Normal probability curve

1. TCS 2: From Unit VI
FUNDAMENTALS OF EDUCATIONAL
REASERCH AND STATISTICS
Prepared By
Bincy K Mathew Reg. No.:20062

2. Introduction
Basic Concepts of Probability
A probability is a number that reflects the chance
or likelihood that a particular event will occur.
Probabilities can be expressed as proportions that
range from 0 to 1, and they can also be expressed
as percentages ranging from 0% to 100%.

3. Indication of Probability
A probability of 0 indicates that there is no chance
that a particular event will occur, whereas a
probability of 1 indicates that an event is certain to
occur. A probability of 0.45 (45%) indicates that
there are 45 chances out of 100 of the event
occurring.

4. A study of obesity in children 5-10 years of age
Age (years)
5 6 7 8 9 10 Total
Boys 5 6 4 5 8 2 30
Girls 4 2 3 1 4 6 20
Totals 9 8 7 6 12 8 50
What is the probability of selecting a boy?
Probability of selecting a boy = No. of Boys
Total Number of students
= 30
50
= 0.6 or 0.6 x 100 = 60%

5. A study of obesity in children 5-10 years of age
Age (years)
5 6 7 8 9 10 Total
Boys 5 6 4 5 8 2 30
Girls 4 2 3 1 4 6 20
Totals 9 8 7 6 12 8 50
What is the probability of selecting a boy?
Probability of selecting a boy = No. of Boys
Total Number of students
What is the probability of selecting a girl?
What is the probability of selecting a 7 year-old?
What is the probability of selecting a boy who is 10 years
of age?

6.  What is a Probability Distribution?
It is a statistical function that describes all the possible
values and likelihoods that a random variable can take
within a given range. This range will be bounded between
the minimum and maximum possible values, but precisely
where the possible value is likely to be plotted on the
probability distribution depends on a number of factors.
These factors include the distribution's mean
(average), standard deviation, skewness, and kurtosis.

7.  What Is Standard Deviation?
It is a statistic that measures the dispersion of a dataset
relative to its mean. The standard deviation is calculated as
the square root of variance by determining each data point's
deviation relative to the mean. If the data points are further
from the mean, there is a higher deviation within the data
set; thus, the more spread out the data, the higher the
standard deviation.

8.  What Is Standard Deviation?
Standard deviation measures the dispersion of a dataset
relative to its mean.

9. What is Normal Distribution?
Normal distribution, also known as the Gaussian
distribution, is a probability distribution that is
symmetric about the mean, showing that data near
the mean are more frequent in occurrence than
data far from the mean. In graph form, normal
distribution will appear as a bell curve.

10. What is Normal Distribution?
A normal distribution is the proper term for a
probability bell curve.
In a normal distribution the mean is zero and the
standard deviation is 1. It has zero skew and a
kurtosis of 3.
Normal distributions are symmetrical, but not all
symmetrical distributions are normal.

11. What Is a t-Distribution?
The T distribution, also known as the Student’s t-
distribution, is a type of probability
distribution that is similar to the normal
distribution with its bell shape but has heavier
tails. T distributions have a greater chance for
extreme values than normal distributions, hence
the fatter tails.

12. The Difference Between
a T Distribution and a Normal Distribution
Normal distributions are used when the population
distribution is assumed to be normal.
The T distribution is similar to the normal
distribution, just with fatter tails. Both assume a
normally distributed population.
T distributions have higher kurtosis than normal
distributions.
The probability of getting values very far from the
mean is larger with a T distribution than a normal
distribution.

13. What Is a t-Distribution?
It is a continuous probability distribution of the z-score
when the estimated standard deviation is used in the
denominator rather than the true standard deviation.
The T distribution, like the normal distribution, is bell-
shaped and symmetric, but it has heavier tails, which
means it tends to produce values that fall far from its
mean.
T-tests are used in statistics to estimate significance.

14. Normal Probability Curve (N.P.C.)
The Normal Probability Curve (N.P.C.)
is symmetrical about the ordinate of the
central point of the curve.
It implies that the size, shape and slope
of the curve on one side of the curve is
identical to that of the other.

15. DEFINITION
The “Graph of the probability density
function of the normal distribution is
a continuous bell shaped curve, symmetrical
about the mean” is called Normal
Probability Curve.

16. Laplace and Gauss (1777-1855), derived
the NORMAL PROBABLITITY CURVE
independently, so the curve is also known
as Gaussian Curve in the honor of Guass.
NPC is the frequency polygon of any
normal distribution.
It is an ideal symmetrical frequency curve
and is supposed to be based on the data
of a population.
APPLICATIONS OF
THE NORMAL PROBABILITY CURVE

17. 1. NPC is used to determine the percentage of
cases in a normal distribution within given
Limits
o The Normal Probability Curve helps us to
determine:
o What percent of cases fall between two scores
of a distribution?
o What percent of scores lie above a
particular score of a distribution?
o What percent of scores lie below a
particular score of a distribution?

