TMGT and doing Chart

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Xbar example problem

Control Chart and Process Capability Data
You don't need full control chart data to calculate process capability indices. What you do need is an estimate of process variation and the tolerance or specificiaton limits. However, the estimate of process variation comes from control chart data. i.e., even if you were not going to create a control chart, to find the process variation estimate you still need the same data that is needed for a control chart. Because of that control chart--process capability relationship, msot authors think it is best to learn about control charts before process capability, However, the ASQ BOK doesn't have it listed that way (presumably because if someone handed you the process variation estimate and the spec limits, you could calculate the process capability indices without creating a control chart). In this set of spreadsheet pages you will see how to find the process variability estimate (the estimated standard deviation or sigma of the process) from raw data. Of course, someone might give you the sigma without you having to calculate it.
Measurements
Sample X1 X2 X3 X4 Mean Std. Dev Range
1 10 13 9 10
2 10 10 11 11
3 12 10 12 4
4 11 9 12 9
5 9 8 10 11
6 12 9 14 11
7 15 8 10 9
8 8 9 12 8
9 14 11 10 10
10 8 10 10 16
11 11 11 11 10
12 11 9 13 12
13 7 8 12 11
14 11 13 14 11
15 8 11 10 14
16 12 8 11 12
17 13 7 15 10
18 13 12 11 9
19 12 13 9 8
20 10 10 7 12
21 11 8 11 7
22 11 10 11 9
23 11 10 8 8
24 12 10 10 11
25 9 14 15 10
Sum
Average
Trial Revised
Center line of X bar chart = Center line of X bar chart =
UCL of X bar chart (s) = UCL of X bar chart (s) =
LCL of X bar chart (s) = LCL of X bar chart (s) =
UCL of X bar chart (R) = UCL of X bar chart (R) =
LCL of X bar chart (R) = LCL of X bar chart (R) =
Center line of s chart = Center line of s chart =
UCL of s chart = UCL of s chart =
LCL of s chart = LCL of s chart =
Center line of R chart = Center line of R chart =
UCL of R chart = UCL of R chart =
LCL of R chart = LCL of R chart =
n = σ0 (s) =
g = σ0 (R) =

Xbar example solution

Control Chart and Process Capability Solution
The process sigma can be found by using the range or the standard deviation. There are advantages to each. The forumulas using each follow. The d2 and c4 values come from a control chart factor table in the text or other reference. The average range or standard deviation is found by averaging the Std. Dev. and Range columns in the data set below, i.e., by averaging all the sample standard deviations and averaging all the sample ranges. Normally you would not use both the std. dev. and range but both are done here for learning purposes.
Instead of sigma hat sub zero, the book uses sigma sub R.
Measurements
Sample X1 X2 X3 X4 Mean Std. Dev Range
1 10 13 9 10 10.50 1.73 4
2 10 10 11 11 10.50 0.58 1
3 12 10 12 4 9.50 3.79 8
4 11 9 12 9 10.25 1.50 3
5 9 8 10 11 9.50 1.29 3
6 12 9 14 11 11.50 2.08 5
7 15 8 10 9 10.50 3.11 7
8 8 9 12 8 9.25 1.89 4
9 14 11 10 10 11.25 1.89 4
10 8 10 10 16 11.00 3.46 8
11 11 11 11 10 10.75 0.50 1
12 11 9 13 12 11.25 1.71 4
13 7 8 12 11 9.50 2.38 5
14 11 13 14 11 12.25 1.50 3
15 8 11 10 14 10.75 2.50 6
16 12 8 11 12 10.75 1.89 4
17 13 7 15 10 11.25 3.50 8
18 13 12 11 9 11.25 1.71 4
19 12 13 9 8 10.50 2.38 5
20 10 10 7 12 9.75 2.06 5
21 11 8 11 7 9.25 2.06 4
22 11 10 11 9 10.25 0.96 2
23 11 10 8 8 9.25 1.50 3
24 12 10 10 11 10.75 0.96 2
25 9 14 15 10 12.00 2.94 6
Sum 263.25 49.88 109
Average 10.53 2.00 4.36
Trial Revised
Center line of X bar chart = 10.53 Center line of X bar chart = 10.53
UCL of X bar chart (s) = 13.78 A3 UCL of X bar chart (s) = 13.78 σ0 (s)
LCL of X bar chart (s) = 7.28 A3 LCL of X bar chart (s) = 7.28 σ0 (s)
UCL of X bar chart (R) = 13.71 A2 UCL of X bar chart (R) = 13.71 σ0 (R)
LCL of X bar chart (R) = 7.35 A2 LCL of X bar chart (R) = 7.35 σ0 (R)
Center line of s chart = 2.00 Center line of s chart = 2.00
UCL of s chart = 4.52 B4 UCL of s chart = 4.52 B6
LCL of s chart = 0.00 B3 LCL of s chart = 0.00 B5
Center line of R chart = 4.36 Center line of R chart = 4.36
UCL of R chart = 9.95 D4 UCL of R chart = 9.95 D2
LCL of R chart = 0.00 D3 LCL of R chart = 0.00 D1
n = 4 σ0 (s) = 2.17
g = 25 σ0 (R) = 2.12
Trial and revised are the same in this case because we used the same groups.

