Understand the calculation of control limits

The control limits are determined by some settings on the characteristic table, which may be overridden for a particular QM specification. The QM specification is the one used by the most recent sample that meets the filter criteria. So once the most recent sample is identified, its record in the Sample_Char_Link table for that sample and characteristic is found (each graph is for a particular characteristic). The qm_spec_id field in that table identifies the particular QM specification to use. Then the Qm_Spec_Char_Link record that goes with this QM specification and characteristic is identified. In it are several fields, which generally have exact counterparts in the characteristic table also, that are needed to calculate the control limits. If a field is null in the Qm_Spec_Char_Link table, the value from the field of the same name in the characteristic table is used instead (which generally will not be null).

The first field to consider is cl_source. If cl_source is 0, the control limits are to be calculated from the data. If cl_source is 1, it means the population average/proportion/defects per unit and population standard deviation (for variables) are given in the database tables. And if cl_source is 2, it means that the exact control limits (and the center line) have been specified in the database for specific chart(s). This last case is the simplest - the control limits are whatever is specified in the appropriate fields (see below) of the Qm_Spec_Char_Link table, or if such a field is null, by the corresponding field in the characteristic table. If that too is null there is then no control limit. In the case of attribute charts, it is normal not to have a lower control limit, though having one is not illegitimate.

The control limit values are in this case represented by a horizontal line at the control limit value; there is no adjustment for changing sample size. The adjustment for DPMO charts mentioned in the second-last line of the chart above means taking the given control limit value (in defects per unit), multiplying by 1 million and dividing by the number of defect opportunities per unit, which is found in the num_defect_opp field of the characteristic table. If the num_defect_opp field is null 1 is used instead.

When cl_source is 1, the field to use for the given average/proportion/defects per unit is determined by the field std_avg_is_target. If true, the target field in the Qm_Spec_Char_Link table supplies this value. (This field is not nullable, and there is no corresponding field in the characteristic table.) If it is false the std_avg field is used instead. The value for the given standard deviation comes from the std_deviation field.

• For charts of the average (the top chart of chart types 2 or 3) the lower control limit is 3 divided by the square root of the sample size times the given standard deviation below the given average, and the upper control limit is this same amount above the given average. 3 divided by the square root of the sample size is known as A and for convenience it is stored it in the a field of the factor table, so it can be read instead of calculated.

• For a range chart (the bottom chart of type 2) the lower control limit is the given standard deviation times D1 for n = the actual sample size. This is found in the d1 field of the factor record for the appropriate sample_size. The upper control limit is the given standard deviation times D2 for n = the actual sample size. This is found in the d2 field of the same factor record.

• For a sigma chart (the bottom of chart type 3) the lower control limit is the given standard deviation times B1 for n = the actual sample size times the square root of the ratio of the sample size to one less than the sample size (B1 times this square root is also known as B5). B1 is found in the b1 field of the factor record for the appropriate sample_size. The upper control limit is the given standard deviation times B2 for n = the actual sample size times the square root of the ratio of the sample size to one less than the sample size (B2 times this square root is also known as B6). B2 is found in the b2 field of the same factor record. That is, if σ' is the given standard deviation, and n is the actual sample size, the lower control limit is and the upper control limit is .

• For an individual X chart (the top of chart type 5) the lower control limit is 3 times the given standard deviation below the given average, and the upper control limit is this same amount above the given average.

• For an individual moving range chart (the bottom of chart type 5) effectively there is no control lower limit, since its value is reported as 0. The upper control limit is the given standard deviation times D2 for n = 2. This is found in the d2 field of the factor record where sample_size = 2, and is equal to approximately 3.686.

• For a moving average chart (the top chart of chart types 7 or 8) the lower control limit is 3 divided by the square root of the moving average span times the given standard deviation below the given average, and the upper control limit is this same amount above the given average. This is the same A factor as is used for charts of the average, but the factor record used is the one where sample_size = the moving average span.

  Every result for a moving average chart produces a new point, regardless of how the results were originally grouped into samples.

• For a moving range chart (the bottom chart of chart type 7) the lower control limit is the given standard deviation times D1 for n = the moving average span, and upper control limit is the given standard deviation times D2 for n = the moving average span. It is analogous to the range chart except that n is not necessarily 2, but rather the moving average span.

