The document outlines the fundamentals of statistics, focusing on measurement scales, test scores, central tendency, and the normal probability curve (NPC). It details various scales of measurement including nominal, ordinal, interval, and ratio, along with methods for analyzing test scores and assessing data distribution using z-scores and percentiles. Additionally, it discusses the importance of measures of central tendency (mean, median, mode) and dispersion (range, quartile deviation, mean deviation, standard deviation) in evaluating educational performance and data interpretation.

1. Statistics in the View point of Evaluation

2. Unit 4 Syllabus-
• 4.2.1- Measuring Scales- Meaning and Statistical Use
• 4.2.2- Conversion and interpretation of Test Score
• 4.2.3- Normal Probability Curve
• 4.2.4- Central Tendency and its importance in Evaluation.
• 4.2.5- Dimensions of Deviation

3. The Unit 4 is all about Statistics…
 Statistics is the study of the collection, analysis, interpretation,
presentation, and organization of data.
 In other words, it is a mathematical discipline to collect,
summarize data.
 Also, we can say that statistics is a branch of applied
mathematics.
 Statistics is simply defined as the study and manipulation of
data. As we have already discussed in the introduction that
statistics deals with the analysis and computation of numerical
data.

4. Projective methods of Evaluation through
Statistics-
• Measurement is a process of assigning numbers to individuals
or their characteristics according to specific rules.” (Eble and
Frisbie, 1991, p.25).
• This is very common and simple definition of the term
‘measurement’.
• You can say that measurement is a quantitative description of
one’s performance. Gay (1991) further simplified the term as a
process of quantifying the degree to which someone or
something possessed a given trait, i.e., quality,
characteristics, or features.

5.  Measurement assigns a numeral to quantify certain aspects of
human and non-human beings.
 It is numerical description of objects, traits, attributes,
characteristics or behaviours.
 Measurement is not an end in itself but definitely a means to
evaluate the abilities of a person in education and other fields
as well.

6. Measurement Scale-
• Whenever we measure anything, we assign a numerical value. This numerical value is
known as scale of measurement. A scale is a system or scheme for assigning values
or scores to the characteristics being measured (Sattler, 1992). Like for measuring
any aspect of the human being we assign a numeral to quantify it, further we can
provide an order to it if we know the similar type of measurement of other
members of the group, we can also make groups considering equal interval scores
within the group.
• Psychologist Stanley Stevens developed the four common scales of measurement:
1. Nominal
2. Ordinal
3. Interval &
4. Ratio
• Each scale of measurement has properties that determine how to properly analyze
the data.

7. Nominal scale-
In nominal scale, a numeral or label is assigned for characterizing
the attribute of the person or thing.
That caters no order to define the attribute as high-low, more-
less, big-small, superior-inferior etc.
In nominal scale, assigning a numeral is purely an individual matter.
It is nothing to do with the group scores or group measurement.
Statistics such as frequencies, percentages, mode, and chi-square
tests are used in nominal measurement.
• Examples include gender (male, female), colors (red, blue, green), or
types of fruit (apple, banana, orange).

8. Ordinal scale-
• Ordinal scale is synonymous to ranking or grading.
• It includes nominal scale and provides an order to the measurement, like; when
we know the achievement scores of students in a group, we can arrange them
either in ascending (lowest to highest) or descending (highest to lowest) order.
• We can also interpret the result like; who stood first, second, 10th in the group,
even the last one in the group.
• In ordinal scale, we can use the statistics such as median (measures of central
tendency), quartile and percentile measures, correlation in rank difference
method, and non-parametric test.
• Examples include rankings (1st, 2nd, 3rd), customer satisfaction ratings (poor,
fair, good, excellent), or educational levels (high school, college, graduate).

9. Interval scale-
• Interval scale carries all the characteristics of earlier scales like nominal and
ordinal and added with an arbitrary zero point.
• That is, there is no absolute zero-point or true zero point.
• In this scale, we can group the scores in to equal intervals like, scores within the
intervals of : 0-5; 5-10; 10-15; 15-20; 20-25; 25-30 etc.
• This is also called as equal interval scale as the size of the classes are equal, i.e.
size of the class 0-5 is 5; 5-10 is 5; 10-15 is 5; 15-20 is also 5.
• As there is no absolute zero point in this measurement and the existing zero value
is an arbitrary one, that’s why all types of measurement done in education and social
sciences are usually done by interval scale.
• The statistics like mean, standard deviation, product moment correlation, t-test
and f-test can be used in interval scale.
• Examples include temperature in Celsius or Fahrenheit, IQ scores, or Likert scales.

