This ppt is a part of Business Analytics course.
Normal distribution : -
The Normal Distribution, also called the Gaussian Distribution, is the most significant continuous probability distribution.
A normal distribution is a
symmetric, bell-shaped curve
that describes the distribution of continuous random variables.
The normal curve describes how data are distributed in a population.
A large number of random variables are either nearly or exactly represented by the normal distribution
The normal distribution can be used to represent a wide range of data, such as test scores, height measurements, and weights of people in a population.
2. What is normal distribution?
The Normal Distribution, also called
the Gaussian Distribution, is the
most significant continuous
probability distribution.
A normal distribution is a
symmetric, bell-shaped curve
that describes the distribution of
continuous random variables.
The normal curve describes how data
are distributed in a population.
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
3. What is normal distribution?
……A normal distribution is a
A large number of random variables are either nearly or
exactly represented by the normal distribution
The normal distribution can be used to represent a wide
range of data, such as test scores, height measurements,
and weights of people in a population.
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
4. What is normal distribution?
The normal distribution has two parameters,
mean (μ ) and standard deviation (σ) .
It is important to know these two parameters because they
are used to calculate probabilities associated with the
normal distribution.
Mean : - It is the measure of central tendency, i.e. it provides
us an idea of the concentration of the observations about
the central part of the distribution.
Standard deviation :- Standard deviation describes the
dispersion or spread the variables about the central value.
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
5. Normal distribution is defined by its
mean and standard dev.
E(X)= =
Var(X)=2 =
Standard Deviation(X)=
dx
e
x
x
2
)
(
2
1
2
1
2
)
(
2
1
2
)
2
1
(
2
dx
e
x
x
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
6. Normal Distribution Definition
The Normal Distribution is defined by the probability density
function for a continuous random variable in a system.
Let us say, f(x) is the probability density function and X is the
random variable.
Hence, it defines a function which is integrated between the
range or interval (x to x + dx), giving the probability of
random variable X, by considering the values between x and
x+dx.
f(x) ≥ 0 ∀ x ϵ (−∞,+∞)
And -∞∫+∞ f(x) = 1
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
7. Normal Distribution Formula
The probability density function of normal or gaussian
distribution is given by;
Where,
x = the variable
μ = the population mean
σ = standard deviation of the population
e = the mathematical constant approximated by 2.71828
π = the mathematical constant approximated by 3.1415
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
8. Normal Distribution Curve
The random variables following the
normal distribution are those whose
values can find any unknown value in a
given range.
For example, finding the height of the
students in the school. Here, the
distribution can consider any value,
but it will be bounded in the range say,
0 to 6ft. This limitation is forced
physically in our query.
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
9. Normal Distribution Curve
Whereas, the normal distribution doesn’t
even bother about the range. The range
can also extend to –∞ to + ∞ and still we
can find a smooth curve.
These random variables are called
Continuous Variables, and the Normal
Distribution then provides here probability
of the value lying in a particular range for a
given experiment.
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
10. Normal Distribution Standard Deviation
Generally, the normal distribution has any positive
standard deviation.
The standard deviations are used to subdivide the
area under the normal curve. Each subdivided
section defines the percentage of data, which falls
into the specific region of a graph.
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
11. The Standardized Normal Distribution
The standard normal distribution, also
called the z-distribution,
is a special normal distribution where
the mean is 0 and the standard deviation
is 1.
Any normal distribution (with any mean
and standard deviation combination) can
be transformed into the standardized
normal distribution (Z)
To compute normal probabilities need to
transform X units into Z units
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
12. The Standardized Normal Distribution
The standard normal distribution, also
called the z-distribution,
is a special normal distribution where
the mean is 0 and the standard deviation
is 1.
Any normal distribution (with any mean
and standard deviation combination) can
be transformed into the standardized
normal distribution (Z)
To compute normal probabilities need to
transform X units into Z units
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
13. The Standardized Normal Distribution
Translate from X to the standardized normal (the “Z” distribution) by
subtracting the mean of X and dividing by its standard deviation:
Z scores tell you how many standard deviations from the mean each
value lies.
