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## Cp Cpk Formulas versus Pp Ppk Formulas

### Formulas for Cp Cpk and Pp Ppk are Similar

People often get confused about the difference between Pp/Ppk and Cp/Cpk.

### USL & LSL = Nominal Value (+)(-) Tolerance

To calculate your Upper and Lower Specification Limits, provide your Nominal Value (+)(-) Tolerance (USL = Nominal Value + Tolerance, LSL = Nominal Value – Tolerance).

Once you have inputted your USL & LSL into the template, input your data – your Process Capability Metrics will then start calculating automatically (a minimum of 2 data values are required).

### One-Sided or Unilateral Specification Limits

Use CpU (USL) or CpL (LSL) for Cpk.

### Pp and Ppk Calculations in Weibull Histogram

Unlike the Pp and Ppk calculations which rely on a Normal distribution, the Weibull histogram uses the WEIBULL (or WEIBULL.DIST) function to calculate the Z-scores for the USL/LSL and from these it calculates Pp and Ppk. There is no Cp or Cpk calculation possible in the Weibull Histogram.

*Pp = (PpU + PpL) / 2*

*Ppk = Min (PpU, PpL)*

*PpU = NORMSINV (WEIBULL (USL, Shape, Scale, TRUE))*

*PpL = NORMSINV (WEIBULL (LSL, Shape, Scale, TRUE))*

If there is only a USL or LSL, the Weibull histogram Ppk is either PpU or PpL.

## Cost Price Formula

The cost price formula is used to calculate the actual price of an item and can also be written as CP. In other words, it is the amount we pay to purchase any commodity. The cost price helps in establishing profitability through the selling price. This means, if the original value is less than the selling price, then you earn a profit and if the original value is greater than the selling price, then we incur a loss. In this section, we will be discussing two cost price formulas, understand the variables involved, and solve a few examples to understand this better.

## Definition of Cost Price

The amount of money that is spent to produce goods or services before any profit is added for the manufacturer or producer is defined as the cost price. It is also referred to many names such as actual cost, last cost or average cost. A cost price includes all the extra pay including production, property costs, materials, power, research and development, testing, worker wages, and anything else that must be paid for. Profit and loss are always calculated on the basis of the cost price and the selling price of any item.

## What is the Cost Price Formula?

Cost price is the amount we pay to buy a commodity and it is computed by two basic formulas given in the image below:

**Formula 1:** If we earn a profit while selling a product, we use the following formula.

Cost price formula = Selling Price – Profit

**Formula 2:** If we incur a loss while selling a product, we use the following formula.

Cost price formula = Selling Price + Loss

**Formula 3:** The formula using gain (profit) percentage and selling price is given as,

Cost price formula = {100/(100 + Profit%)} × SP.

**Formula 4:** The formula using loss percentage and SP is given as,

Cost price formula = {100/(100 – Loss%)} × SP.

## Examples on Cost Price Formula

**Example 1:** The selling price of a toy is $340 and the profit earned by the shopkeeper is $60. Find the cost price of the toy using the cost price formula.

**Solution:**

Here, selling price = $340 and profit = $60

Using the cost price formula, we get

CP = Selling Price – Profit

= $(340 – 60)

= $280

**Answer: The CP of the toy is $280**

**Example 2:** An article was sold for $ 230 at a loss of $ 20. Using the cost price formula can you calculate what was its cost price?

**Solution:**

Here, selling price = $230 and loss = $20

Using the cost price Formula, we get

CP = Selling Price + Loss

= $(230 + 20)

= $250

**Answer: The cost price of the article is $250**

**Example 3:** On selling a chair for $900, Jamie loses 6%. For how much did she purchase it? Calculate the cost price by using the formula.

**Solution:**

Given, Loss = 6%; SP = $900; CP = ?

If loss is 6%, it means that if the cost price is $100, the loss incurred is $6.

If CP is $100, then SP is $94

When SP is 94, CP = $100

When SP is $900

CP = (100/94) × 900 = $957.44

∴ CP = $957.44**Answer: The cost price of the chair is $957.44.**

FAQs on Cost Price Formula

### What is the Cost Price Formula When Gain Percentage Is Given?

Cost price formula when gain (profit) percentage and selling price is given as, Cost price formula = {100/(100 + Profit%)} × SP.

### What is the Cost Price Formula When Loss Percentage Is Given?

Cost price formula when loss percentage and SP is given is expressed as, Cost price formula = {100/(100 – Loss%)} × SP.