10. Ratio scale-
• Ratio scale is called as the highest scale in measurement.
• It carries all the characteristics of earlier discussed scale with a true
or absolute zero point.
• As there is absolute zero point in this measurement, we can say that
zero height means no height. But in the case of interval scale, we can
not say that zero intelligence means no intelligence.
• All types of measurements conducted in Physical Sciences such as
Physics, Mathematics, etc. are done by ratio scale.
• Ratio scale are almost non-existence in psychological and educational
measurement except in the case of psycho-physical measurement.
• Examples include height, weight, age, income, and time in seconds.

11. • We usually use four types or scales of measurement, i.e. nominal, ordinal,
interval and the ratio scale.
• In educational measurement, generally, we use nominal, ordinal and equal
interval scale.
• Ratio scale is used in measurement of physical sciences where there is a
concept of absolute zero point in measurement.
• But in educational measurement, we consider zero point measurement as
the relative zero point.
• The use of different methods of calculating measures of central
tendency is depends upon the nature of the data and its scales of
measurement.

12. 4.2.2- Test Scores
and its Analysis-
-Z-Score
-Percentile
-Percentile Rank

13. Z Score-
• Z-score (or standard score) is a statistical
measure that expresses the relationship of a
data point to the mean of a group of data
points. It is often used in statistics to
standardize scores and make them comparable
across different scales.
• Here's a breakdown of the components:
• Z: The Z-score.
• X: The individual data point.
• μ: The mean of the dataset.
• σ: The standard deviation of the dataset.
• The Z-score tells you how many standard
deviations a particular data point is from the
mean.
• Z-score of 0 indicates that the data point's
score is identical to the mean score, a Z-score
of 1.0 indicates a value that is one standard
deviation from the mean, and so on.
Z-scores are frequently used in fields such as
finance, education, and psychology for comparing
and standardizing data. They are particularly
valuable when dealing with datasets with
different units or scales, as they provide a
common metric for comparison.

14. Percentile-
• A percentile is a statistical measure
used to describe the relative
position of a particular value within
a dataset.
• It indicates the percentage of data
points that are equal to or below a
given value.
• provide a way to express the
relative standing of a particular
value within a group, making it
easier to interpret and communicate
the significance of that value.
• Percentiles are commonly used in
various fields, including statistics,
education, healthcare, and finance.

15. Percentile Rank-
• Percentile rank is a measure that
indicates the percentage of scores in a
dataset that are equal to or below a
particular value.
• It provides information about the
relative standing of a specific data point
within a distribution.
• Percentile rank is commonly used in
various fields, including education,
standardized testing, and healthcare.
• It provides a way to understand and
communicate the relative position of a
particular data point within a
distribution, making it easier to interpret
and compare scores across different
contexts.

16. 4.2.3- Normal
Probability
Curve

17. NPC- Concept and Definition
• The "normal probability curve" refers to a specific type of distribution
known as the normal distribution or Gaussian distribution. This distribution
is often represented as a bell-shaped curve, and it has several important
characteristics.
• Applications: The normal distribution is commonly used in statistics and
research. Many natural phenomena, such as human height, test scores, and
measurement errors, approximate a normal distribution. This makes the
normal distribution a valuable tool for statistical analysis and hypothesis
testing.
• The normal probability curve is fundamental in statistics, providing a
theoretical framework for understanding the distribution of data in many
real-world scenarios. It is widely used in fields such as finance, biology,
psychology, and quality control. The normal distribution is also a key
assumption in many statistical methods and hypothesis tests.

18. Importance of NPC-
• The normal curve is used to characterize complex constructs containing
continuous random variables.
• Many phenomena observed in nature have been found to follow a normal
distribution.
• Some human attributes such as height, weight, intelligence, and even
social skills can be said to be normally distributed.
• Study of natural phenomenon through NPC
• Sampling Theory has its base on NPC
• NPC is useful for large sample tests.