• For a moving sigma chart (the bottom chart of chart type 8) the lower control limit is the given standard deviation times B1 for n = the moving average span, and the upper control limit is the given standard deviation times B2 for n = the moving average span. It is analogous to the sigma chart except that n is not necessarily 2, but rather the moving average span.

• For a percent defective (p) chart (type 16) the lower control limit is 3 times the square root of the ratio of the product of the given average proportion and 1 minus the given average proportion to the sample size above the given average proportion, and the upper control limit is this same amount above the given average proportion. That is, if p' is the given average proportion, and n is the actual sample size, the lower control limit is , and the upper control limit is .

• For a number defective (np) chart (type 17), which as the name suggests is simply a chart of the percent defective times the sample size (n), the lower control limit is 3 times the square root of the product of the sample size, the given average proportion, and 1 minus the given average proportion below the sample size times given average proportion, and the upper control limit is this same amount above the actual sample size times given average proportion.

  Note: Though the chart is not normalized and is showing the number of defects, the given standard is still expressed as a proportion.

  Thus, if p' is the given average proportion, and n is the actual sample size, the lower control limit is and the upper control limit is .

• For a defects per unit (u) chart (type 18) the lower control limit is 3 times the square root of the ratio of the given average number per unit to the sample size below the given mean, and the upper control limit is this same amount above the given average number per unit. That is, if u' is the given average number per unit, and n is the actual sample size, the lower control limit is and the upper control limit is .

• For a number of defects (c) chart (type 19) the given standard is still expressed as the average number per unit. Therefore the lower control limit is 3 times the square root of the product of the actual sample size and the given average number per unit below the product of the actual sample size and the given average number per unit, and the upper control limit is this same amount above the product of the actual sample size and the given average number per unit. That is, if u' is the given average number per unit, and n is the actual sample size, the lower control limit is and the upper control limit is .

• For a defects per million opportunities (DPMO) chart (type 20), use the control limit values for the defects per unit (u) chart multiplied by 1 million and divided by the number of defect opportunities per unit.

When cl_source = 0, control limits are calculated from the data already entered, using the most recent samples_for_cl number of samples that meet the filter criteria. If there are not at least samples_before_cl number of qualifying samples, control limits are not calculated but remain null. When calculating the number of samples for both of these requirements, the number of samples is counted given the default chart type, which may be different than the number of samples originally taken, if grouped data is being plotted on a chart of individuals.

For variables, the sample standard deviation when charting grouped data is calculated as the square root of the sum of the ratio of the sum of the squares of the difference between each result in and the sample average to one less than the number of samples. That is, if s is the sample standard deviation,

where is the ith result_value in the sample, is the average result_value for the sample, and n is the actual sample size. If n = 1, treat s as 0. In the case of a Moving Average and Moving Sigma chart, the "sample" is the set of results within the span, so n can be taken as the value of mov_avg_span, with the current result plus the preceding (n - 1) being considered. s is the value plotted on a sigma/moving sigma chart. Once s is known it can be used to make an estimate of the population standard deviation, also just called the estimated standard deviation or estimated sigma, written as , if sigma_est = 0. It is the average of the sample standard deviation divided by the value of c4 for the actual sample size, for qualifying samples having n > 1. If charting range (which is the difference between the largest and smallest result values in a sample) for a variable, the estimated standard deviation when sigma_est = 0 is the average of the range divided by the value of d2 for the actual sample size n, for qualifying samples having n > 1. As with the Moving Average and Moving Sigma chart, for the Moving Average and Moving Range chart the "sample" is the set of results within the span, so n can be taken as the value of mov_avg_span, with the current result and the preceding (n - 1) being considered. For the IX + MR chart n is always 2, since the moving range is defined as the absolute value of the difference between the current and preceding result values (thus the "sample" in this case comprises the current result and immediately preceding result). c4 and d2 are found in the c_4 and d_2 fields of the factor record respectively for the appropriate sample_size. If sigma_est = 1, then the estimated standard deviation is set equal to the population standard deviation, which is equal to

where j is the total number of results in the qualifying samples, is the value of the ith result, and is the average of all these values.

