The document discusses the normal distribution, which is a symmetrical, bell-shaped distribution. The normal distribution is determined by its mean and standard deviation. Varying the mean shifts the distribution left or right, while varying the standard deviation increases or decreases the spread. The normal distribution is important in research as it represents a set of values around a central peak.

1. Normal distribution
Sudipta Debnath

2. The Normal Distribution
 Bell Shaped
 Symmetrical
 Mean, Median and Mode
are Equal
Location is determined by the
mean, μ
Spread is determined by the
standard deviation, σ
The random variable has an
infinite theoretical range:
+  to  
Mean
= Median
= Mode
X
f(X)
μ
σ

3. A
B
C
A and B have the same mean but different standard deviations.
B and C have different means and different standard deviations.
By varying the parameters μ and σ, we obtain
different normal distributions

4. The Normal Distribution Shape
X
f(X)
μ
σ
Changing μ shifts the
distribution left or right.
Changing σ increases
or decreases the
spread.

5.  A theoretical frequency distribution called normal distribution is of
much importance in research data estimation.
 It is a symmetrical distribution having a set of values in the middle
or peak of the distribution.
 A bell-shaped normal curve graphically represents a normal
distribution.
 Johann Carl Friedrich Gauss (30 April 1777–23 February 1855), a
German mathematician and physical scientist, developed the
concept of normal distribution; hence, it is also known as the
Gaussian curve.

6. Properties
 Used for continuous variables
 Can be used for –ve value variable as well
 Bell shaped
 Mean = Median = mode; on the highest peak of X axis
 Symmetrical; has two parameters, Mean & SD

7.  The shape of the curve depends on the mean and
SD; more the SD, wider is the curve.

8. 7: Normal Probability Distributions
8
Mean and Standard Deviation of
Normal Density
μ
σ

9. 7: Normal Probability Distributions
9
Parameters μ and σ
 Normal pdfs have two parameters
μ - expected value (mean “mu”)
σ - standard deviation (sigma)
σ controls spread
μ controls location

10. 7: Normal Probability Distributions
10
68-95-99.7 Rule for
Normal Distributions
 68% of the AUC within ±1σ of μ
 95% of the AUC within ±2σ of μ
 99.7% of the AUC within ±3σ of μ

11. 7: Normal Probability Distributions 11
Example: 68-95-99.7 Rule
Wechsler adult
intelligence scores:
Normally distributed
with μ = 100 and σ = 15;
X ~ N(100, 15)
 68% of scores within
μ ± σ
= 100 ± 15
= 85 to 115
 95% of scores within
μ ± 2σ
= 100 ± (2)(15)
= 70 to 130
 99.7% of scores in
μ ± 3σ =
100 ± (3)(15)
= 55 to 145

12. 7: Normal Probability Distributions 12
Assessing Departures from Normality
Same distribution on
Normal “Q-Q” Plot
Approximately
Normal histogram
Normal distributions adhere to diagonal line on Q-Q
plot

13. 7: Normal Probability Distributions 13
Negative Skew
Negative skew shows upward curve on Q-Q plot

14. 7: Normal Probability Distributions 14
Positive Skew
Positive skew shows downward curve on Q-Q plot

15. Importance
 Used to find confidence limits
 95% CI is within 1.96 SD on either side
 Basis for test of significance

16. Types
 Standard normal curve
 Normal probability curve

17. Standard normal curve
 When the mean is zero and the SD is one, it is
called as standard normal curve.

18. Normal Probability Curve
 When the total area under the curve is equal to
unity, it is known as normal probability curve

19. The Standardized
Normal Distribution
 Also known as the “Z” distribution
 Mean is 0
 Standard Deviation is 1
Z
f(Z)
0
1
Values above the mean have positive Z-values.
Values below the mean have negative Z-values.

20. Example
 If X is distributed normally with mean of $100 and standard
deviation of $50, the Z value for X = $200 is
 This says that X = $200 is two standard deviations (2
increments of $50 units) above the mean of $100.
2.0
$50
100
$
$200
σ
μ
X
Z 



