Why are features of steel washers normally distributed in a factory making them?

normal_distribution
statistics

#1

Hi All,
I came across a video on youtube.


In this video , Dr. J Maiti from IIT Kharagpur , is giving Introduction to multivariate statistical modeling
In very start , he starts with explaining , meaning of the word "Applied"
The example he took was:
"Imagine 3 machines are used to create steel washers.Each washer has its inner diameter , outer diameter and thickness. Now imagine we produce very large number of washers using these machines.If we take any of the 3 feature of a washers and plot it , they will be normally distributed"
How?

I Hope , my question is clear.Please let me know , if its not.
Thanks


#2

@Rishabh0709

Hi there,

Imagine this, the machines were designed to produce the washers for a fixed size (say 1 cm) of inner diameter. And we expect the machine to do the same too. But what happens is, the number of washers of size 1 cm produced by machine is very large say 1000 . But sometimes due to some uncontrollable factors such as machine error, temperature etc. the size may vary , may be 0.9 cm, 0.8 cm etc but the number of washers of such sizes is small. These deviations are what are called errors and this is what standard deviation( population) or standard error (sample) measures. When plotted they will form a normal distribution.

Hope this helps.

Regards,
Neeraj


#3

Thanks @NSS , for the explanation.
Just to understand it more concretely , can you please explain , why the plot will not form any other distribution?


#4

Hi @Rishabh0709

The logic is based on “Extreme Value Theory”

Extreme value theory or extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the median of probability distributions. It seeks to assess, from a given ordered sample of a given random variable, the probability of events that are more extreme than any previously observed.

Now, if we have configured the machine to build washers of
Height : 1 cm
Inner D : 2 cm
Outer D : 3 cm

Most of the washers built will have the above dimensions. But as @NSS mentioned, due to temperature, many will have Height : 1 +/- .001 cm etc etc

Then due to some major issues, power outage etc some washers will have Height : 2 cm
Now power outages are very rare so Height : 2 cm has very less probability

There are infinite reasons now and all of them will contribute to different measurements, but most of them will be centered around the values at which the machine has been configured to operate

Height : 1 cm
Inner D : 2 cm
Outer D : 3 cm

Regards,
Anant


#5

Hi @Rishabh0709,

Read about Central limit theorem.

It states that with sample size of greater than 30 or more , the sample mean is approximately normally distributed., with mean equivalent to polulation mean and standard deviation of sigma/root(N).

For more appreciation of why this happen -
http://onlinestatbook.com/2/sampling_distributions/clt_demo.html
Open the link and run the simlulation.

Hope this helps.

Regards,
Aayush


#6

Thanks @anantguptadbl for the wonderful explanation
Thanks to @aayushmnit as well.
.