## What do we mean by garage’s width = 12σ and car’s width = 6σ?

Right now we are not in a position of going into the details of the standard normal distribution hence, for the time being let’s assume that my manufacturing process is stabilized, which is represented by a symmetrical curve shown below

The main characteristic of this curve is that the 99.7% of the product would be between LCL & UCL or within ±3σ distance from the mean (μ). Only 0.3% or 3000ppm products would be beyond ±3σ or defective products. So width of the car is equivalent to the width of the process = UCL-LCL = voice of the process = VOP = 6σ = ±3σ.

Second point is that the curves never touches the x-axis à means that there will always be some probability of failure even if you move to infinity from the mean (probability can be negligible but will be there).

Now let’s overlap the above process curve with the customer’s specifications (=12σ = ±6σ) or the garage’s specifications.

We can see that there is a safety margin of 3σ on both side of the process control limits (LCL & UCL). In layman words, in order to produce a defective product, my process has to deviate by another 3σ, which has very remote possibility. Statistically ± (position of LSL & USL) from the mean would account for only ~3.4 ppm failure (don’t bother about the calculation right now, just understand the concept). For this has to happen, someone has to disturb the process deliberately. Compare this failure of 3.4 ppm at ±6σ level with 3000ppm at ±3σ level!

Even if the mean of the process deviate by ±1.5σ, there is enough margin of safety and it will not impact the quality and in regular production, this deviation of ±1.5σ is quite common.

Car Parking & Six-Sigma

What’s the big deal, let’s rebuild the garage to fit the bigger car!

How the garage/car example and the six-sigma (6σ) process are related?

Now Let’s start talking about 6sigma

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## Now it’s important to understand the concept of sigma or the standard deviation

We have seen that we need to restrict the width of the car for a given width of the garage. This is analogous to the with of the process (voice of customer, VOP) Vs the width of the customer’s specification (voice of the customer or VOC).
The width of the process is measured in terms of standard deviation denoted by σ (sigma).

The target of the 6sigma methodology is to reduce this variance (width of the car) to such an extent that even by mistake it should not cross the customer’s specification (or should not hit the wall of the garage).

Before we work towards reducing the σ, we should know about this monster very well as we will be encountering him at every step during the 6sigma journey.

There are two very important characteristics of any data set

Location and the spread of the data set.

Location represents the point in the data set where there is maximum clustering of the data –> Mean and median.

Spread represents the variability in the data set, there will be some observations that will be above the mean and there will be some that will be below the mean. Standard deviation σ measures the average spread of the data from the mean in either direction of the mean.

Office arrival time for last 5 days with average time are given below, deviation of each observation from the mean is also captured.

Let’s calculate the average deviation

Note that sum of all positive deviations = sum of all negative deviations which indicates that the mean divided the data in two equal halves.

Sum of all deviation itself becomes zero, hence we need some other way to calculate this average deviation about the mean.

In order to circumvent the issue, a very simple idea was used

Square of negative number → positive number → square root of this number → ±parent number

Hence square of all the deviations are calculated and summed-up to give sum of squares (simply SS) [1]. This SS is then divided by total number of observations to give average variance around the mean.[2] The square root of this variance gives standard deviation s, the most common measure of variability.

What it typically means that “on an average data is 7.42 units (= 1 standard deviation ±1σ) in either direction of the mean in the given data set. Mean of the data set is at ZERO standard deviation.

If process a stabilized and normally distributed then following holds true

i.e. 99.7 % of the observation in the data set would be between ±3σ.

Now we can understand whey we have taken 12σ as the width of the garage and 6σ as the width of the car!

The concept of ‘σ’ is the most important concept in understanding 6sigma. If we can understand it, downstream we wouldn’t be having any problem in understanding other topics. At this moment one important point to be noted here is that the calculation of σ depend on the type of data or data distribution we are handling.

Calculation of mean and σ would be different depending on whether we are dealing with normal distribution, binomial distribution, Poisson distribution etc. The importance of this would be realized when we would be studying the various types of control charts. At that time we just have to remember that “we must calculate mean and σ according to the distribution”.

[1] Popularly known as sum of squares, this most widely term used in ANOVA and Regression analysis

[2] SS divided by its degree of freedom → mean sum of squares or MSE, these concepts would appear in ANOVA & Regression analysis.

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## How the garage/car example and the six-sigma (6σ) process are related?

Let the width of the garage (D) and that of the car (d) is measured in some units called as sigma or σ, further

Width of the garage is sacrosanct = 12σ (assume for the time being), then following three cases can occur depending on the ratio of D/d = Cp.

The process sigma level is when the Cp =2, this is represented by case ‘C’ given below. Point to be noted is that there is a margin of safety (=3σ) on both side of the car before car touches the garage. The ideal width of car is taken as 6σ (don’t ask why, right now!)

Process capability Cpk: is measured in the terms of the σ distance between the center of the of the car (C1) and the wall of the garage. Cpk tell us “how far is the car from left wall or the right wall of the garage (or customer’s specifications).

Hence, the are two Cpk vales,

Cpk is given by

Car Parking & Six-Sigma

What’s the big deal, let’s rebuild the garage to fit the bigger car!

What do we mean by garage’s width = 12σ and car’s width = 6σ?

Now Let’s start talking about 6sigma

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## What’s the big deal, let’s rebuild the garage to fit the bigger car!

You may argue that what’s the big deal if my existing car doesn’t fit into the garage, I will rebuild the garage that can fit my bigger car.

You can always do it with garage but not with customer’s specifications as you can’t dictate it or simply customer won’t change his processes to fit your defective products. The only way is to improve your existing process or else lose your business to your competitors.

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How the garage/car example and the six-sigma (6σ) process are related?

What do we mean by garage’s width = 12σ and car’s width = 6σ?

Now Let’s start talking about 6sigma

## Car Parking & Six-Sigma

We can easily understand the concept of six-sigma using the analogy of parking a car into the garage and if you want to understand it more seriously, just imagine that you are entering a tunnel (instead of garage) at the speed of 150-200 Km/Hr!

When ever a car is bigger than the garage, it’s not possible to enter the garage. Take another car whose width is slightly less than the width of the car. This time it is possible to enter the garage but we will get frequent scratches unless you are an expert driver! Take a third car whose width is just half of the width of the garage , it’s is now possible for me to enter the garage even if I am drunk.

Car width represents the “current process capability

Garage width represents the “Customer’s specifications

We should remember that

“We need to buy a car according to the dimensions of the garage”

“We need to change our manufacturing processes to meet the customer’s specification”

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What’s the big deal, let’s rebuild the garage to fit the bigger car!

How the garage/car example and the six-sigma (6σ) process are related?

What do we mean by garage’s width = 12σ and car’s width = 6σ?

Now Let’s start talking about 6sigma

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