## Why & How Cpm came into existence? Weren’t Cp & Cpk enough to trouble us?

In the earlier post (see earlier post “what is Taguchi Loss function?”) we end up the discussion stating that Cp need to be penalized for the deviation of the process mean from the specification mean.

If you are producing goods near to LSL or USL hence, the chances of rejection increases which in turn increases the chances of reprocessing and rework thereby increasing the cost. Even if you manage to pass the quality on borderline then your customer has to adjust his process accordingly to accommodate your product thereby, increasing his set-up time and cost involved in readjusting his process. Moreover, the variance from your product and the variance from the customer’s process just get adds up to given final product with more variance (remember! Variance has an additive property).

It’s fine that we need to produce goods and services at the center of the specification, which means that we should know the position of process mean with respect to the center of the customer’s specifications. Hence another index was created called as Cpm was introduced which compensates for the deviation of process mean from the specification mean.

For calculating Cpm, the Cp formula is modified where the total variance of the system becomes

Where μ = process mean & T = specification mean or target specification

Hence, Cp formula

is modified to

This is necessary because if I can keep the process mean and the specification mean near to each other, the chances of touching the specification limits would be less which in turn would reduce the chances of reprocessing and we can control the process in a better way.

If μ = T, then Cpm = Cpk = Cp

Related Posts

What Taguchi Loss Function has to do with Cpm?

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

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

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## What Taguchi Loss Function has to do with Cpm?

The traditional way of quality control can be called as “GOAL-POST” approach where, the possible out-come is goal or no-goal. Similarly, QA used to focus only on the end product’s quality with two possible outcomes, pass or fail.

Later on Taguchi gave the concept of producing products with quality targeted at the center of the customer’s specifications. He stated that as we move away from the center of the specification, we incur cost either at the producer’s end or at the consumer’s end in the form of re-work and re-processing. Holistically, it’s a loss to the society.

For example;

Buying a readymade suit, it is very difficult to find a suit that perfectly matches your body’s contour, hence you end up going for alterations. This incurs cost. Whereas, if you get a suit stitched by a tailor that fits your body contour (specification), it would not incur any extra cost in rework.

Let’s revise what we learned in “car parking” example (see links below). The Cp only focuses on how far the process control limits (UCL & LCL) are from the customer’s specification limits (USL & LSL) …. it doesn’t take into the account the deviation of process mean from the specification mean. Hence, we  require another index which can penalize the Cp for the above deviation and this new index is called as Cpm.

Related Posts

Why & How Cpm came into existence? Isn’t Cpk was not enough to trouble us?

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

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

Kindly provide feedback for our continuous journey

## Understanding the Monster “Variance” part-1

This is one of the ways of calculating the variability in the data set.  Variance helps us in understanding how the data is arranged around the mean. In order to do so, we need to calculate the deviation of each observation from the mean in the data set .

For example: following is the time taken by me during the week to reach the office. The deviation of each  observation from the mean  time is given below.

Now next step is to calculate the average deviation from the mean using well-known formula

Note that the sum of all positive deviations = sum of all negative deviations which indicates that the mean divided the data set in two equal halves. As a result the sum of all deviation becomes zero, hence we need some other way to calculate this average deviation about the mean.

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

Negative number → 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 s² around the mean.[2] The square root of this variance gives standard deviation s, the most common measure of variability.

What it physically means is that on an average data is deviating 7.42 units or simply one standard deviation (±1s) in either of the directions in a given data set.

Above discussion about the sample standard deviation represented by s. For population, variance is represented by σ² and standard deviation by σ.

The sample variance s² is the estimator of the population variance σ². The standard deviation is easier to interpret than the variance because the standard deviation is measured in the same units as the data.

[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.

Related articles:

Why it is so Important to Know the Monster “Variance”? — part-2

You just can’t knock down this Monster “Variance” —- Part-3

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## But How Six-Sigma Tools Compresses the Variation?

In order to understand this, let’s take the following equation

Now if I ask you the value of ‘y’ for x1 = 3 and x2 = 7. The value of ‘y’ would be 38.

Point to be noted here is that “you were in a position to calculate the value of ‘y’ because you have a mathematical equation describing the relationship between ‘y’ and x1 & x2.

Similarly in six-sigma we find out the variables (x) that impact the response (y) and then we find a quantitative relationship between them. In six-sigma language we describe it as “y is a function of x1, x2,….”

For example

Time taken to reach office (y) is a function of following variables

1. When he slept last night?                                      (x1)
2. Did he had drinks last night?                                (x2)
3. When he woke-up?                                                (x2)
4. When he started from the home?                         (x4)
5. How was the traffic in the morning?                    (x5)
6. How fast he was driving?                                      (x6)
7. Which route he took?                                             (x7)

Let’s assume for the time being that x2, x4, x5 and x7 were found to be important during investigation using six-sigma[1] and the relationship between the time taken to reach office and all of the 4 factors can be described arbitrary for the time being as

Using the above equation, the response (time taken to reach office) could be optimized.

[1] This investigation is usually done using a famous methodology called as DIMAC. I hope everyone is acquainted with it. Followed by ANOVA and regression to get the equation.

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## 6sigma is like a clamp that compresses the variability

We have seen that we can’t change the garage’s width (or customer’s specifications), the only way out is to adjust the process variability (car’s width) according to the customer’s specification. This is done by continuous improvement of the process using 6sigma tools.

6sigma tools is like a clamp where we gradually tighten (continuous improvement) the screw to compress a thing (variability in the process)!

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## 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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