Introduction to Educational Statistics

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Introduction to Educational Statistics

• Joseph Stevens, Ph.D., University of Oregon
• [phone], stevensj_at_uoregon.edu

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WHAT IS STATISTICS?

• Statistics is a group of methods used to collect,
  analyze, present, and interpret data and to make
  decisions.

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POPULATION VERSUS SAMPLE

• A population consists of all elements
  individuals, items, or objects whose
  characteristics are being studied. The population
  that is being studied is also called the target
  population.

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POPULATION VERSUS SAMPLE cont.

• The portion of the population selected for study
  is referred to as a sample.

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POPULATION VERSUS SAMPLE cont.

• A study that includes every member of the
  population is called a census. The technique of
  collecting information from a portion of the
  population is called sampling.

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POPULATION VERSUS SAMPLE cont.

• A sample drawn in such a way that each element of
  the population has an equal chance of being
  selected is called a simple random sample.

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TYPES OF STATISTICS

• Descriptive Statistics consists of methods for
  organizing, displaying, and describing data by
  using tables, graphs, and summary measures.

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TYPES OF STATISTICS

• Inferential Statistics consists of methods that
  use information from samples to make predictions,
  decisions or inferences about a population.

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Basic Definitions

• A variable is a characteristic under study that
  assumes different values for different elements.
  A variable on which everyone has the same exact
  value is a constant.

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Basic Definitions

• The value of a variable for an element is called
  an observation or measurement.

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Basic Definitions

• A data set is a collection of observations on one
  or more variables.
• A distribution is a collection of observations or
  measurements on a particular variable.

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TYPES OF VARIABLES

• Quantitative Variables
• Discrete Variables
• Continuous Variables
• Qualitative or Categorical Variables

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Quantitative Variables cont.

• A variable whose values are countable is called a
  discrete variable. In other words, a discrete
  variable can assume only a limited number of
  values with no intermediate values.

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Quantitative Variables cont.

• A variable that can assume any numerical value
  over a certain interval or intervals is called a
  continuous variable.

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Categorical Variables

• A variable that cannot assume a numerical value
  but can be classified into two or more categories
  is called a categorical variable.

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Scales of Measurement

• How much information is contained in the numbers?
• Operational Definitions and measurement
  procedures
• Types of Scales
• Nominal
• Ordinal
• Interval
• Ratio

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Descriptive Statistics

• Variables can be summarized and displayed using
• Tables
• Graphs and figures
• Statistical summaries
• Measures of Central Tendency
• Measures of Dispersion
• Measures of Skew and Kurtosis

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Measures of Central Tendency

• Mode The most frequent score in a distribution
• Median The score that divides the distribution
  into two groups of equal size
• Mean The center of gravity or balance point of
  the distribution

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Median

• The calculation of the median consists of the
  following two steps
• Rank the data set in increasing order
• Find the middle number in the data set such that
  half of the scores are above and half below. The
  value of this middle number is the median.

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Arithmetic Mean

• The mean is obtained by dividing the sum of all
  values by the number of values in the data set.
• Mean for sample data

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Example Calculation of the mean
Four scores 82, 95, 67, 92
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The Mean is the Center of Gravity
95
92
82
67
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The Mean is the Center of Gravity

• X (X X)
• 82 82 84 -2
• 95 95 84 11
• 67 67 84 -17
• 92 92 84 8
• ?(X X) 0

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Comparison of Measures of Central Tendency
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Measures of Dispersion

• Range
• Variance
• Standard Deviation

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Range

• Highest value in the distribution minus the
  lowest value in the distribution 1

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Variance

• Measure of how different scores are on average in
  squared units
• ?(X X)2 / N

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Standard Deviation

• Returns variance to original scale units
• Square root of variance sd

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Other Descriptors of Distributions

• Skew how symmetrical is the distribution
• Kurtosis how flat or peaked is the distribution

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Kinds of Distributions

• Uniform
• Skewed
• Bell-shaped or Normal
• Ogive or S-shaped

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Normal distribution with mean µ and standard
deviation s
Standard deviation s
Mean µ
x
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Total area under a normal curve.
The shaded area is 1.0 or 100
µ
x
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A normal curve is symmetric about the mean
Each of the two shaded areas is .5 or 50
.5
.5
µ
x
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Areas of the normal curve beyond µ 3s.
Each of the two shaded areas is very close to
zero
µ
µ 3s
µ 3s
x
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Three normal distribution curves with the same
mean but different standard deviations
s 5
s 10
s 16
x
µ 50
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Three normal distributions with different means
but the same standard deviation
s 5 s
5 s 5
µ 20 µ 30
µ 40 x
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Areas under a normal curve

• For a normal distribution approximately
• 68 of the observations lie within one standard
  deviation of the mean
• 95 of the observations lie within two standard
  deviations of the mean
• 99.7 of the observations lie within three
  standard deviations of the mean

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99.7
95
68
µ 3s? µ 2s?µ s? µ?? µ s? µ 2s? µ
3s
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Score Scales

• Raw Scores
• Percentile Ranks
• Grade Equivalents (GE)
• Standard Scores
• Normal Curve Equivalents (NCE)
• Z-scores
• T-scores
• College Board Scores

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• Converting an X Value to a z Value
• For a normal random variable X, a particular
  value of x can be converted to its corresponding
  z value by using the formula
• where µ and s are the mean and standard deviation
  of the normal distribution of x, respectively.

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The Logic of Inferential Statistics

• Population the entire universe of individuals we
  are interested in studying
• Sample the selected subgroup that is actually
  observed and measured (with sample size N)
• Sampling Distribution of a Statistic a
  distribution of samples like ours

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The Three Distributions Used in Inferential
Statistics
I. Population
III. Sampling Distribution of the Statistic
II. Sample