.

Big Data & Data Analytics

• Vast amounts of data are available for analysis due to social media and interactive technology.
• Organizations collect large amounts of data daily, referred to as “Big Data”.

Six Defining Terms of Big Data

1. Volume: Vast and huge amount of available data (e.g., terrabytes, yottabytes).
2. Variety: Many different forms of data (e.g., text, audio, video, touch-screen).
3. Velocity: The high speed and frequency at which data is generated and processed.
4. Veracity: The growing abnormalities, biases, and noises affecting data quality.
5. Value: The growing value of data to improve decision-making in organizations.
6. Volatility: The rapid changing and unpredictable flow of data (e.g., social media).

• Many organizations emphasize using data to drive business operations.

• Data alone is meaningless without valuable insights for informed actions.

• Leveraging huge datasets is crucial, making data a critical asset.

• Organizations implement data warehousing to capture, store, and maintain large data.

• Data warehousing makes it possible to store, retrieve, and process extremely large quantities of data in seconds.

• Data is queried using data mining techniques.

• Data mining involves using statistics, mathematics, AI, machine learning, and computers to mine data for better decision-making.

• Dramatic increase in available data, cost-effective storage, faster processing, and recognition of data value have increased the recognition of data analytics.

• Data Analytics: A scientific process of transforming data into critical insight and actionable intelligence to make better-informed decisions.

Four Types of Data Analytics

1. Descriptive Analytics:

   • Looks at past data to explain what happened and why.

2. Diagnostic Analytics: Understand why something happened in the past. Root cause analysis.

3. Predictive Analytics:

   • Uses past data to forecast what will happen in the future, determining necessary actions.

4. Prescriptive Analytics:

   • Helpful in determining what actions need to be taken to affect those outcomes

Descriptive Analytics

• Involves using analytical tools to describe past events, signaling what is right or wrong without explaining why.
• Summarizes findings using data visualization and tabulation methods like data queries, dashboards, descriptive statistics, graphs, and tables.
• For instance, using data to describe the crime rate over the past five years.

Diagnostic Analytics

• Involves using analytical tools to understand why something happened in the past.
• Also known as root cause analysis, it seeks to understand the root causes of events.
• Helpful in determining factors and events that contributed to the outcome by making connections and establishing correlations, providing causal relations but no insight.
• Techniques include data mining, correlation, probabilities, and sensitivity analysis.
• For instance, using data to investigate why the crime rate has increased over the past five years.

Predictive Analytics

• Involves using analytical tools to predict future events.
• Uses analytical models constructed from past data to predict future outcomes or assess the impact of variables.
• Uses findings from descriptive and diagnostic analytics to predict future trends and relies on machine learning.
• For instance, using methods such as mathematical models, linear regression, time-series, and forecasting models to predict future crime rates.

Prescriptive Analytics

• Involves using a set of analytical techniques that yield a best course of action, recommending actions to affect outcomes.
• Aims to prescribe actions to eliminate future problems or take advantage of trends.
• Suggests favorable outcomes for specified actions and various actions to achieve particular outcomes.
• For instance, using optimization models to generate deployment solutions that minimize costs while maximizing efficiencies in reducing crime.

Technology and Statistics

• Given the volume of data, processing speed, and complex calculations, sophisticated calculators and software applications are used for statistical analysis.
• Examples of graphing and statistical software applications are MS Excel, SPSS, SAS, Python, and R.
• Software applications and calculators make statistical calculations easier, faster, and more accurate.
• Focus is now placed on understanding statistical concepts, appropriate techniques, and interpretation of statistical results rather than regurgitating formulas.
• MS Excel is widely used due to its ease of use and prevalence.
• Excel has built-in statistical functions (Analysis ToolPak) that add statistical power.

Ethical Considerations

• It is important to be fair, thorough, objective, and neutral when collecting and analyzing data.

Avoid Unethical Behaviors

• Improper or biased sampling.

• Over-generalization based on small/unrepresentative sampling.

• Considering only data points that reinforce a particular theory.

• Inappropriate analysis of data.

• Development of misleading graphs.

• Fudging data or creating false data.

• Inappropriate summary statistics.

• Running multiple tests until the desired result is obtained.

• Biased interpretation of statistical results.

• The American Statistical Association developed the “Ethical Guidelines For Statistical Practice” to help practitioners make ethical decisions and assist students in performing responsible statistical work.