• For charts of the average (the top chart of chart type 2 or 3), the lower control limit is the estimated standard deviation times 3 divided by the square root of the actual sample size below the overall average, and the upper control limit is this same amount above the overall average.

• For a range chart (the bottom chart of type 2) the center line is at the estimated standard deviation times d2 for the actual sample size. The lower control limit is the estimated standard deviation times 3 times d3 for n = the actual sample size below the center line, and the upper control limit is this same amount above the center line. This is found in the d_3 field of the factor record for the appropriate sample_size.

• For a sigma chart (the bottom of chart type 3) the center line is at the estimated standard deviation times c4 for the actual sample size. The lower control limit is the estimated standard deviation times 3 times the square root of the sum of the ratio of the one less than the actual sample size to the actual sample size, and c4 (for the actual sample size) squared below the center line, and the upper control limit is this same amount above the center line. That is, the lower control limit is and the upper control limit is .

• For an individual X chart (the top of chart type 5) the lower control limit is 3 times the estimated standard deviation below the overall average, and the upper control limit is this same amount above the overall average.

• For an individual moving range chart (the bottom of chart type 5) effectively there is no control lower limit, since its value is reported as 0. The center line is at d2 times the estimated standard deviation, and the upper control limit is 3 times d3 times the estimated standard deviation above this. Both d2 and d3 are for n = 2. Thus the upper control limit is the estimated standard deviation times D2 for n = 2 (approximately 3.686).

• For a moving average chart (the top chart of chart types 7 or 8) the lower control limit is 3 divided by the square root of the moving average span times the given standard deviation below the overall average, and the upper control limit is this same amount above the overall average. This is the same A factor as is used for charts of the average, but the factor record used is the one where sample_size = the moving average span.

• For a moving range chart (the bottom chart of chart type 7) the center line and control limits are calculated the same as for the range chart except that n is not necessarily 2, but rather the moving average span.

• For a moving sigma chart (the bottom chart of chart type 8) the center line and control limits are calculated the same as for the sigma chart except that n is not necessarily 2, but rather the moving average span.

For attributes, the sigma_est field has no effect, because each sample consists of just one reading.

• For a percent defective (p) chart (type 16) the lower control limit is 3 times the square root of the ratio of the product of the actual average proportion and 1 minus the actual average proportion to the sample size above the actual average proportion, and the upper control limit is this same amount above the actual average proportion. The actual average proportion, , is the total number of defectives (= mean = result_value) divided by the total number sampled (= act_sample_size). So , where k is the total number of qualifying samples, is the number of defectives in sample i, and is the actual size of sample i. The lower control limit is, and the upper control limit is , where n is the actual size of the sample.

• For a number defective (np) chart (type 17), the lower control limit is 3 times the square root of the product of the sample size, the actual average proportion, and 1 minus the actual average proportion below the sample size times actual average proportion, and the upper control limit is this same amount above the actual sample size times given actual average proportion. Thus, if is the given average proportion, and n is the actual sample size, the lower control limit is , and the upper control limit is .

• For a defects per unit (u) chart (type 18) the actual average number per unit, is the total number of defects (= mean = result_value) divided by the total number sampled (= act_sample_size). So , where k is the total number of qualifying samples, is the number of defects in sample i, and is the actual size of sample i. The lower control limit is 3 times the square root of the ratio of the given mean to the sample size below the given mean, and the upper control limit is this same amount above the given mean. That is, if u' is the given average number per unit, and n is the actual sample size, the lower control limit is and the upper control limit is .

• For a number of defects (c) chart (type 19) the actual average number of defects is the total number of defects (= mean = result_value) divided by the number of samples. That is, where k is the total number of qualifying samples and is the number of defects in sample i. Therefore the lower control limit is 3 times the square root of the actual average number of defects below the actual average number of defects, and the upper control limit is this same amount above the actual average number of defects. That is, if is the actual average number of defects, and n is the actual sample size, the lower control limit is and the upper control limit is .

• For a defects per million opportunities (DPMO) chart (type 20), use the control limit values for the defects per unit (u) chart multiplied by 1 million and divided by the number of defect opportunities per unit.

When cl_source = 0 or 1 with a default_chart_type of 2 or 3 (an Xbar chart), samples containing just one result should have the plotted point and control limits for the second chart go to 0.