STTN111 Chapter 1: Introduction to Statistics

Why Statistics?

  • Statistics are prevalent in everyday life due to the abundance of numerical information.
  • Examples include weather forecasts (temperatures in Fahrenheit and Celsius), cellphone data (4G, battery percentage), time, and location-specific information.

The Impact of Early Education

  • High-quality early education is crucial for at-risk children.
  • Without it, they are:
    • 25% more likely to drop out of school.
    • 40% more likely to become teen parents.
    • 50% more likely to be placed in special education.
    • 60% more likely to never attend college.
    • 70% more likely to be arrested for a violent crime.

Reasons for Shopping

  • Motivations behind shopping:
    • Necessity: 68%
    • Therapy: 16%
    • Boredom: 8%
    • Addiction: 6%
    • Socializing: 2%

Phone Addiction

  • A significant portion of smartphone users check their phones very frequently.
    • 58% of smartphone users check their phones within an hour of waking up.
  • Age-based breakdown:
    • 18-34: 68%
    • 35-44: 61%
    • 45-54: 55%
    • 55+: 36%

Decision Making and Statistics

  • Understanding data analysis is helpful in decision-making across various lines of work.
  • Statistics involves people, places, and possibilities.
  • Knowledge of statistical methods enhances understanding of how decisions are made and their impact.

Study Unit 1 Objectives

  • Define statistics.
  • Describe the aspects of statistics for managing data.
  • Classify variables as discrete or continuous and by their scale (nominal, ordinal, interval, or ratio).

1. 1 What is Statistics?

  • Statistics is the science of extracting information from data (i.e., making sense of data).
  • Key questions:
    • What can we say about the data?
    • What will we know after analyzing the data?
  • Example data set with values such as 0.00, 1814.4, 372.7, etc.

The Statistical Process

  • The statistical process involves:
    • Knowledge
    • Subject/Characteristic
    • Decision making
    • Data
    • Observation process
    • Measurement process

1. 2 Aspects of Statistics

  • Data collection
  • Summarizing and graphical representation of data
  • Drawing conclusions from data

Data Collection

  • Data collection is a crucial aspect of statistics.
  • Factors influencing results must be considered.
  • Planning is essential.
  • The amount of data needed must be determined.
  • Objectivity is important.

Summarizing and Graphical Representation of Data

  • Descriptive Statistics: Utilizes graphical and tabular methods to summarize and order data.
  • Example: Cellphone usage data
    • Samsung: 1200 (44%)
    • iPhone: 800 (30%)
    • Huawei: 500 (19%)
    • Blackberry: 200 (7%)

Statistical Inference

  • Statistical Inference: Methods used to make conclusions about a population based on sample data.

Self-Evaluation Exercise

  • Question: Which of the following is NOT part of Descriptive Statistics?
    • Graphical representation of the data.
    • The summary of data.
    • The drawing of conclusions about a population from a sample.
    • The ordering of the data.
    • Answer: The drawing of conclusions about a population from a sample.

1. 3. 1 Measurement and Variables

  • Data is obtained through measurement.
  • Definition: Measurement is assigning a numerical value to a property of an observed element.
  • Numerical values can be assigned to any property.

Validity of Measurement

  • Is the measurement valid?
    • Does it provide useful information about the characteristic being studied?
  • Choice of measurement instrument is important.
  • Example: Standardized IQ tests.

Variables

  • Data usually consist of variables.
  • Definition: A variable is any property of an observed element that can vary from one element to the next.
  • Examples: Height, Mass, Gender

Variation in Measurement

  • If a measurement process is repeated, variation is usually observed in the results.
  • Example: Mass of a group of people (81 kg, 65 kg, 74 kg, …).
  • In this case, mass is the variable.

1. 3. 2 Types of Variables

  • Discrete Variable: A variable where possible values are clearly distinguishable and disconnected from one another.
    • Discrete variables assume a fixed value and cannot be specified as a decimal.
    • Example: Number of legs an animal has, number of cars parked outside a building.
  • Continuous Variable: A variable where possible values are not clearly distinguishable.
    • For any two possible values, another value can always be found between them.
    • Example: Height can assume values like 178, 178.1, 178.12, 178.1247….178.9.

Self-Evaluation Exercise: Matching

  • Match column A with column B:
    • Population: The complete group of elements from which one would like to gain information
    • Statistical inference: Make conclusions about a population from sample data.
    • Statistic: The science of extracting information from data.
    • Discrete variable: Total number of spectators at a tennis match at Wimbledon
    • Continuous variable: The velocity of a tennis ball served at a match at Wimbledon

Variable Types: Qualitative vs. Quantitative

  • Variable types are classified as either qualitative or quantitative.
    • Further divisions include Nominal, Ordinal, Interval, and Ratio scales.

