Normal Distribution Curve | Its characteristics and its role in engineering

1. FABRIKAM
METROLOGY & QUALITY ASSURANCE
D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g
UNIVERSITY OF ENGINEERING & TECHNOLOGY LAHORE

2. Normal
Distribution

3. 3
Table of contents
 Normal Distribution Curve
 Non-Normal Distribution
 Mathematical Definition
 Standard Deviation
 Characteristics of Normal Curve
 Normality Test
 Methods for Normality Test
 Example
 Practical Example

4. Normal Distribution Curve
The term “Normal Distribution Curve” is used
to describe the mathematical concept called
normal distribution.
It refers to the shape that is created when a line
is plotted using the data points for an item that
meets the criteria of ‘Normal Distribution’

5. Non-Normal Distribution

6. Mathematical Definition
A continuous random variable X is said to
follow normal distribution with mean ( μ ) and
standard derivation ( σ ), if its probability
function is
f (x) =
1
𝜎 2𝜋
. 𝑒− 𝑥−μ 2
2𝜎2
Where,
μ = mean
σ = Standard deviation

7. Standard Deviation
 The Standard Deviation is a degree of
dispersion from mean value
 It is a measure of how spread out the numbers
are.

8. Characteristics of Normal Distribution
1. Bell Curve
2. Mean, mode and median
3. Symmetry
4. Uni-nodal
5. Standard Deviation
6. The total area under the curve is 1

9. CHARACTERISTICS
1: Bell Curve
• Normal curve often called
bell curve due to its
appearance
• Data follows Bell Curves
closely, but not perfectly

10. • Normal curve is symmetric
about mean
• 50% data is less than mean
and 50% data is greater than
mean
CHARACTERISTICS
2: Symmetry

11. • In normal distribution mean
is always at centre
• In Normal Distribution;
Mean = Median = Mode
CHARACTERISTICS
3: Mean, Mode, Median

12. • Normal curve has only One
mode, so there is only one
peak in a curve which is
called Uni-nodal and lies in
the centre.
CHARACTERISTICS
4: Uni - Nodal

13. • Normal curve have predictable
standard deviation
CHARACTERISTICS
5: Standard Deviation

14. FABRIKAM
NORMALITY
TEST
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15. Normality Test
Normality tests are used to determine if a data
set is well-modeled by a normal distribution
To compute how likely it is for a random
variable underlying the data set to be normally
distributed.
Need to make sure data is normally distributed
before using a normal distribution

16. FABRIKAM
METHODS FOR
NORMALITY TEST
1. Histogram
2. Skew & Kurtosis
3. Probability Plots
4. Chi-Square goodness of Fit
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17. Normality Test Methods
1: Histogram
• It gives visual analyzation of data, either
it looks like bell curve shape i.e. normal
shape or different shape
• It does NOT have to be “Perfect” bell
curve shape
• Data should be symmetrical
• Don’t have several peaks, that is, data
should be unimodal.

18. Normality Test Methods
2: Skew & Kurtosis
Skews
• Normal distributed data has no skew
• In +ve skew we have a tail to right
• In –ve skew we have tail to left
Large Sample Space for better judgement.

19. Normality Test Methods
2: Skew & Kurtosis
Kurtosis
• Kurtosis describes how sharp your peak
is or how flattened it is
• Normal Distribution has Kurtosis of 3.0
• Mesokurtic
• Leptokurtic
• Platykurtic
• Minimum Sample Space = 100

20. Normality Test Methods
3: Probability Plots
• Minimum sample space = 30
• It can be used by hand or statistical
software
• Put data in ascending order
• Start from data 1, and calculate your
plotting position
• Label data scale, draw your points
• Draw line of best fit
• The ‘Best Fit Line’ determines the
normality of curve.

21. Normality Test
Methods
4 : Chi-Square
• Difference between observed and
expected value
• We expect our value to fall in this
distribution, if it falls outside it is not
normally distributed
• Minimum sample size = 125

22. Examples of Normal Distribution
• Height of people
• Measurement errors
• Blood pressure
• Point on a test
• IQ score
• Salaries

23. FABRIKAMFABRIKAM
PRACTICAL EXAMPLE
Problem:
The bottom 30% of students failed an end of semester
exams. The mean for the test was 120, and the
standard deviation was 17. What was passing score?
Data: μ = 120, σ = 17, x = ?
Solution:
z =
𝒙−μ
σ
𝒙 = z σ +μ
𝒙 = z (17) + 120
From table z = -0.52
𝒙 = (-0.52)(17) + 120
𝒙 = 111.16
which is passing score.
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