ARUN PRABHAKARME TQEMPEC UNIVERSITY OF TECHNOLOGY, CHANDIGARH

Properties of Normal Distribution

Data can be "distributed" (spread out) in different ways

Normal Distribution

Normal Distribution

• Many things closely follow a Normal Distribution:

• Heights Of People

• Size Of Things Produced By Machines

• Errors In Measurements

• Blood Pressure

• Marks On A Test

Normal Distribution

• MEAN = MEDIAN = MODE

• SYMMETRY ABOUT THE CENTER

• 50% OF VALUES LESS THAN THE MEAN AND 50% GREATER THAN THE MEAN

Standard Deviationsfor 3 sigma level

 

68% of values are within1 standard deviation of the mean

 

 95% are within 2 standard deviations

 

 

99.7% are within 3 standard deviations

Standard Scores

The number of standard deviations from the mean is also called the "Standard Score", "sigma" or "z-score"

Why Standardize ... ?• It can help you make decisions about your data.

EXAMPLE

95% of students at school are between 1.1m and

1.7m tall. Assuming this data is normally distributed

calculate the mean and standard deviation and find Z

Score for one of your friend who is 1.85 m tall ?

Standard Scores

MEAN

The mean is halfway between

1.1m and 1.7m:

Mean = (1.1m + 1.7m ) = 1.4 m

2

• 95% is 2 standard deviations either side of the mean (a total of 4 standard deviations) so:

1 standard deviation = (1.7m-1.1m) / 4 

= (0.6m / 4) = 0.15 m

Standard Deviations

• z is the "z-score" (Standard Score)

• x is the value to be standardized

• μ is the mean

• σ is the standard deviation

Z - SCORE

( X- μ ) = ( 1.85 - 1.4 ) =  3

Z - SCORE

σ

1.5

• A Practical Example:

Your company packages sugar in 1 kg bags. When you weigh a sample of bags you get these results:

1007g, 1032g, 1002g, 983g, 1004g, ... (a hundred measurements)

Mean = 1010g

Standard Deviation = 20g

Some values are less than 1000g ... How can you fix that?

APPLICATION

• Adjust the mean amount in each bag

• The standard deviation is 20g, and we need 2.5 of them:

• 2.5 × 20g = 50g

• So the machine should average 1050g, like this:

SOLUTION 1

INCREASE THE AMOUNT OF SUGAR

• Adjust the accuracy of the machine

• Or we can keep the same mean (of 1010g), but then weneed 2.5 standard deviations to be equal to 10g:

• 10g / 2.5 = 4g

• So the standard deviation should be 4g, like this:

Reff: www.mathsisfun.com/data/standard-normal-distribution

SOLUTION 1

INCREASE THE AMOUNT OF SUGAR

“Don't cry because it's over, smile because it happened.”

― Dr. Seuss

THANKYOU