# SPC Formula

Standard SPC Formulas

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1. SD = R/D2

2. Cp = USL – LSL / 6 * SD

3. Cpk = min (Cpl, Cpu)

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4. Cpl = X – LSL / 3 * SD

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5. Cpu = USL - X / 3 * SD

6. SD (n-1) (for n-1 samples) = sqrt [((X12 + X22 +…Xn2) – n (XBAR)2 ) / (n-1)]

7. Pp = (USL – LSL) / 6 * SD (n-1)

8. Ppk = min (Ppl, Ppu)

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9. Ppl = (X – LSL) / (3 * SD (n-1))

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10. Ppu = (USL – X) / (3 * SD (n-1))

FOR XbarR Chart :

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11. LCLx = X – (A2 * R)

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12. UCLx = X + (A2 * R)

13. UCLr = (D4 * R)

14. LCLr = (D3 * R)

FOR XbarS Chart :

15. UCLs = B4 * S

16. LCLs = B3 * S

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17. LCLx = X - (A3 * S)

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18. UCLx = X + (A3 * S)

19. S = SD (n-1) for each sub-group

FOR XMR Chart :

20. UCLmr = D4 * R

21. LCLmr = D3 * R

22. UCLx = X + E2 R

23. LCLx = X - E2 R

Chi-Square Goodness-of-Fit Test

In SPC, the chi-square statistic is used to determine how well the actual distribution fits the expected distribution. Chi-square compares the number of observations found in each cell in a histogram (actual) to the number of observations that would be found in an expected distribution. If the differences are small, the distribution fits the theoretical distribution. If the difference are large, the distribution probably does not fit the expected distribution.

Using Chi-square with the assumption of a normal distribution

1.The calculated chi-square is compared to the value in the table of constants for chi-square based on the number of "degrees of freedom."

2.If the calculated chi-square is less than the value in the table, the chi-square test passes. You can keep the assumption that the process has a normal distribution.

3.If the chi-square is larger than the value in the table, the chi-square test fails. At this confidence level, you either do not have enough data to judge the process, or you should reject the assumption that the process has a normal distribution.

For chi-square test:

Good cell defined

A cell where the expected number of observations is at least 5 based on the theoretical distribution (normal curve). If the expected number of observations is not 5 or more as per Duncan), then the cells width is increased until the expected value of 5 or more is reached. If all readings are fitting in only 1 Cell after combining Chi Square test can not be performed.

Degrees of freedom = n - 1. where n is no of classes.

Confidence level can be set from Chart > Options to 95%, 99% and 99.1%.

Formula:

2 K 2

Chi-squarec = å (Oi-Ei) / Ei

K I=0

where K-1 is degree of freedom and K no of class intervals.

Where:

Oi = actual frequency or number of observations in a cell

Ei = expected frequency or number of observations in a cell in the theoretical distribution

å = symbol for "summation."

Skewness Value

Skewness is the measure of the asymmetry of a histogram (frequency distribution). A histogram with normal distribution is symmetrical. In other words, the same amount of data falls on both sides of the mean. A normal distribution will have a skewness of 0. The direction of skewness is to the tail. The larger the number, the longer the tail. If skewness is positive, the tail on the right side of the distribution will be longer. If skewness is negative, the tail on the left side will be "longer."

Formulas

N _ 3

N å ((Xi – X)/S)

Skewness = -------------------- * i =1

(N-1) * (N-2)

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X = mean N= No. of data points S= standard Deviation

Kurtosis value

Kurtosis is a measure of the combined weight of the tails in relation to the rest of the distribution. As the tails of a distribution become heavier, the kurtosis value will increase. As the tails become lighter the kurtosis value will decrease. A histogram with a normal distribution has a kurtosis of 0. If the distribution is peaked (tall and skinny), it will have a kurtosis greater than 0 and is said to be leptokurtic. If the distribution is flat, it will have a kurtosis value less than zero and is said to be platykurtic.

Formulas

N _ 4 2

N*(N+1)* å ((Xi - X)/S) 3*(N-1)

i =1

Kurtosis = --------------------------- - --------------

(N-1)(N-2)(N-3) (N-2)(N-3)

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X = mean, N = No. of data points S= standard Deviation

Non Parametric method for PPK Calculation

Formula:

Pp = (USL - LSL)/(P(0.99875) - P(0.00135))

Ppk = Min {(P(0.5) - LSL)/(P(0.5) - P(0.00135)), (USL –P(0.5)) / (P(0.99875) - P(0.5))}

Box Cox Transformation for CPK Calculation

Box-Cox is a method of transformation of data to make it normal or more nearly normal.

Formula:

l

Yi (l) = (Xi – 1) / l

After transformation of data using above method Cp, Cpk, Pp, Ppk is calculated with standard formulas.

l is Calculated using log likely hood function. Which is

The logarithm of the likelihood function

n _ 2 n

(-n/2)LN [å(Xi(l)-X(l))/n]+( l-1) åLN(Xi)

I =1 I =1

_ n

Where X(l)=1/n å Xi(l)

i=1

Note-

1. l value is restricted from –10 to + 10.

2. If data contains any negative values Box Cox Transformation can not be performed.

3. To run Box Cox transformation Ms-Excel needs to be installed on the machine.

PPM Calculation

Formula:

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1. Calculate Z upper=(USL –X)/ s

2. Locate the value for Zupper in standard normal table.

3. Multiply (1-Z-value) by 100 to get the theoretical % above the upper spec Limit.

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4. Similarly calculate Z Lower=(X-LSL)/ s

5. Locate the value for Z lower in standard normal table.

6. Multiply (1-Z-value) by 100 to get the theoretical % below the lower spec Limit.

7. DPM=(Zupper + Zlower) * 10,000