### What Is the CP Formula When Selling Price and Profit Is Given?

If we earn a profit while selling a product, we use the following CP formula. Cost price formula = Selling Price – Profit

### What Is the CP Formula When Selling Price and Loss Is Given?

If we suffer a loss while selling a product, we use the following CP formula. Cost price formula** **= Selling Price + Loss

### What is the Difference Between Cost Price and Selling Price?

Cost price is the amount paid to purchase an item i.e. the actual price. Whereas the selling price is the price at which the item is sold.

### What is Full Cost Price?

Full cost price is the price of a product that is calculated by a firm on the basis of its direct cost per unit of output plus a markup to cover overhead costs and profits.

## Calculate Cp

Cp is an index used to assess the width of the process spread in comparison to the width of the specification. It is calculated by dividing the allowable spread by the actual spread. The allowable spread is the difference between the upper and lower specification limits. The actual spread is 6 times the estimated standard deviation. Plus or minus 3 times the estimated standard deviation contains 99.73 percent of the data and is commonly used to describe actual spread.

Cp for the example is:

A Cp of one indicates that the width of the process and the width of the specification are the same. A Cp of less than one indicates that the process spread is greater than the specification. This means that some of the data lies outside the specification. A Cp of greater than one indicates that the process spread is less than the width of the specification. Potentially this means that the process can fit inside the specification limits. The following diagrams show this graphically.

In fact, the Cp states how many times the process can fit inside the specification. So a Cp of 1.5 means the process can fit inside the specification 1.5 times. A Cp greater than one is obviously desirable. However, the example has a Cp greater than one and yet it still has data outside the specification. This is due to the position of the overall average relative to the specification. When the overall average is away from the center of the specification, the system can still produce data outside the specification even though the Cp is greater than one, as in the example below:

To overcome this problem, Cpk was created.

## Process Capability Formula

CP means or stands for Process capability. That said the CP calculation formula is used to determine the measurable property of a process to the width of the specification. The ultimate solution of the process capability is usually manifested either in the form of calculations or histograms. The formula to calculate CP is as below:-

CP = (USL–LSL)/6×σ

**Where,**

USL = the upper specification limit

LSL = the lower specification limit and

σ = the standard deviation.

**What is Process Capability Ratio?**

Process capability is used to identify the inconsistency in the process using the upper and lower limits respectively. Elements stated below can be observed from the evaluation and process capability ratio:

- If the value of CP is equivalent to 1, the process is perfectly centered i.e. (CP = 1 is a perfect process)
- When CP is greater than one, it suggests tolerance> the process. This validates that the process seems to be able to do this. (CP> 1 is process compatible)
- If the process length is higher than the tolerance, the process variation is within the range of tolerance, making the process incapable.
- CP
_{k}makes it feasible to compute the closeness of the performance of the process to the condition set upon by permitting for the natural inconsistency of the process. - If the value of CP
_{k}is negative, you are required to ensure that the process you inherited leads to an outcome that is not within the range mentioned by the client and is completely outside. There is a separate CP_{k}calculation formula. - The CP
_{k}of 3 suggests that the output is outstanding and the process has 6σ capacities and that is what we require to achieve. - We require a CP
_{k}of minimum 1.33, which is 4σs, in order to satisfy most customers.

**Solved Examples**

**Example: **Find out CP whose USL, LSL and σ are 12, 7 and 4 respectively?

**Solution:**

From the above cp equation, it is given that

USL = 12

LSL = 7

σ = 4

Now using the process capability ratio formula, which is as below;

CP = (USL − LSL) 6σ

CP = (12 − 7)6 × 4

CP = 0.208333

## FAQs (Frequently Asked Questions)

**Q1. What is the Cpk Formula?**

Answer: Cpk stands for process capability index which is the measure of process capability. It displays how intricately a process is capable of producing the output to its overall specifications. It also judges the level of consistency in our average performance. The cpk calculation formula is:-

Cpk = min(USL−mean/3σ,mean−LSL/3σ)

Where,

USL = the upper specification limit,

LSL = the lower specification limit.

σ = standard deviation

**Q2. What is the Cp and Cpk Formula?**

Answer: An ideally centered process will consist of Cp = Cpk. Both Cpk and Ppk are associated with the standard deviation and centering of the process about the centre point to the permissible tolerance specifications. An approximation for Cpk = Cp (1-k). And since the maximum value for k equals to 1.0, then the value for Cpk will be always equal to or less than Cp. Similar to Ppk, the Cpk capability index is only a function of the mean of the data and standard deviation, not a nominal (target) value that may be historical or given by the customer.

**Q3. Why Do We Divide by 3 to Find the Value of Cpk?**

Answer: With respect to the origin of the 3-sigma limits, it was an idea introduced by Walter Shewhart’s. Since in a normal distribution 99.73% of the population falls within ± 3 sigmas of the mean, he regarded it as a rational point to put the limits.