Xbar example charts

trial X-bar using s trial X-bar using R
trial s chart trial R chart
All groups are in control for X bar and s All groups are in control for X bar and R
10.5 10.5 9.5 10.25 9.5 11.5 10.5 9.25 11.25 11 10.75 11.25 9.5 12.25 10.75 10.75 11.25 11.25 10.5 9.75 9.25 10.25 9.25 10.75 12

1.7320508075688772 0.57735026918962573 3.7859388972001824 1.5 1.2909944487358056 2.0816659994661326 3.1091263510296048 1.8929694486000912 1.8929694486000912 3.4641016151377544 0.5 1.707825127659933 2.3804761428476167 1.5 2.5 1.8929694486000912 3.5 1.707825127659933 2.3804761428476167 2.0615528128088303 2.0615528128088303 0.9574271077563381 1.5 0.9574271077563381 2.9439202887759488

10.5 10.5 9.5 10.25 9.5 11.5 10.5 9.25 11.25 11 10.75 11.25 9.5 12.25 10.75 10.75 11.25 11.25 10.5 9.75 9.25 10.25 9.25 10.75 12

4 1 8 3 3 5 7 4 4 8 1 4 5 3 6 4 8 4 5 5 4 2 3 2 6

p Chart example

P Chart
A P chart plots the proportion nonconforming. It is equivalent in concept to an X-bar chart but instead of plotting a mean it plots the proportion is a sample or subgroup that is nonconforming (is bad, defective, etc.). The p chart is based on the binomial distribution. Note that you could plot any event, e.g., heads, 2s on a die, persons with blue eyes, etc. When a sample plots out of control it tells you that the odds are that you did not get that sample value by random chance; that something is going on other than random chance. Of course it could be the rare instance that random chance did account for the extreme sample value (in which case you just made an alpha or producer's risk mistake).
The p chart comes in two forms: the fraction or proportion nonconforming and the percent nonconforming. A percent chart is a p chart stated in percent (%) instead of proportion. All you do is move the decimal place 2 places to the right (multiply the central line and control limits by 100) and add the percent sign. Many people prefer a percent chart because they are more used to percents than proportions. All the p and np chart varieties (which deal with nonconformities) can be done for conformities. Typically we call the conforming charts q charts. q = 1-p.
The following formulas are used for a p chart (using the proportion).
p is what is plotted on the chart once it is created.
p0 is the estimated proportion in the population.
pbar is the central line.
p = proportion (or fraction) nonconforming in the sample or subgroup
np = the number (count) nonconforming in the sample or subgroup
p bar = the average proportion nonconforming
n = sample size. In the control limit formula the n is the size of the sample being plotted on the chart. This is not necessarily the same n used to create the chart. For instance, the sample sizes used to collect the data could all be 100 (as in my example below), or they could vary. Whether the ns to create the control chart varied or were constant, you can still find pbar. If the samples being plotted on the chart were n = 80, then the n used in the U and L control limit calculation should be 80. The U/LCL formula on this page assume that the sample plotted on the chart (the samples whose quality you want to control) are all the same size (whether or not the ns used to create the chart were all the same or matched the n plotted on the chart). Some practitioners will use the U/LCL formula on this page with varying ns