Quantitative data analysis, Standard Error, Effect Size, Scales of Measurement

Quantitative Analysis
A. Michael J Leo, PhD
Assistant Professor of Education
St. Xavier’s College of Education (Autonomous)
Palayamkottai – 627 002, Tirunelveli, India
9994006762
amjlsxce@gmail.com

OBTAINING QUANTITATIVE DATA
 Closed Questions in Survey
 Experimentation Scores
 Structured Observations
 Psychological Tests
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Dr. A. Michael J Leo, St. Xavier's College of Education(Autonomous), Palayamkottai, Tirunelveli, Tamil Nadu,
India

Quantitative Data or Variables
• The characteristics or traits for which
numerical value can be assigns are called
variables.
• E.g., Intelligence, Attitude, Height, Weight
and so on
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Dr. A. Michael J Leo, St. Xavier's College of Education(Autonomous), Palayamkottai, Tirunelveli, Tamil
Nadu, India

SCALES OF MEASUREMENT
• Measurement is assigning numbers to objects
and events according to logically accepted
values.
• A numerical method of describing observations
of materials or characteristics.
• S.S. Stevens (1951) in his book ‘Scales of
Measurement’
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Dr. A. Michael J Leo, St. Xavier's College of Education(Autonomous), Palayamkottai, Tirunelveli,
Tamil Nadu, India

The Properties
• Identity
• Magnitude
• Equal intervals
• A minimum value of zero
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India

Nominal Scale
Identity/Counting
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ORDINAL SCALE
Magnitude/Ordering
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India

INTERVAL SCALE
Equal Interval
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RATIO SCALE
True Zero
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Types of Graph
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Graphical Representation of Data
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Descriptive Statistics
The characteristics (attributes) of a sample thus
limit generalization to the particular group
(sample). No conclusions are extended beyond
Inferential Statistics
To draw generalizations beyond the sample
with a known degree of accuracy.
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Descriptive Data Analysis
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The measures of Descriptive Analysis
(i) Central Tendencies : Mean, Median, Mode
(ii) Variability: Range, Average Deviation, QD and SD
(iii) Relative Positions : Z, T Scores & Percentile Rank
(iv) Relationship : Rank order and Product moment
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Testing of Hypothesis
Five ingredients to test Hypotheses
Null Hypothesis
Alternate Hypothesis
Type I & II Errors
Statistical Significance
Test Statistic
Critical Region
Interpretation
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Tamil Nadu, India

Type I & II Errors
• Type I error : Reject H0 when H0 is true.
• Type II error : Accept H0 when H0 is false.
Probabilities of the errors
Probability of Type I error = P (Reject H0 / H0 is true) = α
Probability of Type II error = P (Accept H0 / H1 is true) = β
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Nadu, India

Power of a Statistical Test
• Power is the probability of avoiding a Type II
error.
• The probability of a Type I error is known as
Alpha
• The probability of a Type II error is known as
Beta
Alpha
Beta
(1-Beta) is known as the Power of the test
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1. Increase the effect size (the difference between
the null and alternative values) to be detected
2. Increase the sample size(s)
3. Decrease the variability in the sample(s)
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Statistical Significance (Alpha level)
– Three common levels of significance
• 0.01 (1 chance out of 100)
• 0.05 (5 chances out of 100)
• 0.10 (10 chances out of 100)
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Standard Error
• The standard deviation of the sampling
distribution is called the standard error.
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Standard Error
• Standard Error of the mean,
SE= σ/ √N
Where
N - Total number of cases in the sample
σ - Standard deviation of the
distribution of the sample means.
(Actually the SD of the Sample)
Note:
N>=30, √N / N<30, √N-1
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Setting up confidence Interval
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Estimation of Population Mean
Formula,
• At 5% level of significance Mp=Ms ± 1.96 × S.E
• At 1% level of significance Mp=Ms ± 2.58 ×S.E
Where,
Mp –> Population mean
Ms –> Sample mean
S.E –> Standard Error
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Tamil Nadu, India

Interpretation
• Accept or Reject (At 5% level)
• If p value <0.05, Reject the Null Hypothesis
• If p value > 0.05, Accept the Null Hypothesis
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Parametric Tests : Assumptions
• The variables expressed should be in interval or
ratio scales.
• The samples have equal or nearly equal
variances.
• The population values are normally distributed.
• The observations are independent.
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Tamil Nadu, India

5/22/2020
Dr. A. Michael J Leo, Assistant Professor of
Education, St. Xavier’s College of Education
(Autonomous), Palayamkottai - 627002
26

Sample size table for known
population size
• Krejcie and Morgan, 1970
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Dr. A. Michael J Leo, Assistant Professor of Education, St. Xavier’s College of Education
(Autonomous), Palayamkottai - 627002
27

t – test : Difference between the mean
scores of two groups
Group Statistics
SEX N Mean Std. Deviation Std. Error Mean
SS BOY 150 259.84 39.831 3.252
GIRL 150 259.13 37.692 3.078
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Dr. A. Michael J Leo, St. Xavier's College of Education(Autonomous), Palayamkottai,
Tirunelveli, Tamil Nadu, India

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Nadu, India

Effect Size Calculator for T-Test
Cohen's d = (M2 - M1) ⁄ Sdpooled
Where SDpooled = √((SD1
2 + SD2
2) ⁄ 2)
Interpretation
d = 0.2 Small effect
d = 0.5 Medium effect
d = 0.8 Large effect
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India
31

Paired t-test
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Tamil Nadu, India

• Cohen's d = (Msample - µpopulation) ⁄ σ
Interpretation
d = 0.2 Small effect
d = 0.5 Medium effect
d = 0.8 Large effect
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Tamil Nadu, India
34

ANOVA
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Non-Parametric tests
• The nature of the population from which
samples are drawn is not known to be normal.
• The variables are expressed in nominal form.
• The data are measures which are ranked or
expressed in numerical scores which have the
strength of ranks.
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Chi-Square test
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Correlation Analysis
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India
41

Thank You
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Tamil Nadu, India
42

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