Educational Statistics: Activities of Statisticians

1
Educational StatisticsActivities of
Statisticians

• Major activities in statistics involve
• design of experiments and surveys to test
  hypotheses
• exploration and visualization of sample data
• summary description of sample data
• modeling of uncertainty (e.g. flipping a coin)
• forecasting based on suitable models
• hypothesis testing and statistical inference

2
Educational Statistics Very Brief History

• Chevalier de Meres gambling problems and Blaise
  Pascals solutions (mid-1600s).
• Abraham de Moivres publication (in English) of
  his Doctrine of Chance in mid 1700s.
• William Sealy Gossetts development of the
  formula, in the early 1900s, for the standard
  error of the mean.
• Development of the t-test, analysis of variance,
  and non-parametric statistics in the first
  quarter of the 1900s.

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Educational StatisticsStatistical Terms and
Vocabulary

• Statistics a set of methods, procedures and
  rules for organizing, summarizing, and
  interpreting information.
• This is a general definition.
• Later, a distinction between statistics and
  parameters will be made.
• Here, it would be better to speak of statistical
  methods.

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Educational StatisticsStatistical Terms and
Vocabulary

• Use of symbols in statistics
• Statisticians (and statistical books) use symbols
  as shorthands for complex concepts and
  constructs.
• Symbols are typically either Arabic or Greek
  letters.
• For example
• µ (the Greek letter, mu) typically represents
  the mean (arithmetic average) of a set of values.
• s (the lower-case Greek letter, sigma) typically
  represents the standard deviation of a set of
  values.

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Educational StatisticsStatistical Terms and
Vocabulary

• Two Types of statistical methods
• Descriptive statistics methods used to
  summarize, organize, and simplify data.
• Inferential statistics methods that allow us to
  make generalizations about populations based on
  data obtained from samples.

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Educational StatisticsStatistical Terms and
Vocabulary

• Population vs Sample
• Population all members of a particular group
  (e.g., all Appstate freshman, all males over the
  age of 21, all of the schools in NC).
• Sample a subgroup of a population that is
  usually assumed to be representative of the
  population (e.g., 10 Appstate freshman selected
  at random).

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Educational StatisticsStatistical Terms and
Vocabulary

• Variable any characteristic that can vary across
  individuals, groups, or objects. For example
• Weight
• Occupation
• Grade-point average
• Level of test anxiety
• Later we will look at various types of variables

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Educational StatisticsStatistical Terms and
Vocabulary

• Values the numerical value of a particular
  realization of a variable.
• For instance if the variable is weight and
  Mortimer weighs 147 lbs. Then the value of the
  variable for Mortimer is 147.
• Make sure you can distinguish between variables
  and values

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Educational StatisticsStatistical Terms and
Vocabulary

• Parameters and Statistics
• Parameter the value of a variable in a
  population.
• Statistic the value of a variable in a sample.
• Statistics are often used to estimate or draw
  inferences about parameters.

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Educational StatisticsStatistical Terms and
Vocabulary

• Sampling error the difference between a sample
  statistic and its corresponding population
  parameter.
• The values of sample statistics vary from sample
  to sample, even when all samples are drawn from
  the same population.

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Educational StatisticsStatistical Terms and
Vocabulary

• Statistical procedures are the tools of research.
• There are several types (or methods) of research
  studies and the type of statistical procedure
  used will often vary from one type of research to
  another.

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Educational StatisticsStatistical Terms and
Vocabulary

• The correlational method of research.
• Examines relationships among two or more
  variables.
• For example What is the relationship between
  hours of TV watched per day and the number of
  calories consumed per day?
• Note that there no cause-effect relationship is
  postulated.
• Correlation does not imply causation.

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Educational StatisticsStatistical Terms and
Vocabulary

• The experimental method is used when the
  researchers wants to establish a cause and effect
  relationship.
• The researcher manipulates one variable (the
  independent) variable, and
• Observes (or measures) what happens to a second
  variable (the dependent variable),
• while Controlling for all other variables
  (extraneous variables).

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Educational StatisticsStatistical Terms and
Vocabulary

• A quasi-experiment is similar to a (true)
  experiment except that here the independent
  variable is not manipulated by the researcher.
• For example, in studying the effects of sex on
  mathematics achievement a researcher compares
  boys and girls (the independent variable).

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Educational StatisticsMeasurement

• Another tool of quantitative research.
• Definition A rule for the assignment of numbers
  to attributes or characteristics of individuals,
  or things.
• Eg.
• 1 if Male, 2 if Female.
• Score on a test.
• A judges rating (on a scale of 1 to 10) of
  physical attractiveness.

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Educational StatisticsScales of Measurement

• Types of measurement.
• The type of measurement scale has implications
  for the type of statistical procedure employed.
• Some statistical procedures assume a certain
  level of measurement.
• Three types of measurement can be distinguished
  nominal, ordinal, and scale.

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Educational StatisticsScales of Measurement

• Types of measurement nominal.
• Coarse level of measurement used for
  identification purposes.
• Substitutes numbers for other categorical labels.
  