plotted on the chart as long as the ns do not vary too much. But how much variation in sample size is too much? If the sample sizes plotted on the chart varied by a percent or two, the control limits would not vary to one or two decimal places. If some ns were twice as big as others, you would make a lot of producer and consumer risks. It is a judgment call as to whether the ns plotted on the chart vary enough that a different control limit should be used (because the control limits are effected by the sample size plotted on the chart). See the next tab about what to do if the sample sizes plotted on the p chart vary too much.
Remember that p0 is the population parameter and is estimated from the average sample statistic after any out of control trial samples with assignable causes are eliminated. Recall the Xbar trial and revised procedures. The negative lower limits would be set to zero.
Below are some example UCL and LCL calculations. Note that each of these example is for a different control chart. For any one p control chart, the pbar would be the same for all samples. As discussed in a note above, it is simplest to interpret if only one sample size is plotted on a p chart. See the next tab concerning p charts on which are plotted samples of varying sizes.
You can also add your own pbar and n values
Note what happens as pbar and n changes.
Σ np Σ n pbar n UCL LCL
67 134 0.5 100 0.650 0.350
119 1190 0.1 10 0.385 -0.185
119 1190 0.1 100 0.190 0.010
119 1190 0.1 1000 0.128 0.072
119 1190 0.1 10000 0.109 0.091
10 200 0.05 500 0.079 0.021
4 200 0.02 500 0.039 0.001
2 200 0.01 500 0.023 -0.003
2 2000 0.001 500 0.005 -0.003
1 10000 0.0001 500 0.001 -0.001
138 7500 0.0184 300 0.042 -0.005
13 2000 0.0065 100 0.031 -0.018
ERROR:#DIV/0! ERROR:#DIV/0! ERROR:#DIV/0!
Where do the summations come from? The text has examples. Following is another one. It is best to use 20 or more samples to find the summations (statistically, the more the better, also the larger the summation of n the better). Things are simplified when all the sample sizes are the same but the sample sizes do not have to be the same. There is a rule of thumb that the smallest sample size not be less than 1/2 the average sample size. The number of defects can be any sort of observation or occurrence.
Lot or sample Sample size (n) Number of defects in the sample (np) Lot or sample Sample size (n) Number of defects in the sample (np)
1 100 0 1 150 2
2 100 1 2 50 0
3 100 0 3 99 0
4 100 0 4 75 0
5 100 0 5 115 0
6 100 0 6 100 0
7 100 0 7 99 0
8 100 2 8 137 3
9 100 2 9 100 0
10 100 1 10 100 1
11 100 0 11 77 0
12 100 0 12 89 0
13 100 1 13 100 1
14 100 2 14 140 3
15 100 0 15 125 1
16 100 0 16 100 0
17 100 0 17 69 0
18 100 1 18 95 1
19 100 2 19 66 0
20 100 1 20 114 1
Σ of n and np respectively = 2000 13 Σ 2000 13
mean of n and np respectively = 100 0.65 100 0.65
0.0065 0.0065
0.65% 0.65%
Actual data plotted on a p chart after it is constructed would look something like the example below.
n = constant sample size = 100
chart line values
pbar UCL LCL
0.1 0.19 0.01
Lot or sample Number of defects in the sample (np) p = np/n = what is plotted
1 0 0
2 1 0.01
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 2 0.02
9 2 0.02
10 1 0.01

These are example summations.

This row uses the data from the example below.

Note the same pbar can result from different sample sizes.