Converting a normal distribution into a z-distribution allows you to
calculate the probability of certain values occurring and to compare
different data sets.
The Z distribution always has mean = 0 and
standard deviation = 1
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
14. The Standardized Normal Probability
Density Function
The formula for the standardized normal probability
density function is
Wheree = the mathematical constant approximated by 2.71828
π = the mathematical constant approximated by 3.14159
Z = any value of the standardized normal distribution
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
15. Finding Normal Probabilities
Probability is measured by the area under the curve
a b X
f(X) P a X b
( )
≤
≤
P a X b
( )
<
<
=
(Note that the
probability of any
individual value is zero)
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
16. Probability as Area Under the Curve
Probability is measured by the area under the curve
The total area under the curve is 1.0, and the curve is
symmetric, so half is above the mean, half is below
f(X)
X
μ
0.5
0.5
1.0
)
X
P(
0.5
)
X
P(μ
0.5
μ)
X
P(
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
17. Comparing X and Z units
Z
₹100
2.0
0
₹200 ₹X (μ = ₹100, σ = ₹50)
(μ = 0, σ = 1)
Note that the shape of the distribution is the same, only the scale has changed. We can express
the problem in the original units (X in Rs. ) or in standardized units (Z)
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.
𝑍 =
𝑋 − 𝜇
𝜎
=
₹200 − ₹100
₹50
= 2.0
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
18. General Procedure
for Finding Normal Probabilities
To find P(a < X < b) when X is distributed
normally:
Draw the normal curve for the problem in terms of X
Translate X-values to Z-values
Use the Standardized Normal Table
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
19. The value within the
table gives the
probability from Z =
up to the desired Z
value
.9772
2.0
P(Z < 2.00) = 0.9772
The row shows
the value of Z
to the first
decimal point
The column gives the value of
Z to the second decimal point
2.0
.
.
.
Z 0.00 0.01 0.02 …
0.0
0.1
The Cumulative Standardized Normal table gives the probability less
than a desired value of Z (i.e., from negative infinity to Z)
The Standardized Normal Table
Z
0 2.00
0.9772
Example:
P(Z < 2.00) = 0.9772
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
20. The Standardized
Normal Table
What is the area to the
left of Z=1.51 in a
standard normal curve?
Z=1.51
Z=1.51
Area is 93.45%
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
21. Finding Normal Probabilities
Let X represent the time it takes
(in seconds) to download an image
file from the internet.
Suppose X is normal distribution
with a mean of18.0 seconds and a
standard deviation of 5.0 seconds.
Find P(X < 18.6)
18.6
X
18.0
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
22. Finding Normal Probabilities
Let X represent the time it takes, in seconds to download an
image file from the internet.
Suppose X is normal with a mean of 18.0 seconds and a
standard deviation of 5.0 seconds. Find P(X < 18.6)
Z
0.12
0
X
18.6
18
μ = 18
σ = 5
μ = 0
σ = 1
(continued)
0.12
5.0
8.0
1
18.6
σ
μ
X
Z
P(X < 18.6) P(Z < 0.12) @Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
24. Finding Normal
Upper Tail Probabilities
Suppose X is normal with mean 18.0 and
standard deviation 5.0.
Now Find P(X > 18.6)
X
18.6
18.0
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
26. Finding a Normal Probability Between
Two Values
Suppose X is normal with mean 18.0 and standard deviation 5.0.
Find P(18 < X < 18.6)
P(18 < X < 18.6)
= P(0 < Z < 0.12)
Z
0.12
0
X
18.6
18
0
5
8
1
18
σ
μ
X
Z
0.12
5
8
1
18.6
σ
μ
X
Z
Calculate Z-values:
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
27. Z
0.12
0.0478
0.00
= P(0 < Z < 0.12)
P(18 < X < 18.6)
= P(Z < 0.12) – P(Z ≤ 0)
= 0.5478 - 0.5000 = 0.0478
0.5000
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.02
0.1 .5478
Standardized Normal Probability
Table (Portion)
Solution: Finding P(0 < Z < 0.12)
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
28. Probabilities in the Lower Tail
Suppose X is normal with mean 18.0 and
standard deviation 5.0.
