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Wize AP Statistics Textbook > Collecting Data and Sampling Methods

Introduction to Statistics

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Introduction to Statistics



Statistics is the science of collecting, presenting, analyzing, and interpreting data.




A sample is a portion of the population that it is drawn from. The data in a sample is used to describe or infer about a population.
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Two Branches of Statistics

  • Descriptive Statistics
  • A summary of your data.
  • Refers to describing numbers in the data you collected in various ways such as displaying data using charts and graphs.
  • Merely describing.
  • Examples
  • The midterm grade average is 58%, ranging from 12% to 85%.
  • The distribution of grades is skewed to the left, and one-third of the class failed.
  • Seth made $70,000 in commissions selling condos, putting him in the top 25% percentile amongst his colleagues.
  • The average life expectancy is 75 years, an increase of more than 56% since 1960.

  • Inferential Statistics
  • Making estimations and drawing conclusions after analyzing your data.
  • Refers to using the data in the sample to make general conclusions about the population.
  • More interesting but more challenging than describing data.
  • Examples
  • Students in the morning class performed better than those in the evening class.
  • A student's midterm grade is a good predictor of their final exam grade.
  • Sales significantly increased after real estate agents attended Stella's seminar.
  • Homes with waterview sell at higher prices than homes without waterview.
  • Women live longer than men, on average.
  • There is strong evidence that exercise increases life expectancy.


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Sample vs. Population

What is the Point of Statistics?

  • To try to describe the real world.
  • This may not be easy to do when the population of interest is large.
  • We draw conclusions about the population using statistics.
In most cases, we do not need to survey every single individual in the population.
  • Usually impossible
  • Costly
  • Time-consuming
  • Overkill and unnecessary

Drawing a Sample From the Population

  • Drawing a large enough sample that is representative of the population is usually sufficient.
  • The statistics in a sample is used to describe or infer about the parameter of a population.

A population is defined as all people or items with a certain characteristic that one wishes to understand.
  • Measurable characteristics of a population are called parameters

A sample consists only of observations drawn directly from the population.
  • Measurable characteristics of a sample are called statistics

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Wize Tip
Population parameters →\rightarrow usually Greek letters (i.e. μ, σ, ρ\mu,\ \sigma,\ \rho)
Sample statistics →\rightarrow usually lowercase English letters (i.e. x‾, s, p\overline{x},\ s,\ p)


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Sampling Frame

A sampling frame is the available or accessible source of data from which the sample is drawn from.

Examples

Census, human resources database, street map, telephone directory, etc.

A census is data collected about every individual in the entire population of interest.

Practice: Sample vs. Population

We want to determine the average salary of assistant professors at UCLA. Based on a random sample of 100 from the HRMS database of all assistant professors, the mean salary is $100,000.


(i) What is the population?
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Measurement Scales

Measurement scales are used to quantify or categorize variables.
  • Quantitative variables can take on different numerical values.
  • Categorical variables can take on different qualities or characteristics.

Wize Tip
What types of charts and graphs to use depends on whether your are displaying quantitative variables or categorical variables.


There are four common levels of data measurement:

Nominal

  • Categorical or numerical values
  • Simply used to classify or identify
  • You can’t rank or put them in any order.
  • You can’t do any math (+,−,×,÷) with it!
  • Examples
  • religion, occupation, ethnicity, brand, dog breed

More Examples

  • Binary data
  • Unit can only take on only two possible states
  • Examples
  • 0/1, yes/no, pass/fail, win/lose
  • Identifier (or String)
  • unique string of numbers and/or letters representing that name of the variable
  • Examples
  • zip code, jersey number, student number
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Ordinal
  • Categorical or numerical values
  • Simply used to classify or identify
  • You can rank or put them in order.
  • You can’t do math (+,−,×,÷) with it!
  • Examples
  • level of education (BA < MA < PhD)
  • level of pain (mild < moderate < severe)
  • Olympic medals (bronze < silver < gold)
  • dress size (S < M < L < XL)

Interval

  • Always quantitative
  • You can rank or put them in order.
  • Distances between consecutive intervals are equal.
  • You can do some math (+,−) with it! Can’t (×,÷).
  • The value “0” has little meaning.
  • Examples
  • temperature (°F, °C\degree F,\ \degree C), IQ

Ratio

  • Always quantitative
  • Numbers go from lowest to highest.
  • Distances between consecutive intervals are equal.
  • You can do all math (+,−,×,÷) with it!
  • The value “0” has significant meaning (i.e. absolute zero)
  • Examples
  • age, height, weight, income
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Summary


Practice: Types of Variables

Classify each of the following variables as nominal, ordinal, interval, or ratio.

1) Height of a student