• No order of magnitude is implied.
• Examples sex (male or female), student
  classification (freshman, sophomore, junior,
  senior), etc.

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Educational StatisticsNominal-scale Data

• Also called categorical data (or variables).
• Represent lowest level of measurement.
• Classify individuals into one of two or more
  mutually exclusive categories.
• The categories are usually represented by
  numbers. Eg
• 1 if Male, 2 if Female.
• 1 if Democrat, 2 if Republican, 3 if Independent,
  4 if Other.
• The numbers DO NOT indicate more or less of an
  attribute.

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Educational Statistics Scales of Measurement

• Types of measurement ordinal.
• Objects measured on an ordinal scale differ from
  each other in terms of magnitude, but the units
  of magnitude are not equal.
• The objects can be ordered in terms of their
  magnitude (more or less of an attribute.
• Examples class rank, seeding in golf or tennis,
  percentiles, level of motivation.
• Do not allow common mathematical opperations.
• What about grades or GPA?

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Educational Statistics Ordinal-scale Data

• In addition to classifying individuals, ordinal
  scales rank individuals in terms of the degree to
  which they possess measures characteristics of
  attributes.
• Ordinal scales allow us to compare individuals in
  terms of who has more (or) less of a
  characteristic or attribute.
• Do NOT indicate HOW MUCH more or less.
• Eg.
• Class rank
• Acrobatic competition judgments.
• What about grades or GPA?

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Educational Statistics Scales of Measurement

• Types of measurement scale.
• On an interval scale, objects are not only
  ordered by magnitude, but the distance between
  any two adjacent units is equal to the distance
  between any other two adjacent units.
• Examples SAT scores, Celsius and Fahrenheit
  scales, developmental scale scores (e.g.,
  EOG/EOC).
• Allow common mathematical opperations.

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Educational StatisticsScaled Data

• Includes both interval and ratio level scales.
• Scale measurement yield equal intervals between
  adjacent scale points.
• The difference between 5-6 and 6 is the same
  as the difference between 3 and 3-6.
• The difference between an SAT-V score of 435 and
  445 is the same as the difference between a score
  of 520 and 530.
• Most scores obtained form achievement tests,
  aptitude tests, etc. are treated as scaled data.

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Educational StatisticsVariables

• Any event, category, behavior, or attribute that
  can
• take on different values, and
• can be measured.
• Examples
• age type of instruction
  achievement
• test score group assignment motivation
• class size size of print
  creativity

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Educational StatisticsTypes of Variables

• Discrete and Continuous variables
• Variables can also be described in terms of the
  types of values they can be assigned.
• Discrete variables are categorical. No values
  between two adjacent values are permissible.
• Continuous variables can (theoretically) have an
  infinite number of values.

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Educational StatisticsTypes of Variables

• Independent variables.
• Dependent variables.
• Attribute variables.
• Extraneous variables.
• Confounding variables.
• Intervening variables.

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Educational StatisticsIndependent Variables

• True independent variables
• Experimental.
• Manipulated.
• Controlled.
• Quasi-independent variables
• Naturally occurring.
• Organic or biological.
• Quasi-experimental

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Educational StatisticsDependent Variables

• Effects.
• Outcomes.
• Measured variables.
• Dependent variables are functions of independent
  variables.

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Educational StatisticsAttribute Variables

• Characteristics or Attributes.
• May effect the dependent variables.
• Examples
• age experience
• sex attitude
• race advantagement

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Educational StatisticsExtraneous Variables

• Nuisance or controlled variables.
• Irrelevant to the focus of the study.
• Can affect interpretation of results.
• Examples
• time of day sequence of events
• side of building sex of investigator
• age of school building current events

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Educational StatisticsConfounding Variables

• Extraneous variables whose effects on the
  dependent variables cannot be distinguished from
  those of the independent variable(s).
• Usually occurs when an extraneous variables is
  correlated with one or more independent variables.

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Educational StatisticsIntervening Variables

• Black box variables.
• Invented to account for internal, unobservable
  psychological processes that intervene between
  independent and dependent variables.
• E.g. learning intervenes between teaching and
  achievement

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Educational StatisticsContinuous Variables

• Numerical data in research can be classified as
  either continuous or discrete.
• Variables that can take on any of a continuously
  ordered set of values within some specified
  range.
• Examples
• age dogmatism
• experience motivation
• achievement intelligence

33
Educational StatisticsDiscrete Variables

• Variables whose values can only be whole numbers.
• Characterized by gaps in the measurement scale.
• Typically represent counts of things.
• number of children school enrollment
• size of family number of books

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Educational StatisticsContinuous or Discrete
Can you tell?

• How would you classify the following variables?
  Continuous or discrete?
• Grade Level College classification
• Occupation Time on task
• Actually it depends upon how these variables are
  defined.
• What is the underlying characteristic or trait?
• Is it continuous or naturally discrete?

35
Educational Statistics

• The END!

36
Educational StatisticsStatistical Terms and
Vocabulary

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