Basics of Statistical Analysis Statistics Dr Rekha Sharma

1. BASICS OF
STATISTICAL ANALYSIS
DR REKHA SHARMA
ASSOCIATE PROFESSOR
UGC- HUMAN RESOURCE DEVELOPMENT CENTRE,
RASHTRASANT TUKADOJI MAHARAJ NAGPUR UNIVERSITY, NAGPUR

2. STATISTICS
•A branch of applied mathematics concerned with
– collection and interpretation of quantitative data and
– the use of probability theory to estimate population
parameters concerned with treatment of quantitative
information from groups of individuals.

3. DATA
• The data collected from various sources through the use of
different tools and techniques generally consists of numerical
figures, ratings, narrations, responses to open-ended questions
comprising a questionnaire or an interview schedule, quotations,
field notes etc.

4. TYPES OF DATA
QUALITATIVE DATA
• Qualitative data are verbal or other symbolic
materials.
• The detailed descriptions of observed behaviours,
people, situations and events
• These data are gathered by a variety of methods
and techniques such as:
• (i) observation; (ii) interviews; (iii) questionnaires,
opinionnaires and inventories; and (iv) recorded
data from newspapers and other documents
maintained by educational institutions, courts,
clinics, government and non-government agencies
etc
• Provide data of qualitative nature in the form of
emotions of respondents, their thoughts and
experiences
QUANTITATIVE DATA
• obtained by using various tools and tests based
on scale of measurement: nominal, ordinal,
interval or ratio.
• The quantitative data are either parametric or
non-parametric
• Parametric data undergo metric scale
measurement
• Non-parametric data are obtained by applying
nominal or ordinal scales of measurement. These
data are either coded, counted or ranked.

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6. LEVELS OF MEASUREMENT
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7. STATISTICAL METHODS
• •Statistical methods are extensively used in analysing
quantitative data in educational research.
• •These methods are classified as:
• • Descriptive statistics
• • Inferential statistics

8. DESCRIPTIVE
STATISTICS
• •Describes the characteristics of a sample.
• •No conclusions are extended beyond this group.
• •Method of organizing and summarizing data in Tables, graphs or
numbers.
• –Frequencies, percent, cumulative percent
• –Measures of Central Tendency: Mean ,mode, median
• –Measures of Dispersion: Range, SD, SE of mean, QD
• –Normal Probability Curve, Skewness, Kurtosis
• –Measures of Relative Positions: Sigma Score (Z)), T score,
Percentile Rank
• –Measures of Relationship: Product Moment Correlation, Rank
order correlation

9. MEASURES OF DESCRIPTIVE
STATISTICS
(i) Central Tendency or
Averages
(ii) Variability or Dispersion
(iii) Relative Position
(iv) Relationship

10. MEASURES OF CENTRAL
TENDENCY OR AVERAGES
•We use averages to describe the characteristics of samples.
•But in statistical language, use of the term 'average’ does not
convey any specific meaning, because there are a number of
types of averages, only one of which may be appropriate to
use in describing given characteristics of a sample/group.
•Three are mostly used in the analysis of educational data.
These include: Mean, Median and Mode

11. THE MEAN
• •The Mean of a distribution is the arithmetic average.
• •It is perhaps the most familiar, most frequently used
and well understood average.
• •It is computed by dividing the sum of all the
observations by the total number of observation.

12. SELECTION AND APPLICATION OF
MEAN
• •Mean is a more useful measure of central tendency than
its other measures.
• •Mean is an appropriate statistic when
• •i) the observations or measures are distributed
symmetrically about the centre of the distribution;
• •ii) the most stable measure of central tendency is
desired;
• •iii) additional statistics are to be computed later.

13. THE MEDIAN
•The Median is a point in an ordered
arrangement of observations, above and below
which one-half of the observations fall.
•It is a measure of a position rather than
size.

14. SELECTION AND APPLICATION OF
MEDIAN
Median is
computed when
i) the distribution of
the observation
measures is markedly
skewed;
Ii) there is not
sufficient time to
compute a Mean;
Iii) an incomplete
distribution is given;
iv) we are interested in
whether cases fall within
the upper or lower halves
of distribution and not
particularly in how far
they are from the central
point.

15. THE MODE
The mode is defined as
the most frequently
occurring observation
in a distribution.
It is located by
inspection rather than
by computation.
If there is only one
value which occurs a
maximum number of
times, then the
distribution is said to
have one mode or to
be unimodal.
In some distributions
there may be more
than one mode. A two
mode distribution is
bimodal; more than
two, multimodal.