Now Find P(17.4 < X < 18)
X
17.4
18.0
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
29. Probabilities in the Lower Tail
Now Find P(17.4 < X < 18)…
X
17.4 18.0
P(17.4 < X < 18)
= P(-0.12 < Z < 0)
= P(Z < 0) – P(Z ≤ -0.12)
= 0.5000 - 0.4522 = 0.0478
(continued)
0.0478
0.4522
Z
-0.12 0
The Normal distribution is
symmetric, so this probability
is the same as P(0 < Z < 0.12)
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
30. Empirical Rule
μ ± 1σ encloses about
68.26% of X’s
f(X)
X
μ μ+1σ
μ-1σ
What can we say about the distribution of values
around the mean? For any normal distribution:
σ
σ
68.26%
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
31. The Empirical Rule
μ ± 2σ covers about 95.44% of X’s
μ ± 3σ covers about 99.73% of X’s
x
μ
2σ 2σ
x
μ
3σ 3σ
95.44% 99.73%
(continued)
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
32. Given a Normal Probability
Find the X Value
Steps to find the X value for a known
probability:
1. Find the Z value for the known probability
2. Convert to X units using the formula:
Zσ
μ
X
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
33. Finding the X value for a Known
Probability
Example:
Let X represent the time it takes (in seconds) to
download an image file from the internet.
Suppose X is normal with mean 18.0 and standard
deviation 5.0
Find X such that 20% of download times are less than
X.
X
? 18.0
0.2000
Z
? 0
(continued)
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
34. Find the Z value for
20% in the Lower Tail
20% area in the lower tail
is consistent with a Z
value of -0.84
Z .03
-0.9 .1762 .1736
.2033
-0.7 .2327 .2296
.04
-0.8 .2005
Standardized Normal Probability
Table (Portion)
.05
.1711
.1977
.2266
…
…
…
…
X
? 18.0
0.2000
Z
-0.84 0
1. Find the Z value for the known probability
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
35. Finding the X value
2. Convert to X units using the formula:
8
.
13
0
.
5
)
84
.
0
(
0
.
18
Zσ
μ
X
So 20% of the values from a distribution with
mean 18.0 and standard deviation 5.0 are less
than 13.80
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
36. Exercise
Q.1 The mean of the weight of the students in this
class is 80 kg, the standard deviation (SD) is 10.5 kg,
find the probability for :-
• Find P(X < 85.5)
• Find P(X > 85.5)
• Find P(80 < X < 85.5)
• Find P(75 < X < 80)
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
37. Exercise
Q.2 IQ tests are measured on a
scale which is N (100, 15). A woman
wants to form an 'Eggheads Society'
which only admits people with the top
1% of IQ scores. What would she have
to set as the cut-off point in the test to
allow this to happen?
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
38. Exercise
Q3. A manufacturer does not know the mean and SD
of the diameters of ball bearings he is producing.
However, a sieving system rejects all bearings larger
than 2.4 cm and those under 1.8 cm in diameter. Out
of 1000 ball bearings 8% are rejected as too small and
5.5% as too big. What is the mean and standard
deviation of the ball bearings produced?
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM
39. Exercise
Solution 3 :
Assume a normal distribution of N( 1 − 0.08)
So 1-0.08 = 0.92 = 92% area in the lower tail is consistent
with a Z value of =
Z value for 0.92 = 1.4
so 1.8 is 1.4 standard deviations below mean
Similarly for N ( 1 − 0.055) , area in the upper tail is
1-0.055 = 0.945, so from z-table the value of z = 1.6 ,
so 2.4 is 1.6 standard deviations above the mean.
@Ravindra Nath Shukla (PhD Scholar) ABV-IIITM