You can see that none of these sample are out of control (because p is not above the UCL) even without plotting on a chart. Notice that whether or not individual items are in tolerance or spec is another matter.

variable p chart example

P chart with variable subgroup size
The only difference between the P chart with a constant sample size and a p chart with a variable sample size is that with the variable sample size p chart you have to plot different upper and lower control limits for each sample size. An alternative is to calculate an average UCL and LCL based on an average sample size. Hayden doesn't like this because you have to mentally consider the actual sample size when visually checking a plotted sample and determine if the sample is out of control based on (a) if the plotted sample is above or below the control limit in consideration of (b) what the sample size is (so in a way you are really using different upper and lower control limits). Another alternative is to plot multiple upper and lower control limits based on certain common sample sizes, e.g., n = 100, 150, or whatever. This is pretty easy to interpret and very useful; especially, if the samples sizes are actual sample sizes. For example, maybe on 1st shift 500 parts are produced, 300 are produced on 2nd shift, and 125 are produced on Saturday. You can but several different U/L control limits on the same chart that correspond to different sample sizes. Obviously, you would have to keep in mind the sample size that was plotted and use the correct U/L control limits. If there are too many control limits, interpretation can be confusing. Another tactic is to plot the exact upper and lower control limits for each sample plotted. This would be the most accurate method and is easy to interpret. However, you would have to use the formula below each time you plotted a sample (the n in the formula would be the sample size that is plotted). This is not cumbersome when software is used to plot the p chart.
You will notice that the following formula is slightly different from the one on the previous tab. The following formula would be used if you were going to plot ns of varying sizes on a p chart. This formula would also work for a fixed sample size P chart on the previous tab.
The centerline for a variable sample size p chart would still be pbar (as calculated on the previous page). The pbar and numerator would be constants for a given process; therefore, only n in the formula above would have to be plugged and jugged to calculate the upper and lower control limits for variable sample sizes (and software may be doing that for you). Variable or constant sample P charts can be displayed as a percent chart. The example U/LCL calculations also pertain to a p chart with variable sample sizes.
Actual data plotted on a variable p chart after it is constructed would look something like the example below
n varies
pbar = 0.01
Lot or sample sample size (n) Number of defects in the sample (np) p = np/n what is plotted on the chart UCL LCL
1 100 0 0.000 0.040 -0.020
2 80 1 0.013 0.043 -0.023
3 100 0 0.000 0.040 -0.020
4 120 0 0.000 0.037 -0.017
5 100 0 0.000 0.040 -0.020
6 100 0 0.000 0.040 -0.020
7 80 0 0.000 0.043 -0.023
8 120 2 0.017 0.037 -0.017
9 120 2 0.017 0.037 -0.017
10 80 1 0.013 0.043 -0.023

You can see that none of these sample are out of control even without plotting on a chart because none of the individual p values plotted on the chart are above their respected upper control limits.

np chart example

nP charts
nP charts are P charts (P charts are proportions or percents) that have been converted to charts that plot the actual number (np) nonconforming. These charts are usually easier to interpret because you can compare the number of nonconforming parts to a control limit of nonconforming parts, i.e., to the number of nonconformities allowed. There is only a slight difference in formulas for the central line and control limits.
As for a p chart, p0 is the estimated proportion in the population. n is the number in the sample. It makes sense that np0 would be the central line. If the proportion nonconforming (p0) was estimated to be .1 in the population (the p0 is .1) and the sample size was 10, you would expect the average number nonconforming (the centerline of the chart) to be 1. n times p0 = 10 x .1 = 1. Or if p0 is stated as a percent, 10 x 10% = 1.
Central line = np0 = n x p0
p0 = Just like for a p chart on the first tab.
Below are some example UCL and LCL calculations. Note that each of these example is for a different control chart. For any one np control chart, the centerline would be the same for all samples.
Σ np Σ n pbar, p0 n centerline np0 UCL LCL
67 134 0.5 100.00 50.00 65.00 35.00
119 1190 0.1 10.00 1.00 3.85 -1.85
11 1100 0.01 100.00 1.00 3.98 -1.98
119 1190 0.1 1,000.00 100.00 128.46 71.54
119 1190 0.1 10,000.00 1,000.00 1,090.00 910.00
10 200 0.05 500.00 25.00 39.62 10.38
4 200 0.02 500.00 10.00 19.39 0.61
2 200 0.01 500.00 5.00 11.67 -1.67
2 2000 0.001 500.00 0.50 2.62 -1.62
1 10000 0.0001 500.00 0.05 0.72 -0.62
138 7500 0.0184 200.00 3.68 9.38 -2.02
ERROR:#DIV/0! ERROR:#DIV/0! ERROR:#DIV/0!
Actual data plotted on a np chart after it is constructed would look something like the example below. This is the same data used in the example in the p chart tab.
n = constant sample size = 100
pbar = p0 = 0.01
chart line values
center-line = np0 UCL LCL
1 3.98 0
Lot or sample Number of defects in the sample (np); what is plotted on the control chart
1 0
2 1
3 0
4 0
5 0
6 0
7 0
8 2
9 2
10 1