16. SELECTION AND APPLICATION OF MODE
The mode is
the most
appropriate
statistics
when
i) the
quickest
estimate of
central
tendency is
wanted;
ii) a very
rough
estimate of
central
tendency
will suffice;
iii) the
purpose is
to know the
most typical
case.

17. MEASURES OF VARIABILITY
OR DISPERSION
(i) Range
(ii) Average Deviation
(iii) Standard Deviation

18. THE RANGE
The range is the
simplest measure of
variability. It is the
difference between
the highest and
lowest observation
or score.
The range is used
when:
i) the data are too
scant or too
scattered;
ii) only an idea of
extreme scores or of
total spread is
wanted.

19. SELECTION AND APPLICATION OF RANGE
Although the range has the advantage of being
easily calculated, it has some limitations.
I) The value of range is based on only two extreme
scores in the total distribution and thus it does not
provide us any information of the variation of many
other scores of the distribution.
II) it is not a stable statistics as its value can differ
from sample to sample drawn from the same
population

20. AVERAGE DEVIATION (AD)
• The average deviation or AD is the mean of the
deviations of all of the separate observations or scores in
a series taken from their mean.
• However, in averaging deviations to find average
deviation (AD), the signs (+ and -) are not taken into
consideration i.e. all the deviations, whether plus or
minus, are treated as positive.
• The formula for computing average deviation of an
ungrouped data is:

21. SELECTION AND APPLICATION OF
AVERAGE DEVIATION (AD)
Average deviation is used when:
i) It is desired to consider all deviations from the mean
according to their size;
ii) Extreme deviations would affect standard deviation
unduly.

22. QUARTILE DEVIATION (Q)
The quartile deviation
or Q is one-half the
scale distance
between the 75th and
25th percentiles in a
frequency
distribution.
The 25th percentile or
Q, is the first quartile
on the score scale, the
point below which lie
25 per cent of the
scores.
The 75th percentile or
Q, is the third quartile
on the score scale, the
point below which lie
75 per cent of the
score

23. SELECTION AND
APPLICATION OF
QUARTILE DEVIATION
1) The mean is the measure of central tendency;
2) There are scattered or extreme scores which would
influence the standard deviation disproportionately;
3) The concentration around the median, the middle
50 per cent of cases, is of primary interest

24. STANDARD DEVIATION (SD)
• The standard deviation (SD) is the most general and
stable measure of variability.
• It is the positive square root of variance.
• The average of the squared deviations of the
measures or scores from their mean is known as
variance which is generally denoted as sigma .

25. SELECTION AND APPLICATION OF STANDARD
DEVIATION
(a) Standard
deviation is used
when:
i) the statistics having
greatest stability is
sought;
ii) extreme deviations
exercise a
proportionally
greater effect upon
the variability;
iii) co-efficient of
correlation and other
statistics are
subsequently
computed.

26. NORMAL PROBABILITY CURVE
• One of the most common probability distribution in statistics is the
normal distribution, commonly represented by the normal curve.
• This mathematical and theoretical distribution describes the
expected distribution of sample means and many other chance
occurrences.
• The standardized normal distribution is a purely theoretical
probability distribution, but it is the most useful distribution in
inferential statistics.

27. NORMAL PROBABILITY CURVE
• The normal probability curve is based upon the law of
probability or probable occurrence of certain events.
• The 'probability' of a given event is defined as the
expected frequency of occurrence of this event among
events of a like sort.

28. NORMAL PROBABILITY CURVE
• For example, the probability of
an unbiased coin falling heads
is half and that of a dice
showing a three-sport is 1/6.
• When any set of observations
confirms to this mathematical
form, it can be represented by a
bell shaped curve with given
characteristics:
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29. CHARACTERISTICS OF NORMAL
PROBABILITY CURVE
• 1. The curve is symmetrical around
its vertical axis. It implies that the
size, shape and slope of the curve on
one side of the curve is identical to
that on the other side of the curve.
• 2. The values of mean, mode and
median computed for a distribution
following this curve always coincide
and have the same value. Source: https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.simplypsychology.org%2Fnormal-
distribution.html&psig=AOvVaw21IW4oXJ3rsYNQWzkve2e9&ust=1619209734801000&source=images&cd
=vfe&ved=0CAIQjRxqFwoTCND2_cfYkvACFQAAAAAdAAAAABAD

