Data Hours Period Cycle Time

Datahoursperiodcycle Time14523334443055164773983496910431129123813451

Data: Hours Period Cycle Time. Calculate R-Bar? 2. Upper Control Limit for the Range? 3 Upper control limit for the individuals? 4 Lower control limit for the individuals? 5. What is the control chart telling you? 6. Is there adequate discrimination?

Paper For Above instruction

The analysis of process control data is fundamental to quality management, enabling organizations to monitor, control, and improve their processes effectively. In this scenario, the task involves calculating key statistical control chart parameters—namely the average range (R-Bar), the upper and lower control limits for the range and individual measurements—and interpreting what these charts reveal about the process.

Given the raw data sequence: 14, 52, 33, 34, 44, 30, 55, 16, 47, 73, 98, 34, 96, 91, 04, 31, 12, 91, 23, 81, 34, 51, it appears these numbers reflect cycle times over specified periods. To proceed, data must be organized into subgroups, typically in batches of 4-5 measurements, allowing for the calculation of ranges and individual control limits. Assuming subgroups of size five for illustration, the data can be partitioned as follows:

• Subgroup 1: 14, 52, 33, 34, 44
• Subgroup 2: 30, 55, 16, 47, 73
• Subgroup 3: 98, 34, 96, 91, 04
• Subgroup 4: 31, 12, 91, 23, 81
• Subgroup 5: 34, 51

Calculating within these subgroups allows us to determine the range (the difference between the maximum and minimum within each subgroup). The ranges for each subgroup are:

• Range 1: 52 - 14 = 38
• Range 2: 73 - 16 = 57
• Range 3: 98 - 04 = 94
• Range 4: 91 - 12 = 79
• Range 5: 51 - 34 = 17

The average range (R-Bar) is calculated as:

R̄ = (38 + 57 + 94 + 79 + 17) / 5 = 657 / 5 = 131.4

Next, control limits for the range are computed using the appropriate constants for the subgroup size, n=5, from standard control chart constants. For n=5, the chart constants are typically: A2 = 0.577, D3 = 0, D4 = 2.114.

Upper Control Limit for the Range (UCL_R) is:

UCL_R = D₄ × R̄ = 2.114 × 131.4 ≈ 278.4

Lower Control Limit for the Range (LCL_R) is:

LCL_R = D₃ × R̄ = 0 × 131.4 = 0

since D₃ equals zero for this subgroup size.

Moving to individual measurements, the average (mean) process time can be computed by averaging all the measurements:

Sum of all data points = 14 + 52 + 33 + 34 + 44 + 30 + 55 + 16 + 47 + 73 + 98 + 34 + 96 + 91 + 04 + 31 + 12 + 91 + 23 + 81 + 34 + 51 = 1171

Number of observations = 22

Mean (X̄) = 1171 / 22 ≈ 53.2

Standard deviation estimation is necessary for control chart calculations, but for individual control limits, the average moving range (R̄_m) is a better estimator. Calculating the range between consecutive data points:

• |14 - 52|= 38
• |52 - 33|= 19
• |33 - 34|= 1
• |34 - 44|= 10
• |44 - 30|= 14
• |30 - 55|= 25
• |55 - 16|= 39
• |16 - 47|= 31
• |47 - 73|= 26
• |73 - 98|= 25
• |98 - 34|= 64
• |34 - 96|= 62
• |96 - 91|= 5
• |91 - 04|= 87
• |04 - 31|= 27
• |31 - 12|= 19
• |12 - 91|= 79
• |91 - 23|= 68
• |23 - 81|= 58
• |81 - 34|= 47
• |34 - 51|= 17

Sum of these ranges: 38 + 19 + 1 + 10 + 14 + 25 + 39 + 31 + 26 + 25 + 64 + 62 + 5 + 87 + 27 + 19 + 79 + 68 + 58 + 47 + 17 = 929

Number of ranges: 21

Average moving range R̄_m = 929 / 21 ≈ 44.24

The individual control limits are then calculated as:

UCL_individual = X̄ + 3 × (R̄_m / d₂)

LCL_individual = X̄ - 3 × (R̄_m / d₂)

Here, d₂ is a constant for n=2, which is 1.128.

Thus,

UCL_individual = 53.2 + 3 × (44.24 / 1.128) ≈ 53.2 + 3 × 39.22 ≈ 53.2 + 117.66 ≈ 170.86

LCL_individual = 53.2 - 117.66 ≈ -64.46

Since process times cannot be negative, LCL is adjusted to zero if necessary.

Interpretation of these control charts indicates whether the process is stable and predictable. If most data points lie within the calculated control limits and no patterns or trends are evident, the process is considered to be in control. Any points outside these limits or suddenly recurring patterns could suggest special causes of variation needing investigation.

In this particular case, the high ranges, notably the maximum range of 94, and some points exceeding the UCL based on individual control limits might suggest variability issues. The process may not be entirely stable, requiring further analysis or process improvements. The control chart likely indicates that while most data are within limits, there are instances of excessive variation, signaling the need for process adjustments or further investigation to enhance stability and consistency.

Ultimately, these calculations and interpretations assist in quality control management by highlighting areas where the process may be drifting out of control, prompting corrective actions to maintain quality and efficiency.

References

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