Look how much easier the interpretation of a constant sample size np chart is compared to p charts (fixed or variable n). You are plotting whole defects on the chart. You can see none of the samples are out of control; none have more than 3.98 defects.

Use below in an example.

c chart excample

c chart
A c chart counts nonconformities in a unit (thing, piece, part, jug of milk, carpet, work shift, object, product, job, etc.). c charts are based on the Poisson distribution. The unit must be consistent, e.g., all shifts are the same amount of time, all carpets are the same size, the oil is in the same size drum, etc. Note that the unit might be one egg, a dozen eggs, a 1,000 pounds of eggs, one nail, a pound of nails, etc. For a c chart, the nonconformities are often larger than the unit because the unit is one thing and that one thing might have 7 scratches, 3 cracked eggs, two rusty nails, etc.
The following formulas are used. Note that sample size is not part of the formulas because it is assumed constant in the determination of cbar and the c chart only plots a constant unit (one size, one thing, whatever that thing is). Note that the unit does not have to be a recognizable whole object, e.g., a car. The unit is constant but it can be constant amount of liquid or weight or volume, etc. Always know what the unit is.
cbar is the central line of the control chart
c = the count of nonconformities
g = the number of subgroups (not the size of the subgroup, the size of the subgroup is constant--one unit)
Example calculations of control limits follow. Plug in some of your own. Negative LCL values would be set to zero on the control chart. Note that each of these example is for a different control chart. For any one control chart, the centerline would be the same.
Σc g centerline cbar UCL LCL
13 20 0.65 3.07 -1.77
1000 100 10.00 19.49 0.51
790 79 10.00 19.49 0.51
320 32 10.00 19.49 0.51
1000 200 5.00 11.71 -1.71
475 95 5.00 11.71 -1.71
50 10 5.00 11.71 -1.71
200 200 1.00 4.00 -2.00
95 95 1.00 4.00 -2.00
10 10 1.00 4.00 -2.00
ERROR:#DIV/0! ERROR:#DIV/0! ERROR:#DIV/0!
Where does the summation of nonconformities (c) come from? From real data. You may be give the summation of c value and asked to construct a chart, or given the data and must calculate summation of c, or you may have to collect the data also. Similar to proportions, the samples used to find summation of c do not have to be the same size or the same size as samples plotted on the c chart. They most often are the same though because a c chart is only for plotting samples of the same size and similar sample sizes were probably used to collect the data from which the c chart was constructed.
Lot or sample Number of non-conform-ities in the sample (c).
1 0
2 1
3 0
4 0
5 0
6 0
7 0
8 2
9 2
10 1
11 0
12 0
13 1
14 2
15 0
16 0
17 0
18 1
19 2
20 1
g = 20 13 = Σ c
cbar = 0.65
Actual data plotted on a c chart after it is constructed would look something like the example below.
Unit size must be 1.
centerline = 0.65
ULC = 3.07
LCL = 0
Lot or sample Number of non-conform-ities in the sample (c).
1 0
2 1
3 0
4 2
5 0
6 0
7 0
8 4
9 2
10 1

This row uses the data from below. The control chart values are used in the example below.