19. Characteristics of NPC-
• it has a bell shape,
• the mean and median are equal-First, its mean (average), median (midpoint),
and mode (most frequent observation) are all equal to one another. Moreover,
these values all represent the peak, or highest point, of the distribution. The
distribution then falls symmetrically around the mean, the width of which is
defined by the standard deviation.
• The Empirical Rule - 68% of the data falls within 1 standard deviation. For all
normal distributions, 68.2% of the observations will appear within plus or minus
one standard deviation of the mean; 95.4% of the observations will fall within
+/- two standard deviations; and 99.7% within +/- three standard deviations.
This fact is sometimes referred to as the "empirical rule," a heuristic that
describes where most of the data in a normal distribution will appear.

20. Application of NPC in Evaluation-
1. Grading and Assessment
2. Standardized Testing
3. Student Performance Analysis
4. Setting Benchmarks and Goals
5. Assumption in Statistical Tests
6. Placement and Intervention Programs
7. Growth and Development Monitoring
8. Grading Curves
• In summary, the normal probability curve
is a valuable tool in education for
understanding, analyzing, and interpreting
student performance. It provides a
framework for making informed decisions
about assessment, grading, and
intervention strategies. Additionally, it is
a key component in the statistical analysis
of educational research data.

21. 4.2.4-
Measures of
Central
Tendency and
Evaluation

22. Mean-
• In statistics, the "mean" is a
measure of central tendency
that represents the average
of a set of values. T
• There are different types of
means, but the most common
one is the arithmetic mean.
• The arithmetic mean is
calculated by summing up all
the values in a dataset and
dividing the sum by the total
number of values.
1. Grading and Assessment
2. GPA Calculation
3. Program Evaluation
4. Benchmarking
5. Teacher Evaluation
6. Research and Data Analysis
7. Standardized Testing
8. Resource Allocation
9. Identification of Trends
10. Needs Assessment
11. Programmatic Decision-Making
• The mean is a fundamental concept in statistics and is widely
used in various fields to describe and summarize data.

23. Median-
• The median is a measure of
central tendency that
represents the middle value
in a dataset when the values
are arranged in numerical
order.
• It is one of the three main
measures of central
tendency, along with the
mean and mode.
• The median is less sensitive
to extreme values (outliers)
compared to the mean,
making it a robust statistic in
some situations.
In educational contexts, the median is commonly
used in a variety of scenarios, including-
1. grading
2. analyzing test scores
3. and evaluating student performance.
• It provides a useful summary statistic that
helps educators understand the central
position of values in a dataset, especially when
dealing with non-normally distributed data or
when extreme values may distort the
interpretation of the mean.

24. Mode-
• The mode is a measure of
central tendency that
represents the most
frequently occurring value in
a dataset.
• Unlike the mean and median,
which represent the average
and middle values,
respectively, the mode
identifies the value that
appears with the highest
frequency.
• Applicability-
• The mode is suitable for both numerical and categorical data.
For numerical data, it represents a specific value, while for
categorical data, it represents the category with the highest
frequency.
• While the mode provides information about the most common
value, it may not always be a sufficient statistic for
summarizing a dataset. In such cases, the mean and median
may also be considered to provide a more comprehensive
picture of the dataset's central tendency.

25. 4.2.5- Dimensions of
Deviation
-Range
-Quartile Deviation
-Mean Deviation
-Standard Deviation

27. Range-
• Range may be defined simply as that interval between the lowest and the
highest scores. It is very common and general measure of spread and is
computed when we need to know at a glance comparison of two or more groups
for variability. Since, it is based on two extreme values and tells nothing about
the variation of the intermediate values, it is not an authentic measure of
dispersion. Range may be computed by the following formula:
• (Range = Highest Score – Lowest Score) .... (3)
• It is deceptive and not authentic. For example, in any class of 40 students, 1
student got 20 marks, while all the rest 39 students scored between 70 to 80
marks out of 100. Range informs you that there exist a range of 60 marks,
while the majority secured between 70 to 80 and the more appropriate and
near variation is of 100 marks only. Therefore, it may be used as a quick and at
a glance measure of dispersion, but you cannot be dependent on “Range” in
order to know true dispersion.

