Normal Distribution

Introduction

In real life normally the process output follows the pattern of normal distribution. This can be best explained with the help of dice.

A dice has six faces. When the dice is rolled on, chance of appearing a number from 1, 2, …5, 6 on top of the dice is one in six.

A Dice

Chance of appearing 3 on top, is one in six or 1/6

Assumptions

Each dice represents a cause.

Number appearing on top of the dice represents contribution

of that dice (cause) to the total effect.

Sum of the numbers appearing on top of each dice represents the total effect.

Case - When Only One Factor/Dice is present

Let us roll the dice say 300 times.

Since there is only one dice, the number appearing on top and the sum are same.

Out of 300 rolling of dice, number of times we get the number as 1, 2, 3, 4, 5 and 6 is approximately 50 (300/6). If we make histogram of sums then we get uniform distribution.

Total Effect With only One Dice ( Factor )

Sum of numbers

N=300

10

2 3 4 5 6

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1

Freq

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Case - When Only Two Factors are present

The sum of the numbers appearing on top varies between 2, 3, 4 ….10, 11, 12.

Chances of getting sum as 2 (1+1) and 12 (6+6) is (1/6)×(1/6) = 1/36, which is low.

Chances of getting sum as 7 is highest because there are several combinations; (6+1), (2+5), (3+4).

Histogram for the sums is triangular.

Total Effect With two Dices ( Factors )

Sum of numbers

N=300

10

20

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40

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2 4 6 8 10 12

Frequency

Case - When 3 Factors are Present

The sum of the numbers appearing on top varies between 3, 4, 5,6,7,8,9 ….16, 17, 18.

Chances of getting sum as 3 (1+1+1) and 18 (6+6+6) is (1/6)×(1/6)×(1/6) = 1/216, which is quite low.

Chances of getting sum as 10 is more because there are several combinations.

Distribution pattern for the sum is a curve with hump.

Total Effect With 3 dices ( Factors )

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Sum of Numbers

3 6 9 12 15 18

Freq

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Case - Total Effect when 4 Dices ( Factors )

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Sum of Numbers

4 8 12 16 20 24

With only 4 factors, the distribution of outcome is near normal

Freq

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Systems In Real Life

In real life we have several factors which effect the output of a system.

For example, in manufacturing setup, we have as bearing play, coolant

quality, voltage fluctuation, variation in material property, changes in

humidity, operator’s mood, inspection methods etc. Each one of them has

different pattern. They act on the process simultaneously with different

combinations; hence the process output follows the normal distribution.

In Real Life, We have Normal Distribution

Hydraulic Fluctuation

Voltage Fluctuation

Coolant

Bearing Play

Operator

Material

Humidity

Time

Inspection Methods

Output of Processis normal Distribution

Normal distribution

Normal Distribution

In practice generally we come across data which is of variable type and is measured on continuous scale. If there are no extraneous factors it follows a systematic pattern known as “Normal Distribution”. It can be derived after plotting histogram as explained in the next slide.

Histogram

CTQ

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Thickness of output in mm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Normal curveF

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Thickness of output in mm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Normal Curve is a smooth symmetrical bell shaped curve as shown below.

CTQ

Most commonly used intervals

In normal distribution, the most common intervals are :

1. Process mean - SD and process mean + SD i.e 1 Sigma

2. Process mean - 2 SD and process mean + 2 SD i.e 2 Sigma

3. Process mean - 3 SD and process mean + 3 SD i.e 3 Sigma

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Mean = 10, SD = 2

- SD + SD

Mean+SD =12Mean-SD =8

% Population

% Population between Mean - SD & Mean + SD (1Sigma)

% population between Mean - SD and Mean + SD or 1 Sigma = 68.3,

This implies that 68.3% population lies between dimensions of 8 and 12

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Mean = 10, SD = 2

% Population

Mean-2 SD=6Mean+2 SD=14

-2 SD +2 SD

% Population between Mean - 2 SD & Mean + 2 SD (2Sigma)

% population between Mean - 2 SD and Mean +2 SD or 2 Sigma = 95.5This implies that 95.5% population lies between dimensions 6 &14

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Mean = 10, SD = 2

Mean + 3SD =16Mean - 3SD =4

% Population

- 3 SD + 3 SD

% population between Mean - 3 SD and Mean +3 SD or 3 Sigma = 99.73This implies that 99.73 % population lies between 4 & 16

% Population between Mean - 3 SD & Mean + 3 SD (3 Sigma)

Range of observations for nearly 100% population (to be precise 99.73%) depends upon, mean of the process and standard deviation of the process. If it follows normal distribution the population lies between -

Range of observations for nearly 100% population

Mean - 3 SD & Mean + 3 SD i.e ( 3Sigma )