Sample 8 indicates an out of control process.

u chart example

u chart
A u-chart compared to a c-chart is analogous to a variable subgroup p-chart compared to a regular p-chart because the control limits change as the number of units change. As with a c chart, you have to be clear about what a unit is because all ns (the sample size, the number inspected) will vary but must be multiples of the same unit. This is simple if you are counting whole identifiable things like cars or hamburgers. But what if the thing is eggs, what is the unit of eggs? One egg, one kilogram of eggs, one liter of eggs? What is the product is rope? What is unit of rope? One foot, 100 yards? What about oil or a gas? Is the unit going to be based on a volume or a weight or something else. If the unit was a barrel of oil all ns must be put in terms of that unit. Therefore, you can't use kilograms of oil and quarts, and barrels. You must use the same unit measurement that varies for the different samples. For example, there are 42 gallons of oil in an oil barrel and 4 quarts in gallon. Therefore, there are 168 quarts in a barrel of oil. If you unit was quarts and you sampled a pint, your n would be .5 (1/2 quart). If you unit was quart and you sampled a barrel, you n would be 168. A u-chart is a count of nonconformities per unit. U-charts are useful when the unit changes. For example, the carpets are different sizes, the rope is of different lengths, the number of people change, the quantity of milk tested changes. If you were going to inspect eggs and the unit was always going to be 1 dozen (1 carton of 12 eggs), you would use a c-chart. If you were going to inspect eggs but the unit might be 1 egg (1/12 dozen) or 12 1/2 dozen, or a gross (12 cartons), etc. you would use a u-chart.
ubar = the average count of nonconformities for many subgroups and is centerline for the chart.
c = count of nonconformities in a subgroup
n = number inspected in a subgroup
u= the count of nonconformities per unit in a subgroup
Example calculations of control limits follow. Plug in some of your own. Negative LCL values would be set to zero on the control chart. Note that each of these example is for a different control chart. For any one control chart, the centerline would be the same.
Σc Σn centerline ubar n UCL LCL
13 29.75 0.44 5.00 1.32 -0.45
1000 100 10.00 0.08 42.87 -22.87
790 79 10.00 1.00 19.49 0.51
320 32 10.00 1.00 19.49 0.51
1000 200 5.00 2.50 9.24 0.76
475 95 5.00 7.50 7.45 2.55
50 10 5.00 12.00 6.94 3.06
200 200 1.00 0.08 11.39 -9.39
95 95 1.00 1.00 4.00 -2.00
10 10 1.00 2.50 2.90 -0.90
50 10 5.00 7.50 7.45 2.55
100 10 10.00 12.00 12.74 7.26
ERROR:#DIV/0! ERROR:#DIV/0! ERROR:#DIV/0!
As with other types of charts, the estimates of process parameters (the average u or ubar in this case) comes from real data.
Lot or sample number of units (n) Number of non-conform-ities in the sample (c) number of non-conform-ities per unit (u)
1 0.5 0 0.00
2 1 1 1.00
3 2 0 0.00
4 3 0 0.00
5 2.5 0 0.00
6 0.75 0 0.00
7 1 0 0.00
8 2 2 1.00
9 3 2 0.67
10 0.5 1 2.00
11 0.5 0 0.00
12 0.5 0 0.00
13 5 1 0.20
14 1 2 2.00
15 1 0 0.00
16 1 0 0.00
17 1 0 0.00
18 1 1 1.00
19 1 2 2.00
20 1.5 1 0.67
Summations of n and c respectively 29.75 13
ubar = 0.44
Actual data plotted on a u chart after it is constructed would look something like the example below.
ubar = 0.44
Lot or sample number of units (n) Number of non-conform-ities in the sample (c) number of non-conform-ities per unit (u); what is plotted on the chart. UCL LCL
1 1 0 0.00 2.42 -1.55
2 0.5 1 2.00 3.24 -2.37
3 1.5 0 0.00 2.06 -1.18
4 2 0 0.00 1.84 -0.97
5 2 0 0.00 1.84 -0.97
6 1 0 0.00 2.42 -1.55
7 0.5 0 0.00 3.24 -2.37
8 2 2 1.00 1.84 -0.97
9 3 4 1.33 1.58 -0.71
10 0.5 1 2.00 3.24 -2.37