30. CHARACTERISTICS OF
NORMAL PROBABILITY CURVE
• 3. The curve has no boundaries in
either direction, for the curve never
touches the base line, no matter
how far it is extended i.e. the curve
is asymptotic and extends from - m
(minus infinity) to + m (plus
infinity).
• 4. The data 'cluster' around the
mean.
• 5. The percentages in a given
standard deviation are
greatest around the Mean and
decrease as one moves away from
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31. KURTOSIS
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32. SKEWNESS
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33. APPLICATIONS OF THE
NORMAL CURVE (i) to find the percentage of cases, or the number
when N is known, that fall between the mean and
a given 0 distance from the mean;
(ii) the percentage of the total area included
between the mean and a given 0 distance from the
mean.
2. The normal curve is used to convert a raw score
into standard score.
3. Normal curve is useful in calculating the
percentile rank of scores in a normal distribution.
4. For normalising a given frequency distribution,
the normal curve is used.
(i) to find the percentage of cases, or the number
when N is known, that fall between the mean and
a given 0 distance from the mean;
(ii) the percentage of the total area included
between the mean and a given 0 distance from the
mean.
2. The normal curve is used to convert a raw score
into standard score.
3. Normal curve is useful in calculating the
percentile rank of scores in a normal distribution.
4. For normalising a given frequency distribution,
the normal curve is used.

34. INFERENTIAL
STATISTICS
• •Statistical methods used to draw
generalizations beyond the sample
with a known degree of accuracy.

35. INFERENTIAL
STATISTICS
• •It is a decision to choose the right way to do analysis
according to
• – Parametric vs nonparametric
• – Level of confidence
• – Hypothesis
• –Type of dependent /independent variables
• – Number of groups, means
• – Correlation, significance, associations, relationships
• – Reliability and validity of scales

36. TYPES OF TESTS
•Independent t test
•T test
•Paired t test
•F test (ANOVA)
•Pearson’s coefficient
of Correlation
•CR test, Z test
Parametric
tests
•Chi Square(Unrelated Samples)
•Median Test(Unrelated
•Mann-Whitney U test
Samples)
•Sign Test (Related Samples)
•Wilcoxon Matched Paired
Signed Ranks Test (Related
Samples)
Non
Parametric
tests

37. HYPOTHESIS
• •A hypothesis is a proposed
explanation for the occurrence of the
phenomena that a researcher
formulates prior to conducting an
experiment.
• Hypothesis is an assumption that is
made on the basis of some evidence.
• A research hypothesis is a hypothesis
that is used to test the relationship
between two or more variables.

38. TYPES OF HYPOTHESIS
• Null Hypothesis: States that there is no relationship between the two
variables being studied. Denoted by H0
• Alternate Hypothesis: The alternative hypothesis states that there is a
relationship between the two variables being studied. Denoted by H1 or HA
• Non Directional Hypothesis: predicts that the independent variable will have an effect
on the dependent variable, but the direction of the effect is not specified.
• Directional Hypothesis: It predicts the nature of the effect of the independent variable
on the dependent variable.

39. LEVEL OF STATISTICAL SIGNIFICANCE
(ALPHA - ERROR)
• Two types of error can occur.
• Wrongly rejecting a true null hypothesis (HO) is called Type-I error.
• The probability of this error is popularly known as P-Value.
• The maximum P-value allowed in a study problem is called the level
of significance or sometimes 𝛂 level.

40. P VALUE, SIGNIFICANCE VALUE
Statistical Decision Null Hypothesis
True False
Rejected Type I Error ✔️
Not Rejected ✔️ Type II Error

41. P VALUE
• In null hypothesis significance testing, the p-value is the probability
of obtaining test results at least as extreme as the results actually
observed, under the assumption that the null hypothesis is correct.
• The P-value is kept at a low level, mostly less than 5% or P<0.05.
• A result is called statistically significant when P<0.05 and highly
significant when P<0.01.
• The complement of this is 95% used in confidence interval (CI).
• Significance level + Confidence Interval = 100

42. REFERENCES
• IGNOU Study Material MFN 009 Research Methods and BioStatistics, MSc
Dietetics and Food Service Management. http://www.ignouhelp.in/ignou-mfn-09-
study-material/
• IGNOU Study Material MES-016 Educational Research, Data Analysis and
Interpretation
http://egyankosh.ac.in/handle/123456789/2863
• Statistics In Psychology And Education by Henry E. Garrett, Paragon International
Publishers, New Delhi