Introduction to Educational Statistics.pptx

I Introduction to
Educational Statistics
Joseph Stevens, Ph.D., University of Oregon
(541) 346-2445, stevensj@uoregon.edu

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

I POPULATION VERSUS SAMPLE
A vooulation consists of all elements -
A. A
individuals, items, or objects - whose
characteristics are being studied. The
population that is being studied is
also called the target population.

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

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 sam.
.
.v. le.

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

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

I Basic Definitions
A variable is a characteristic under study
that assumes different values for
clifferent elements. A variable on which
everyone has the same exact value is a cons
tant.

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

I 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.

I TYPES OF VARIABLES
■ Quantitative Variables
□ Discrete Variables
□ Continuous Variables
■ Qualitative or Categorical Variables

I 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.

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

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

I Scales of Measurement
■ How much information is contained in
the numbers?
■ Operational Definitions and measurement
procedures
■ Types of Scales
o Nominal
□ Ordinal
□ Interval
□ Ratio

I Descriptive Statistics
Variables can be summarized and displayed
usi•
ng:
□ Tables
□ Graphs and figures
□ Statistical summaries:
■ Measures of Central Tendency
■ Measures of Dispersion
■ Measures of Skew and I<.urtosis

I 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

I 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
ofthis middle number is the median.

I 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:
n

I Example: Calculation of the mean
Four scores: 82, 95, 67, 92
n

I
T
h
e
Mean is the Center of Gravity
82
92 95
67

I
T
h
e
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
L(X - X ) =
0

Comparison of Measures of
Central Tendency
>,
(.)
C:
G)
::,
o-
uf
. variable
Mean Median Mode

I Measures of Dispersion
■ Range
■ Variance
■ Standard Deviation

I Range
Highest value in the distribution minus
the lowest value in the
distribution + 1

I Variance
■ Measure of how different scores are
on average in squared units:
I (X - X )2
/ N

I Standard Deviation
■ Returns variance to original scale units
Square root of variance = sd

I Other Descriptors of Distributions
■ Skew - how symmetrical is the
distribution
■ I<urtosis - how flat or peaked is
the distribution

I I<inds of Distributions
■ Uniform
■ Skewed
■ Bell-shaped or
Normal
■ Ogive or S-shaped

>,
0
C
>,
0
C
Cl)
Cl)
:::J :::J
CT Ca
,
T
a,
.
.
.
.
u
.
u
.
Variable
(a)
Variable
(b)
>,
0
>,
0
C
a
,
:::J
C
Cl)
:::J
CT CT
a, Cl)
.
.
..
.
u
.
u..
Vari abl
e
(c)
Vari able
(d)

I Normal distribution with mean µ and
standard deviation a
Standard
deviation =
rJ
Mean=µ X

I 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
µ X

I Areas of the normal curve beyond µ +
3a.
Each of the two shaded areas is
very close to zero
µ - 3 0 µ µ + 30 X

Three normal distribution curves with the
same mean but different standard deviations
- 0 =
5
µ = 50 X

Three normal distributions with different
means but the same standard deviation
a= 5 a= 5 a= 5
µ =20 µ =30 µ =40 X

I Areas under a normal curve
For a normal distribution approximately
1. 68°/o of the observations lie within
one standard deviation of the mean
2. 95°/o of the observations lie within
two standard deviations of the mean
3. 99.7°/o
of
the observations lie within three
standard deviations of the mean

99.7°/4- -
-
-
-
-
-
-

- - - - - - 95 %
µ - 30 µ
-
20 µ - 0 µ µ +0 µ +
20
µ +
30

I Score Scales
■ Raw Scores
■ Percentile Ranks
■ Grade Equivalents (GE)
■ Standard Scores
□ Normal Curve Equivalents
(NCE)
□ Z-scores
□ T-scores
□ College Board Scores

Approximately 34% of the Approximately 34% of the
scores fall between the mean scores fall between the mean
and one standard devlaUon and one standard deviation
below the mean. above the mean.
Approximately 14¾ of the Approximately 14¾ of the
scores fall between one scores fall between one
standard deviation and two standard deviation and two
standard deviations below standard deviations above
the mean. the mean.
13.59°1• 13.sgo.
2.14o/.
-3s -2s -1s x +1s +2s
+3s
/ / I
' '
Two standard One standard Mean (median One standard Two standard
deviations below deviation below and mode) deviation above
devlatlons above the mean the mean the mean the mean
If the meanIs 3.0 (Xa 3 .0 ) and the standard deviatlon is 1.0 (s 1.0) the scores a re as
follows:
0.0
(-3 s)
1.0
(-2 s)
2.0
(- 1s)
3.0
ex>
4 . 0
( + 1 $)
5.0
(+ 2 s )
6.0
(+
3s )
Normal Curve

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
z ==
X-µ
(J
whereµ and cr are the mean and standard
deviation of the normal distribution of x,
respectively.

I T
h
e 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

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

Introduction to Educational Statistics.pptx

More Related Content

Similar to Introduction to Educational Statistics.pptx (20)

More from mubasherakram1 (11)

Recently uploaded (20)

Introduction to Educational Statistics.pptx