Introduction to Educational statistics and measurement

UNIT 1
REVIEW OF BASIC STATISTICS I
Introduction
Importance of studying statistics
• Decide on the statistical tool before the data collection.
It helps in the instrument design and data collection.
• Select statistical technique that is appropriate for
proposed analysis.
• Check assumptions before using the statistical technique.
• Know the difference between statistical significance and
practical significance.
• Garbage in, garbage out.

Scales of measurement
 Nominal, Ordinal, Interval and Ratio guide in the
selection of the right tools for statistical analysis.
 Depending upon the traits/attributes/characteristics
and the way they are measured, different kinds of
data result representing different scales of
measurement.
 For example, the numbers, 3, 8 and 15 could be
interpreted differently depending on the source or
scale of measurement.
 There are 4 types of measurement scales. These are
Nominal, Ordinal, Interval and Ratio.

Scale Arithmetic Function
Addition
(+)
Subtraction
(─)
Multiplication
(X)
Division
÷
Nominal
Ordinal
Interval √ √
Ratio √ √ √ √
Arithmetic
Comparisons

Frequency distributions
1. Used to check data entry.
Research question:
What are the major reasons for pre-service teachers
wanting to become teachers?
Responses:
Strongly agree (4), Agree (3), Disagree (2), Strongly
disagree (1)
Run an SPSS frequency distribution and check the
output after the data entry.

1. In teaching I can acquire knowledge and become more
resourceful and achieve self-growth. 20
2. Teaching is a noble profession. 18
3. I love working with children. 17
4. I love teaching. 16
5. Teaching provides immediate employment after training. 15
6. The teaching profession provides future career opportunities. 12
7. Teaching is intellectually stimulating. 11
8. I have an inborn talent for teaching. 10
9. I would like to influence young lives. 10
10. I want to help the rural communities. 9
2. Used to make a frequency of responses for answers to
research questions.
Research question:
What are the major reasons for pre-service teachers wanting to
become teachers?
Reason Frequency

Skewness
• This concept shows deviation from normal distribution.
• It could be negative or positive.
• It is determined by the construction of a frequency
polygon or by calculating from formulae.
• However, statistical softwares (e.g., SPSS) make it easier
to obtain the degree of skewness.
• Where the degree of skewness is 0 or very close to 0,
the distribution is normal.
• Where the degree of skewness is negative, implying
that there are more high scores than low scores.
• Where the degree of skewness is positive, implying that
there are more low scores than high scores.

Using polygon
Positive skewness Negative skewness
Uses in research
1. Determine the shape of a distribution especially whether normal
or not. This helps to know which distribution and statistical tool to
use for the analysis of data.
2. Helps to know which measure of location/variability/variation to
use in the reporting of descriptive data. (i.e. standard
deviation/variance or quartile deviation)
If the distribution is normal, the standard deviation and mean are
reported but if the distribution is skewed, the median and quartile
deviation are reported.

Using SPSS procedure:
1. Open SPSS
2. Enter data, if data is not already entered
3. Click Analyze
4. Click Descriptive Statistics
5. Click Frequencies
6. Highlight the variable in the left window and
click the arrow ( ) in the middle to move the variable to
the ‘Variables’ window on the right.
7. Click, Statistics.
8. Click Skewness under the Distribution box.
9. Click Continue.
10. Unclick the box for, Display frequency tables, and remove
the (√ ) mark.
11. Click OK

Measures of central tendency/location
These are descriptive statistics that should be reported with
every test of statistical significance.
The Mean and the Median
1. The Mean is reported as part of summary statistics when
data is normal but when data is skewed, Median should be
reported.
2. They are used to determine the shape of a statistical
distribution. If the mean = median, then the distribution is
normal, otherwise it is skewed.
3. The Mean is used to find the differences between/among
groups in terms of variables of interest. E.g., differences
between males and females in terms of weight in a class.

Measures of variability/variation/dispersion
• The standard deviation and variance, and quartile
deviation (semi-interquartile range) are the most
useful measures in statistical inference and
decision making.
• The standard deviation/variance is reported as
part of summary statistics when data is normal
but when data is skewed, quartile deviation (semi-
interquartile range) is reported.

Measures of relative position
(percentiles, percentile ranks, standard scores)
• These are percentiles and percentile
ranks, standard scores(Z , T, and Stanine )
• The main purpose of these measures is to
describe an individual’s position in
relation to a known group or the norm
group.

Percentiles and Percentile Ranks
 Notation and Interpretations:
 = 60. Sixty is the score below which 40% of the scores lie in a
specific group after the scores have been arranged sequentially.
This means that a student who obtains a score of 60 has done
better than 40% of the members in the specific group.
 PR of 60 = 75. Seventy-five is the position for a score of 60 when
the distribution is divided into 100 parts. This means that a
student who obtains a score of 60 has 75% of the scores falling
below him/her in the group.

Standard Scores (Z):
It indicates the number of standard deviation units an
individual score is above or below the mean of each
group.
It represents an individual score that has been
transformed into a common standard using the mean
and the standard deviation.
,
Z =
X − X
S

Standard Scores (T):
T is a transformed form of the Z, (using mean of 50 and
standard deviation of 10) intended to deal with the
negative signs and fractional/decimal values to ease
interpretation of test scores.
T = 50 + 10Z,
where mean is 50 and standard deviation is 10.

Parametric vs. non-parametric statistics and tests
Parametric tests make certain assumptions about a data
set; namely, that the data are drawn from a population with
a specific (normal) distribution. Non-parametric tests make
fewer assumptions about the data set.
The majority of elementary statistical methods are
parametric, and parametric tests generally have higher
statistical power. If the necessary assumptions cannot be
made about a data set, non-parametric tests can be used.
As the table below shows, parametric data has an
underlying normal distribution which allows for more
conclusions to be drawn as the shape can be
mathematically described. Anything else is non-parametric.

Description Parametric Non-parametric
Assumed distribution Normal Any
Assumed variance Homogeneous Any
Typical data Ratio or Interval Ordinal or Nominal
Usual central measure Mean Median
Benefits Can draw more conclusions Simplicity; Less affected by outliers
Tests
Independent measures, Independent-measures t-test Mann-Whitney test
2 groups
Independent measures, One-way, ind.-measures ANOVA Kruskal-Wallis test
More than 2 groups
Repeated measures, Matched-pair t-test Wilcoxon test
2 conditions

Using SPSS procedure:
1. Open SPSS
3. Click Analyze
6. Highlight the variable in the left window and
click the arrow ( ) in the middle to move the variable to
the ‘Variables’ window on the right.
8. Click Skewness under the Distribution box.
9. Click Continue.
10. Unclick the box for, Display frequency tables, and
remove
the (√ ) mark.
11. Click OK

Using SPSS procedure to obtain the Measures:
1. Open SPSS
3. Click Analyze
6. Highlight the variable in the left window and click the arrow (
in the middle to move the variable to the ‘Variables’ window
on the right.
8. Click Quartiles, Percentiles (put in the values needed in the
box and click Add, each time), Mean, Median, Std. deviation,
Variance
9. Click Continue.
10. Unclick the box for, Display frequency tables, and remove the
(√ ) mark.
11. Click OK

Exercise
Enter data for College A (English, Maths,
Educ.) into SPSS and obtain
1. Degree of skewness
2. Mean, Median
3. Standard deviation, Variance
4. Quartiles, Quartile deviation
5. 27, 70, 95, 98 Percentiles

Introduction to Educational statistics and measurement

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