The LCL below zero would be set to zero. No u values are out of control, i.e. above their respective UCL.

VIIA Xbar problem data

X-bar problem data
Measurements
Sample X1 X2 X3 X4
1 5.00 4.99 5.00 5.01
2 5.01 5.02 4.98 5.00
3 5.01 4.99 4.99 4.98
4 4.98 4.99 4.99 4.98
5 4.99 5.00 5.00 5.00
6 5.00 5.00 5.01 5.00
7 5.00 4.99 5.02 5.01
8 5.02 4.99 5.02 4.98
9 5.01 5.01 5.02 5.00
10 4.99 5.01 5.00 5.01
11 4.99 4.99 4.98 5.00
12 5.00 5.00 5.00 5.02
13 4.98 4.99 4.97 5.01
14 4.99 4.99 5.01 5.00
15 5.01 5.01 4.99 4.99
16 5.01 5.00 4.99 5.00
17 5.00 5.01 5.00 4.99
18 5.02 5.00 5.00 5.01
19 5.01 4.99 4.99 5.01
20 4.99 4.98 5.01 5.01
21 5.02 5.01 5.01 5.00
22 5.00 5.00 5.00 4.99
23 4.98 5.01 5.00 5.01
24 5.00 4.99 5.01 5.00
25 4.99 4.99 5.00 5.00
Dimensional Spec is 5 +/- .05
You can do an R or S chart for the variation.
You can use R directly for the R chart or estimate by S.

VIIA p chart problem data

P chart problem data
Constant lot size of 100
Lot or sample Number of defects in the sample
1 0
2 1
3 0
4 0
5 0
6 0
7 0
8 2
9 2
10 1
11 1
12 1
13 1
14 1
15 1
16 0
17 0
18 0
19 0
20 0
21 0
22 3
23 0
24 0
25 0

VIIA c chart problem data

c chart problem data
Lot or sample Number of non-conform-ities in the sample (c).
1 0
2 1
3 0
4 0
5 0
6 0
7 0
8 0
9 0
10 0
11 0
12 1
13 1
14 1
15 1
16 1
17 2
18 3
19 4
20 5

VIIA u chart problem data

u chart problem data
Lot or sample number of units (n) Number of non-conform-ities in the sample (c)
1 5 2
2 10 1
3 15 0
4 5 0
5 5 0
6 5 2
7 5 0
8 10 2
9 15 1
10 15 1
11 10 0
12 10 0
13 10 1
14 5 2
15 15 1
16 10 0
17 5 0
18 10 1
19 15 3
20 5 1

4

0

ˆ

c

s

=

s

4

0

ˆ

c

s



2

0

ˆ

d

R

=

s

2

0

ˆ

d

R



p

p

=

0

pp

0

(

)

n

p

p

p

LCL

U

-

±

=

1

3

/



n

pp

pLCLU



1

3/

n

np

p

=

n

np

p

n

np

p

S

S

=

n

np

p

(

)

n

p

p

p

LCL

U

-

±

=

1

3

/



n

pp

pLCLU



13

/

(

)

0

0

0

1

3

/

p

np

np

LCL

U

-

±

=

 

000

13/ pnpnpLCLU 

g

c

c

S

=

g

c

c

c

c

LCL

U

3

/

±

=

ccLCLU 3/ 

n

c

u

=

n

c

u

n

u

u

LCL

U

3

/

±

=

n

u

uLCLU 3/ 

n

c

u

S

S

=

n

c

u