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Introduction to Educational Statistics

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Title: Introduction to Educational Statistics


1
Introduction to Educational Statistics
  • Joseph Stevens, Ph.D., University of Oregon
  • (541) 346-2445, stevensj_at_uoregon.edu

2
WHAT IS STATISTICS?
  • Statistics is a group of methods used to collect,
    analyze, present, and interpret data and to make
    decisions.

3
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.

4
POPULATION VERSUS SAMPLE cont.
  • The portion of the population selected for study
    is referred to as a sample.

5
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.

6
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.

7
TYPES OF STATISTICS
  • Descriptive Statistics consists of methods for
    organizing, displaying, and describing data by
    using tables, graphs, and summary measures.

8
TYPES OF STATISTICS
  • Inferential Statistics consists of methods that
    use information from samples to make predictions,
    decisions or inferences about a population.

9
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.

10
Basic Definitions
  • The value of a variable for an element is called
    an observation or measurement.

11
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.

12
TYPES OF VARIABLES
  • Quantitative Variables
  • Discrete Variables
  • Continuous Variables
  • Qualitative or Categorical Variables

13
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.

14
Quantitative Variables cont.
  • A variable that can assume any numerical value
    over a certain interval or intervals is called a
    continuous variable.

15
Categorical Variables
  • A variable that cannot assume a numerical value
    but can be classified into two or more categories
    is called a categorical variable.

16
Scales of Measurement
  • How much information is contained in the numbers?
  • Operational Definitions and measurement
    procedures
  • Types of Scales
  • Nominal
  • Ordinal
  • Interval
  • Ratio

17
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

18
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

19
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.

20
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

21
Example Calculation of the mean
Four scores 82, 95, 67, 92
22
The Mean is the Center of Gravity
95
92
82
67
23
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

24
Comparison of Measures of Central Tendency
25
Measures of Dispersion
  • Range
  • Variance
  • Standard Deviation

26
Range
  • Highest value in the distribution minus the
    lowest value in the distribution 1

27
Variance
  • Measure of how different scores are on average in
    squared units
  • ?(X X)2 / N

28
Standard Deviation
  • Returns variance to original scale units
  • Square root of variance sd

29
Other Descriptors of Distributions
  • Skew how symmetrical is the distribution
  • Kurtosis how flat or peaked is the distribution

30
Kinds of Distributions
  • Uniform
  • Skewed
  • Bell-shaped or Normal
  • Ogive or S-shaped

31
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32
Normal distribution with mean µ and standard
deviation s
Standard deviation s
Mean µ
x
33
Total area under a normal curve.
The shaded area is 1.0 or 100
µ
x
34
A normal curve is symmetric about the mean
Each of the two shaded areas is .5 or 50
.5
.5
µ
x
35
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
36
Three normal distribution curves with the same
mean but different standard deviations
s 5
s 10
s 16
x
µ 50
37
Three normal distributions with different means
but the same standard deviation
s 5 s
5 s 5
µ 20 µ 30
µ 40 x
38
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

39
99.7
95
68
µ 3s? µ 2s?µ s? µ?? µ s? µ 2s? µ
3s
40
Score Scales
  • Raw Scores
  • Percentile Ranks
  • Grade Equivalents (GE)
  • Standard Scores
  • Normal Curve Equivalents (NCE)
  • Z-scores
  • T-scores
  • College Board Scores

41
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42
  • 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.

43
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

44
The Three Distributions Used in Inferential
Statistics
I. Population
III. Sampling Distribution of the Statistic
II. Sample
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