4.2 Variable Control Charts

4.2 Variable Control Charts

Variable control charts monitor continuous quantitative metrics measured on an interval or ratio scale (e.g., dimension, shaft diameter, weight, tensile strength, temperature, or processing time). Effective process monitoring requires tracking two distinct statistical properties simultaneously: process central tendency (location) and process variability (dispersion).

Overview of Variable Control Chart Selection

Selecting the appropriate variable control chart depends primarily on subgroup size ((n)) and sampling economics:

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Control Chart Factors Lookup Table

Control chart factors are derived from normal distribution theory to convert sample statistics into 3-sigma control limits. The table below provides standard ASQ CQE factor values:

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$\bar{X}$ and (R) Control Charts ($2 \le n \le 9$)

The $\bar{X}$ and (R) chart pair is the standard tool for sample sizes from 2 to 9. The dispersion chart ((R) chart) must always be evaluated first. If the (R) chart is out of control, the sample variation is unstable, rendering the $\bar{X}$ control limits mathematically invalid.

Formulas:

• Subgroup Mean: $\bar{X}_i = \frac{\sum_{j=1}^n X_{ij}}{n}$
• Subgroup Range: $R_i = X_{\max} - X_{\min}$
• Overall Grand Mean (Centerline for $\bar{X}$): $\bar{\bar{X}} = \frac{\sum_{i=1}^k \bar{X}_i}{k}$
• Average Range (Centerline for (R)): $\bar{R} = \frac{\sum_{i=1}^k R_i}{k}$

Control Limits:

$$\text{Range Chart Limits: } UCL_R = D_4 \bar{R}, \quad LCL_R = D_3 \bar{R}$$ $$\text{X-bar Chart Limits: } UCL_{\bar{X}} = \bar{\bar{X}} + A_2 \bar{R}, \quad LCL_{\bar{X}} = \bar{\bar{X}} - A_2 \bar{R}$$ $$\text{Estimated Process Standard Deviation: } \hat{\sigma} = \frac{\bar{R}}{d_2}$$

Worked Example 1: $\bar{X}$ and (R) Calculation

A CNC precision shop collects (k = 25) subgroups of size (n = 5) shaft diameters. The grand mean is $\bar{\bar{X}} = 50.020\text{ mm}$, and the average range is $\bar{R} = 0.150\text{ mm}$.

1. Calculate (R) Chart Limits: From factor lookup for (n = 5): $D_3 = 0$, $D_4 = 2.114$.

   $$UCL_R = 2.114 \times 0.150 = 0.3171\text{ mm}$$ $$LCL_R = 0 \times 0.150 = 0.0000\text{ mm}$$

2. Calculate $\bar{X}$ Chart Limits: From factor lookup for (n = 5): $A_2 = 0.577$.

   $$UCL_{\bar{X}} = 50.020 + (0.577 \times 0.150) = 50.020 + 0.08655 = 50.10655\text{ mm}$$ $$LCL_{\bar{X}} = 50.020 - (0.577 \times 0.150) = 50.020 - 0.08655 = 49.93345\text{ mm}$$

3. Estimate Process Standard Deviation ($\hat{\sigma}$): From factor lookup for (n = 5): $d_2 = 2.326$.

   $$\hat{\sigma} = \frac{0.150}{2.326} = 0.06449\text{ mm}$$

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$\bar{X}$ and (S) Control Charts ($n \ge 10$)

When subgroup size $n \ge 10$, the sample range (R) becomes an inefficient estimator of process dispersion because it depends solely on extreme values. Sample standard deviation (S) utilizes every data point in the subgroup, making $\bar{X}$ and (S) charts statistically superior.

Formulas:

• Subgroup Standard Deviation: $S_i = \sqrt{\frac{\sum_{j=1}^n (X_{ij} - \bar{X}_i)^2}{n - 1}}$
• Average Standard Deviation: $\bar{S} = \frac{\sum_{i=1}^k S_i}{k}$

Control Limits:

$$\text{S Chart Limits: } UCL_S = B_4 \bar{S}, \quad LCL_S = B_3 \bar{S}$$ $$\text{X-bar Chart Limits: } UCL_{\bar{X}} = \bar{\bar{X}} + A_3 \bar{S}, \quad LCL_{\bar{X}} = \bar{\bar{X}} - A_3 \bar{S}$$ $$\text{Estimated Process Standard Deviation: } \hat{\sigma} = \frac{\bar{S}}{c_4}$$

Worked Example 2: $\bar{X}$ and (S) Calculation

An automated stamping process with subgroup size (n = 10) collects (k = 20) subgroups, yielding $\bar{\bar{X}} = 12.500\text{ mm}$ and $\bar{S} = 0.040\text{ mm}$. From factor table for (n = 10): $A_3 = 0.975$, $B_3 = 0.284$, $B_4 = 1.716$, $c_4 = 0.9727$.

1. (S) Chart Limits:

   $$UCL_S = 1.716 \times 0.040 = 0.06864\text{ mm}$$ $$LCL_S = 0.284 \times 0.040 = 0.01136\text{ mm}$$

2. $\bar{X}$ Chart Limits:

   $$UCL_{\bar{X}} = 12.500 + (0.975 \times 0.040) = 12.500 + 0.0390 = 12.5390\text{ mm}$$ $$LCL_{\bar{X}} = 12.500 - (0.975 \times 0.040) = 12.500 - 0.0390 = 12.4610\text{ mm}$$

3. Estimate Process Standard Deviation ($\hat{\sigma}$):

   $$\hat{\sigma} = \frac{0.040}{0.9727} = 0.04112\text{ mm}$$

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Individual and Moving Range ((I-MR)) Charts ((n = 1))

Used when rational subgrouping is impossible or impractical—such as automated 100% inspection, continuous chemical processing, low-volume production, or destructive testing.

Formulas (Moving Range Span = 2):

• Moving Range: $MR_i = |X_i - X_{i-1}|$ for $i = 2, 3, \dots, k$
• Average Moving Range: $\bar{MR} = \frac{\sum_{i=2}^k MR_i}{k - 1}$
• Centerline for Individuals: $\bar{X} = \frac{\sum X_i}{k}$

Control Limits:

$$\text{Moving Range Limits: } UCL_{MR} = D_4 \bar{MR} = 3.267 \bar{MR}, \quad LCL_{MR} = 0$$ $$\text{Individual Limits: } UCL_X = \bar{X} + E_2 \bar{MR} = \bar{X} + 2.660 \bar{MR}, \quad LCL_X = \bar{X} - 2.660 \bar{MR}$$ $$\text{Estimated Process Standard Deviation: } \hat{\sigma} = \frac{\bar{MR}}{d_2} = \frac{\bar{MR}}{1.128}$$

Note: The factor $E_2 = \frac{3}{d_2} = \frac{3}{1.128} \approx 2.660$.