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  Preliminary Concepts

  1. Some Definitions
  2. Types of Characteristics
  3. Scales of Measurement
    1. Nominal or Catagorical
    2. Ordinal or Rank
    3. Interval
    4. Ratio
  4. Some Mathematical Concepts
    1. Real or Exact Limits
    2. Rounding
    3. Summation Sign
    4. Summation Rules

Practice Problems (Answers)

I. Some Definitions
  1. Data
    Measurements collected during a scientific observation.

  2. Statistics
    A branch of mathematics concerned with describing and interpreting data.
    There are actually two functions to statistics:
    Serves to organize, summarize, and describe data.
    Serves to make inferences or generalizations about a total set of individuals or events on the basis of data from a smaller group.
  3. Population
    A collection of subjects or events that share a common characteristic.

  4. Sample
    A portion of the population.
    That is:

    Note population and sample are relative terms. While this class might be a sample from the population of all students at the college, all students at the college could be a sample from the population of all students at public universities in the USA.

  5. Parameter
    A description of some characteristic of a population.
    Is fixed or constant & symbolized with Greek letters.
    Note that this term has a different meaning to the lay public and to statisticians.

  6. Statistic
    A description of some characteristic of a sample.
    May vary & typically symbolized with English letters.

    In light of these later definitions, we can now be more precise in our definitions of the two functions of statistics:

  7. Descriptive Statistics
    Serves to describe or summarize the parameters of a population.

  8. Inferential Statistics
    Serves to infer or generalize about the parameters of a population based on statistics from samples.

  9. Random Sampling
    A procedure of selecting a sample whereby each member of the population has an equal chance of being chosen into the sample.

  10. Stratified Random Sampling
    Involves subdividing the population into strata or layers and randomly sampling from each. Consider an example of sampling undergraduate college students.

    Population    Sample
    Freshman 40% Freshman 40%
    Sophomore 30% Sophomore 30%
    Junior 20% Junior 20%
    Senior 10% Senior 10%

II. Types of Characteristics
  1. Constant - Does not vary. Always has the same value.
    Exs. pi (π), the number of members of a baseball team on the field.

  2. Variable - occurs in differing amounts or kinds.
    Exs. IQ, height, eye color.

    There are actually two kinds of variables:
    1. Qualitative - differs in kind or quality but not amount.
      Exs. eye color
    2. Quantitative - differs in amount but not in quality.
      Ex. IQ

      There are two kinds of quantitative variables:
      1. Discrete or discontinuous - can only occur in integer (whole number) amounts.
        Ex. number of children in a family.
      2. Continuous - conceivable or possible in any amount.
        Ex. height.

III. Scales of Measurement

Are a set of procedures for assigning numbers to things. Note that the act of measurement discretizes (rounds off) a continuous variable because one can never measure a continuous variable exactly. Ex. I am 5'8" tall. However, if you measured me, I would probably be something like 5'8.21332. . . . inches tall.

There are four scales of measurement that you should be familiar with. The first two are sometimes called nonparametric because they have nothing resembling a zero point.

  1. Nominal or Categorical
    Used to distinguish different categories for qualitative variables. Gives the same number to members of the same category and different numbers to members of different categories. In other words, the numbers here are essentially "dummy codes."
    Exs. Gender, ethnic background.

  2. Ordinal or Rank
    Uses numbers in a manner such that the numbers are in the same relationship as the characteristic is among the different members of the group of people or things. In this case, the numbers indicate position in an ordered series and not how much of a difference exists between the successive positions on the scale.
    Exs. Hardness of rocks, Beauty.

  3. Interval
    Is a parametric scale (i.e., it has a fixed unit of measurement) and has an arbitrary zero point (which means the zero point does not truly reflect absence of the characteristic).
    Exs. Celsius or Fahrenheit temperature.

  4. Ratio
    Is also a metric scale because it has an absolute zero (which truly reflects absence of the characteristic).
    Exs. Kelvin temperature, speed, height.
What follows is a figure showing the 3 temperature scales in order to clarify their similarities and differences, as well as a table that summarizes the properties of the 4 scales of measurement as well as the mathematical statements that one can make with them.

IV. Some Mathematical Concepts
  1. Real or Exact Limits

  2. Rounding
  3. Summation Sign (∑)


  4. Summation Rules (shortcuts):

    1. The sum of a constant (c) times a variable (X) equals the constant times the sum of the variable.

    2. The sum of a constant (c) taken N times equals N times the constant.

    3. The sum of two variables (X + Y) equals the sum of each variable summed.

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