## CP

### Definition of Cp

Cp is a common measure used in process capability analysis, which allows you to know how well your process is meeting your customers’ expectations and specifications. This article will explain how Cp is calculated and interpreted. We will explore its application, benefits and provide some tips and best practices so that you can begin assessing how well your process is doing when measured against the requirements specified by your customer.

## Overview: What is Cp?

Process capability is the quantifiable comparison of the Voice of the Customer (specs, requirements or expectations of your customer) and the Voice of the Process (control limits). The goal of process capability is to come up with a quantified measure that informs you as to whether you are meeting your customer expectations or not — as well as a relative value of how well you are meeting or how badly you are missing.

Besides Cp, or capability potential, there are a number of other process capability metrics. They are Cpk, Pp, and Ppk.

Cp, or capability potential, is a measure of the potential capability with the assumption that the distribution of the sample process data is normally distributed and centered within the customer’s upper specification limit and lower specification limit. It represents the best the process capability can be for the specs and process distribution. If there is only a one-sided spec, then Cp cannot be calculated.

The formula is:

You can say this as *Cp* *is equal to the customer’s upper specification limit minus the lower specification limit divided by 6 standard deviations. *

We are assuming that the sample distribution is normally distributed, and its width is calculated by adding and subtracting 3 standard deviations on either side of the sample data mean. The standard deviation that is used is called a “pooled standard deviation” or “within variation.”

Cp can take on values equal to 1, greater than 1, or less than 1.

- A value of 1 would mean the width of the distribution is equal to the width of the specifications, or specs.
- A value greater than 1 would mean the specs are wider than the distribution of the process, so we are comfortably within spec.
- If the value is less than 1, it means the width of the process distribution is wider than the specs, and items will be outside the specs.

But, all of this assumes that the process is centered within the specs — this is rarely the situation. More often than not, the process mean (x-bar) will not be centered within the specs. Interestingly, if the process mean is off-center, the Cp will still be the same since the Cp does not consider the location of the process. That’s when we need to use the Cpk to assess the process capability.

## 3 benefits of Cp

Cp is a simple and useful metric for comparing your process with your internal or external specifications, targets, expectations, or any other desired outcome for your process.

### 1. Simple yet powerful calculation

To compute your Cp value, you only need to know the standard deviation of your data and the specs that you want to test against.

### 2. Can account for a process centered within the specs

If the process is centered within the specs, you would use the calculation of Cp to assess the potential of your process capability. But, as it is in most cases, if the process is shifted off-center, then the Cpk will account for that shift.

### 3. Can be used to compare different processes

The Cp allows you to compare the performance of disparate functions since it is a generic measure of the process’ ability to meet specs. If a manufacturing line had a Cp of 0.82, and a finance department had a Cp of 1.89, you could say that finance is doing a better job of meeting their customer’s requirements than is manufacturing.

## Why is Cp important to understand?

Meeting your customers’ expectations and requirements is vital to your organization staying in business. By understanding how to compute Cp and interpret its value, you will be able to know what to do to meet and possibly exceed your customers’ expectations.

### Know what variables are needed to calculate Cp

If you intend to use Cp for decision making purposes, you need to know what variables are needed and what their values are so you can do the proper calculations.

### Cp represents your current state process capability

If your Cp is not high enough to satisfy your customers’ specs, you need to take action so that out-of-spec products are not produced and shipped.

### You might want to adjust the specs

If you’re not meeting specs, you might want to speak with your customer to see if they can be adjusted without any negative impact. You can use the current Cp calculation to back into what level of specs you’re able to meet.

## An industry example of Cp

A coffee company was having a problem with overfill and underfill on their jars during manufacturing. Underfill is a big problem because if the quantity is below the required amount, there could be a recall and significant fines. Overfill is also an issue because it means you’re giving away extra product that the company doesn’t get paid for. The process Cp was 0.79, which meant that the width of the process distribution was wider than the allowable specs. Out-of-spec product was being produced on both ends.

The company put two of its Lean Six Sigma Black Belts on a project to reduce the variation of the process with the goal of narrowing the process distribution and pulling both ends into spec. The project was successful, and the Cp was now calculated as 1.74.

Since the process was now tight within the specs and still centered, the plant manager decided to adjust the process, so the mean of the distribution moved closer to the lower spec. The process distribution was still well within the specs, and nothing fell outside the lower spec. Less coffee was being put into the jars, so the profit per jar increased.

## 3 best practices when thinking about Cp

Since Cp is merely a calculation, there aren’t really a lot of best practices but more of a series of hints that allow you to get the maximum value out of the calculation.