29. Quartile Deviation and its Interpretation-
• The second measure of measures of dispersion is Quartile Deviation. It is also
known as ‘semi-interquartile range’. As you know the interval between highest and
lowest score is known as ‘range’, in a similar way distance between first and third
quartile divided by two is known as ‘Quartile deviation’. Therefore, it may be
expressed as: QD or Q = (Q3 -Q1) / 2 ............... (4)
• Since you are already aware on the concept of percentile, therefore, you may
simply infer the formula in the following manner: QD or Q = (P75 - P25) / 2 ..............
(5)
• In order to calculate Q, it is clear that we must first compute the 75th and 25th
Percentile and, therefore, the formula (1) previous discussed for calculating
percentile may be used in the following manner: Q1 = L + [(N/4 - Cfb) /fq] i
....................... (6)
• and Q3 = L+[(3N/4 - Cfb)/fq] i .................. (7)
• Note: You must remember that formula (1), used for Percentile has been taken
from the formula which you used in calculating the median.

30. Example: Calculate the Quartile Deviation for the data given in Table ’15.1’.
• Solution: Using the data given in Table ’15.1’ and Formula (6) and (7):
• (i) Q1 = P25 = 49.5+[(25-21)/35] 10 = 50.64
• (ii) Q3 = P75 = 59.5 + [(75-56)/20]10 = 69
• Therefore, QD = (Q3-Q1)/2 = (69-50.64)/2 = 9.18
• Q = 9.18
• Interpretation of Quartile Deviation
• Quartile deviation is easy to calculate and interpret, it is independent of the problem of extreme
values and, therefore, it is more representative and authentic than range. In distribution where
we prefer median as a measure of central tendency, the quartile deviation is also preferred as
measure of dispersion. However, both the measures are not suitable to algebraic operations,
because both do not consider all the values of the given distribution. In case of symmetrical
distribution, mean and median are equal and median lies at an equal distance from the two quartiles
i.e.
• Q3 – Median = Median – Q1
In case of non-symmetrical distribution, two possibilities may arise:
• I. Q3 – Median > Median – Q1 (Positive Skewed Curve)
• II. Q3 – Median < Median – Q1 (Negative Skewed Curve)

31. Mean Deviation-
• The mean deviation is defined as a statistical measure that is used to
calculate the average deviation from the mean value of the given data
set.
• It quantifies the dispersion or spread of data points in a dataset. Mean
deviation is also known as average deviation.
• Use in Descriptive Statistics:
• Mean deviation is used in descriptive statistics to quantify the average
amount by which individual data points deviate from the mean.
• Limitation:
• One limitation of mean deviation is that it does not have a clear mathematical
interpretation, especially when comparing datasets with different means.

32. Standard Deviation-
• The standard deviation is the average amount of variability in your
dataset. It tells you, on average, how far each value lies from the mean.
A high standard deviation means that values are generally far from the
mean, while a low standard deviation indicates that values are clustered
close to the mean.

33. Skewness-
• Skewness measures the degree of symmetry of a distribution.
The normal distribution is symmetric and has a skewness of zero.
• If the distribution of a data set instead has a skewness less than
zero, or negative skewness (left-skewness), then the left tail of
the distribution is longer than the right tail; positive skewness
(right-skewness) implies that the right tail of the distribution is
longer than the left.

34. Kurtosis-
• Kurtosis measures the thickness of the tail ends of a distribution in
relation to the tails of a distribution. The normal distribution has a
kurtosis equal to 3.0.
• Distributions with larger kurtosis greater than 3.0 exhibit tail data
exceeding the tails of the normal distribution (e.g., five or more
standard deviations from the mean). This excess kurtosis is known in
statistics as leptokurtic, but is more colloquially known as "fat tails."
The occurrence of fat tails in financial markets describes what is known
as tail risk.
• Distributions with low kurtosis less than 3.0 (platykurtic) exhibit tails
that are generally less extreme ("skinnier") than the tails of the normal
distribution.

35. Questions for Self Study-
1. What problems may be encountered while interpreting the test scores?
2. How is Frequency Distribution Table prepared?
3. What are Z Test and T Test? How are they calculated?
4. What is Percentile and Percentile Rank?
5. What are the various categories for classifying measurement scales?
6. Discuss the importance and Characteristics of Normal Probability Curve.
7. Explain the uses and characteristics of Mean.
8. What are the advantages of Mode?
9. What are the various Dimensions of Deviation? Discuss the Standard
Deviation with example.