Introduction to Educational Statistics

- Joseph Stevens, Ph.D., University of Oregon
- (541) 346-2445, stevensj_at_uoregon.edu

WHAT IS STATISTICS?

- Statistics is a group of methods used to collect,

analyze, present, and interpret data and to make

decisions.

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.

POPULATION VERSUS SAMPLE cont.

- The portion of the population selected for study

is referred to as a sample.

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.

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.

TYPES OF STATISTICS

- Descriptive Statistics consists of methods for

organizing, displaying, and describing data by

using tables, graphs, and summary measures.

TYPES OF STATISTICS

- Inferential Statistics consists of methods that

use information from samples to make predictions,

decisions or inferences about a population.

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.

Basic Definitions

- The value of a variable for an element is called

an observation or measurement.

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.

TYPES OF VARIABLES

- Quantitative Variables
- Discrete Variables
- Continuous Variables
- Qualitative or Categorical Variables

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.

Quantitative Variables cont.

- A variable that can assume any numerical value

over a certain interval or intervals is called a

continuous variable.

Categorical Variables

- A variable that cannot assume a numerical value

but can be classified into two or more categories

is called a categorical variable.

Scales of Measurement

- How much information is contained in the numbers?
- Operational Definitions and measurement

procedures - Types of Scales
- Nominal
- Ordinal
- Interval
- Ratio

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

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

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.

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

Example Calculation of the mean

Four scores 82, 95, 67, 92

The Mean is the Center of Gravity

95

92

82

67

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

Comparison of Measures of Central Tendency

Measures of Dispersion

- Range
- Variance
- Standard Deviation

Range

- Highest value in the distribution minus the

lowest value in the distribution 1

Variance

- Measure of how different scores are on average in

squared units - ?(X X)2 / N

Standard Deviation

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

Other Descriptors of Distributions

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

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

Total area under a normal curve.

The shaded area is 1.0 or 100

µ

x

A normal curve is symmetric about the mean

Each of the two shaded areas is .5 or 50

.5

.5

µ

x

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

Three normal distribution curves with the same

mean but different standard deviations

s 5

s 10

s 16

x

µ 50

Three normal distributions with different means

but the same standard deviation

s 5 s

5 s 5

µ 20 µ 30

µ 40 x

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

99.7

95

68

µ 3s? µ 2s?µ s? µ?? µ s? µ 2s? µ

3s

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.

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

The Three Distributions Used in Inferential

Statistics

I. Population

III. Sampling Distribution of the Statistic

II. Sample