18. Example:
 Given a distribution of scores with a mean of 24
and σ of 8. Assuming normality what percentage
of the cases will fall between 16 and 32.
Solution: Here first of all we have to convert both
the scores 16 and 32 into a standard score.
It is found that 34.13
cases fall between
mean and – 1σ and
34.13 cases fall
between mean and +
1σ. So ± σ covers
68.26% of cases.
So that 68.25% cases
will fall between 16
and 32.

19. 2. NPC is used to determine the value of
a score whose percentile rank is given:
By using NPC table we can determine the raw
score of the individual if the percentile
rank is given.
Example:
In a distribution of scores of a doss Pinky’s
percentile rank in statistics is 65. The
mean of the distribution is 55 with a
standard deviation of 10. Find but the raw
score of Pinky in Statistics.

20.  Solution:
As Pinky’s percentile rank is 65 so in a normal
distribution her position is 35% above the
mean. By entering in to the table ‘A’ we found
that 35% from the mean is + 1.04 σ.
By putting the value in ‘Z’ score.

21. 3. NPC is used to find the limits in a normal
distribution which include a given percentage
of cases:
When a distribution is normally distributed and
what we know about the distribution is Mean and
the Standard deviation at that time by using the
table area under NPC we can determine the
limits which include a given percentage of cases.
 Example:
Given a distribution of scores with a mean of 20
and σ of 5. If we assume normality what limits
will include the middle 75% of cases.

22.  Solution:
In a normal distribution the middle 75% cases
include 37.5% cases above the mean and 37.5%
cases below the mean. From the Table-A we can
say that 37.5% cases covers 1.15 σ units.
Therefore the middle 75% cases lie between mean
and ± 1.15 σ units. So in this distribution middle
75% cases will include the limits 14.25 to 25.75.

23. 4. It is used to compare two distributions
in terms of- overlapping:
If scores of two groups on a particular
variable are normally distributed. What
we know about the group is the mean and
standard deviation of both the groups.
And we want to know how much the first
group over-laps the second group or vice-
versa at that time we can determine this
by using the table area under NPC.

24. 5. NPC helps us in dividing a group into sub-
groups according to certain ability and
assigning the grades:
When we want to divide a large group in to certain sub-
groups according to some specified ability at that time
we use the standard deviation units of a NPC as units of
scale.
 Example:
An achievement test was administered to the 600 8th
grade students. The teacher wants to assign these
students in to 4 grades namely A, B, C and D according
to their performance in the test. Assuming the normality
of the distribution of scores calculate the number of
students can be placed in each group.

25.  Solution:
The area under a NPC is divided in to ± 3σ units
or 6σ units. Here we have to divide the students
in to 4 sections. So each section has
So if we shall distribute the section in order of
merit. The section-A will be within 1.5σ to 3σ
Section B will be within Mean to 1.5σ
Section C will be within Mean to —1.5σ and
Section D will be with in —1.5σ to – 3σ.

27. 6. NPC helps to determine the relative
difficulty of test items or problems:
When it is known that what percentage of
students successfully solved a problem
we can determine the difficulty level of
the item or problem by using table area
under NPC.

28. 7.NPC is useful to normalize a frequency
distribution:
In order to normalize a frequency
distribution we use Normal Probability
Curve. For the process of standardizing
a psychological test this process is very
much necessary.

29. 8.To test the significance of
observations of experiments we use
NPC:
In an experiment we test the relationship
among variables whether these are due
to chance fluctuations or errors of
sampling procedure or it is real
relationship. This is done with the help
of table area under NPC.

30. 9. NPC is used to generalize about
population from the sample:
We compute standard error of mean,
standard error of standard deviation and
other statistics to generalize about the
population from which the samples are
drawn. For this computation we use the
table area under NPC.

31. CONCLUSION:
Hence, we come to a conclusion that NPC is
used to determine the percentage of
cases in normal distribution within given
limits. It is also used to compare two
distributions in terms of- overlapping. It
also helps us in dividing a group into sub-
groups according to certain ability and
assigning the grades etc.

32. It can be summarized as:
 To normalize a frequency distribution. It is
an important step in standardizing a
psychological test or inventory.
 To test the significance of observations in
experiments, findings them relationships with
the chance fluctuations or errors that are
result of sampling procedures.
 To generalize about population from which the
samples are drawn by calculating the standard
error of mean and other statistics.
Continue…

33. Continue…
 To compare two distributions. The NPC is used
to compare two distributions.
 To determine the difficulty values. The Z scores
are used to determine the difficulty values of
test items.
 To determine the level of significance. The
levels of significance of statistics results are
determined in terms of NPC limits.
 To scale responses to opinionnaires, judgment,
ratings or rankings by transforming them
numerical values.