1. 3. 3 Types of Scale

  • Nominal Scale: Values indicate classes or categories.
  • Ordinal Scale: Values indicate classes or categories with an order.
  • Interval Scale: Properties of an ordinal scale, with meaningful differences between values.
  • Ratio Scale: Properties of the interval scale, with meaningful ratios between values.

Nominal Scale: Examples

  • Values indicate classes or categories.
  • Example: Eye color (“blue”, “grey”, “green” associated with numbers 0, 1, and 2).
  • Numbers are only used to distinguish between colors.
  • Other examples: male/female, yes/no.

Ordinal Scale: Examples

  • Values indicate classes or categories with a specific order.
  • Example: Attitude (“poor”, “reasonable”, “well”, or “excellent” with values 0, 1, 2, 3).
  • Numbers distinguish categories, and there is an associated order.
  • The differences between the values bear no significance.

Interval Scale: Examples

  • Properties of an ordinal scale, but meaning is assigned to differences between respective values.
  • Example: Time of day (08:00 – 09:00 | 14:00 – 15:00 | 14:00 – 17:00).
  • Zero-point is arbitrary (could be anything).
  • “0” does not imply the “absence of time”.
  • Other example: temperature.

Interval Scale Continued

  • Differences have a meaning.
  • Example: Thermometer
  • The difference between 10∘C10^{\circ}C and 20∘C20^{\circ}C has the same interpretation as the difference between 40∘C40^{\circ}C and 50∘C50^{\circ}C.
  • 0∘C0^{\circ}C does not mean the absence of temperature.
  • The interval between 7:00 and 9:00 has the same interpretation as the interval between 10:00 and 12:00.

Ratio Scale: Examples

  • Like the interval scale, but the ratios between the values have meaning.
  • Examples: Mass, Height, Speed
  • Ratio scale has a clear definition of zero (starting point).
  • Ratios can be meaningfully interpreted (e.g., for length, we can say 10m is twice as long as 5m).
  • Data is also numerical (or quantitative).

1. 3. 4-5 Discrete / Continuous Data

  • Discrete variables yield discrete data.
  • Continuous variables yield continuous data.
  • Graphical representation and analysis techniques differ.

1. 3. 4-5 Discrete/Continuous Data: Self-Evaluation Exercises

  • Question 1: Which of the following definition(s) is/are correct?
    • i. Statistics is the science of extracting information from data, i.e., statistics makes sense of data.
    • ii. Statistical inference refers to methods used to draw conclusions about the population from sample data.
    • iii. Descriptive statistics refers to methods to collect data.
    • iv. Measurements are invalid if they lead to useful information concerning the property being studied.
    • v. Measurement involves the process of assigning a numerical value to a property of an element.
    • Correct Options: i, ii, v
  • Question 2: Consider the following descriptions of variables. Which of these statements is/are false?
    • i. "The color of the shirt that I am wearing" (where blue: "1", green: "2", …) is an example of a discrete variable that is measured on the nominal scale.
    • ii. "The number of subjects I take this semester" is an example of a discrete variable and is measured on the nominal scale.
    • iii. “The amount of water I drink daily" is an example of a continuous variable that is measured on the ratio scale.
    • iv. "The time of day at which I walk to the Student Centre for a Chicken Wrap" is an example of a continuous variable that is measured on the interval scale.
    • v. "My height” is an example of a continuous variable that is measured on the ratio scale.
    • False Options: ii

Homework

  • Exercises 1, 2, and 3 on pages 11-12.

Textbook Exercises

  • Exercise 1: If we collect data on people's taste in music (Rock 'n Roll, Rap, R&B, Jazz, Classical), what type of data are we working with?
    • Answer: v. Discrete Data on a nominal scale
  • Exercise 2: Consider the following variables. Indicate whether the variable is discrete or continuous:
    • (a) The number of spectators at a soccer match. Discrete
    • (b) The amount of soda-pop (in ml) that a student drinks in one day. Continuous
    • (c) The mass of a boxer that competes in the heavy-weight division. Continuous
    • (d) The number of clients that visit a supermarket daily. Discrete
  • Exercise 3: Consider the following variables. Indicate the scale used to measure the variable:
    • (a) The position in which a person places in the Comrades marathon. Ordinal
    • (b) The type of vehicle preferred by an individual, e.g.,