### 1. Do both Cp and Cpk calculations

By comparing the potential capability with the actual capability, you will understand what, if anything, you need to do to be sure you remain in spec.

### 2. Use the Cp to compare different parts of the organization

The use of the generic Cp metric allows you to compare various functions in the organization. It is a measure of how well each is doing relative to their specs and allows for an apples-to-oranges comparison.

### 3. Use the Cp to establish strategies to take advantage of a good value

You can consider revising specs or shifting the mean of the process within the specs to take advantage of a good process and high Cp.

## Frequently Asked Questions (FAQ) about Cp

**What is the difference between Cp and Cpk?**

Cp assumes that the process is centered within the specs, while Cpk adjusts for a dynamic shift in the mean of the process.

**Can I compare Cp across different departments of the company? **

Yes. Cp is a generic measure of how well a department is doing relative to its required specs. You can state that Department A is doing better at meeting specs than Department B if Department A’s Cp is higher.

**Other than Cp, are there other common measures of process capability? **

Yes. You can also calculate Cpk, Pp, and Ppk to measure process capability.

## So, what is Cp again?

Cp is a common measure of a process’s ability to meet customer expectations, primarily specifications. It assumes that the process data is normally distributed and that the mean of the process be centered if there are upper and lower specifications.

It is calculated by finding the difference between the upper and lower specification. That difference is then divided by 6 standard deviations. The higher the value is, the better the process capability.

Knowledge and understanding of your Cp value can be a powerful tool for gaining competitive advantage and improved profitability. It can be used for any process with stated specs or qualitative expectations and can also be used as a comparative tool across different functional areas.

*Cost Price:*

The price, at which an article is purchased, is called its *cost price*, abbreviated as *C.P.*

**Selling Price:**

The price, at which an article is sold, is called its *selling prices*, abbreviated as *S.P.*

*Profit or Gain:*

If S.P. is greater than C.P., the seller is said to have a *profit* or *gain*.

**Loss:**

If S.P. is less than C.P., the seller is said to have incurred a *loss*.

**IMPORTANT FORMULAE**

- Gain = (S.P.) – (C.P.)
- Loss = (C.P.) – (S.P.)
- Loss or gain is always reckoned on C.P.
- Gain Percentage: (Gain %)

## Cpk Formulas

Process capability index i.e. Cpk is an important statistical tool. It is used to measure the ability of a process to produce desired output within the customer’s specification limits. In other words, it measures the producer’s capability to give a product within the tolerance range. Also, it is used to estimate how close we are to a given target as well as about the consistency around the average performance. We will learn the basic concept of cpk as well as Cpk formulas with examples.

**Concept of Cpk Formulas**

Cp and Cpk both are used for Process Capability measurement. Generally, we may use this when a process is under statistical control. This generally happens with a mature process that will be around for a while. Process capability will use the process of sigma value determined from either the Moving Range, Range or Sigma control charts.

Therefore, Process capability index i.e. cpk is the measurement of process capability. It shows how closely a process is able to produce the required output in comparison to its overall specifications. It also decides about the consistency around the average performance.

Cpk gives us the best-case scenario for the existing process. It may also estimate the future process performance, assuming performance is consistent over time. In Six Sigma we have to describe the quality of processes in terms of sigma. This is because it gives us an easy way to talk about how capable different processes are with a common mathematical model.

### Important Points about Cpk

- Cpk is a short term process index that describes the potential capability of a process assuming it was analyzed and stays in the control.
- It’s an option along with z-score in statistics.
- We may use time-series and SPC charts to determine process control. If the process is out of control, then assessing the current process is unlikely to reflect the long term performance.
- The Cp is the best a process can perform if that process is centered on its midpoint.
- The addition of “k” in Cpk quantifies the amount of which a distribution is centered. A perfectly cantered process where the mean is the same as the midpoint will have a “k” value of 0.
- The minimum value of “k” is 0.0 and the maximum is 1.0. A proper centered process will have Cp = Cpk.
- An estimate for Cpk = Cp(1-k). Since the max value for k is 1.0, so the value for Cpk will always be less or equal to Cp.
- Input is required from the customer regarding the lower specification limit (LSL) and the upper specification limit (USL).

**The Formula for Cpk**

**Solved Examples for Cpk Formulas**

Q.1: Food served at a restaurant should be between 38^{\circ}C and 49^{\circ}C when it is delivered to the customer. The process used to keep the food at the correct temperature has a process standard deviation of 2^{\circ}C and the mean value for these temperatures is 40. Find out the process capability index of the process?

Solution:

USL (Upper Specification Limit) =49∘C

LSL (Lower Specification Limit) =39∘C

Standard Deviation =2∘C

Mean = 40

Cpk formula is:

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