9.2 Control Charts for Variables Data

Variables control charts monitor continuous quantitative process measurements (e.g., diameter, weight, temperature, tensile strength, cycle time) over time. They are the primary tool in the Control phase for detecting special cause variation and maintaining statistical process control.

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Statistical Principles of Statistical Process Control (SPC)

Continuous process variation consists of two components:

1. Common Cause Variation: Inherent natural background noise present in a stable process.
2. Special Cause Variation: Unassignable external disturbances (e.g., raw material lot change, tool wear, operator error) that shift the process mean or increase variance.

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$\bar{X}$ and $R$ Chart (Sample Subgroups $2 \le n \le 8$)

The $\bar{X}$ and $R$ chart is the most widely deployed variables chart for small subgroup sample sizes.

Mathematical Formulas & Limit Factors

Given $k$ subgroups of size $n$:

1. Subgroup Mean ($\bar{x}_i$) & Range ($R_i$): $$\bar{x}_i = \frac{\sum x}{n}, \quad R_i = x_{\max} - x_{\min}$$
2. Grand Mean ($\bar{\bar{x}}$) & Average Range ($\bar{R}$): $$\bar{\bar{x}} = \frac{\sum \bar{x}_i}{k}, \quad \bar{R} = \frac{\sum R_i}{k}$$

Control Limit Formulas

• $\bar{X}$-Chart:

$$UCL_{\bar{X}} = \bar{\bar{x}} + A_2 \bar{R}, \quad LCL_{\bar{X}} = \bar{\bar{x}} - A_2 \bar{R}$$

• $R$-Chart:

$$UCL_R = D_4 \bar{R}, \quad LCL_R = D_3 \bar{R}$$

Where $A_2, D_3, D_4$ are standard SPC constants derived from subgroup size $n$.

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$\bar{X}$ and $S$ Chart (Sample Subgroups $n \ge 9$)

When subgroup size $n \ge 9$, sample range $R$ loses statistical efficiency because extreme values dominate $R$. The sample standard deviation $S$ utilizes all $n$ data points.

Control Limit Formulas

• $\bar{X}$-Chart: $UCL_{\bar{X}} = \bar{\bar{x}} + A_3 \bar{S}, \quad LCL_{\bar{X}} = \bar{\bar{x}} - A_3 \bar{S}$
• $S$-Chart: $UCL_S = B_4 \bar{S}, \quad LCL_S = B_3 \bar{S}$

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Individual and Moving Range ($I$-$\text{MR}$) Chart ($n = 1$)

Used when process data consists of individual observations (e.g., continuous chemical process batch outputs, low-volume production, automated 100% inspection).

Moving Range ($MR_i$) Formula

$$MR_i = |x_i - x_{i-1}|$$

Using $d_2 = 1.128$ for $n = 2$ moving range pairs:

$$UCL_I = \bar{x} + 2.66 \bar{MR}, \quad LCL_I = \bar{x} - 2.66 \bar{MR}$$ $$UCL_{MR} = D_4 \bar{MR} = 3.267 \bar{MR}, \quad LCL_{MR} = 0$$

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Construction Algorithms & Calculation Formulas for Variables Charts

1. Construction of $\bar{X}$ and $R$ Charts

Given $k$ historical subgroups ($k \ge 20-25$) each containing $n$ observations ($2 \le n \le 8$):

Step 1: Calculate Subgroup Statistics

For each subgroup $i = 1, 2, \dots, k$:

$$\bar{x}_i = \frac{1}{n} \sum_{j=1}^{n} x_{ij}, \quad R_i = x_{i,\max} - x_{i,\min}$$

Step 2: Calculate Overall Averages

$$\bar{\bar{x}} = \frac{1}{k} \sum_{i=1}^{k} \bar{x}_i, \quad \bar{R} = \frac{1}{k} \sum_{i=1}^{k} R_i$$

Step 3: Compute Control Limits Using SPC Tabular Constants

• Range Chart ($R$-Chart): $$\text{UCL}_R = D_4 \bar{R}, \quad \text{Center Line}_R = \bar{R}, \quad \text{LCL}_R = D_3 \bar{R}$$
• Mean Chart ($\bar{X}$-Chart): $$\text{UCL}_{\bar{X}} = \bar{\bar{x}} + A_2 \bar{R}, \quad \text{Center Line}_{\bar{X}} = \bar{\bar{x}}, \quad \text{LCL}_{\bar{X}} = \bar{\bar{x}} - A_2 \bar{R}$$

Standard SPC Tabular Constants Table ($n = 2$ to $10$)

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Worked Calculation Example: $\bar{X}$ and $R$ Chart

A Black Belt monitors shaft turning diameters. Subgroup size $n = 5$, number of subgroups $k = 20$. Baseline calculations yield grand mean $\bar{\bar{x}} = 25.040\text{ mm}$ and average range $\bar{R} = 0.080\text{ mm}$. From the table for $n = 5$: $A_2 = 0.577$, $D_3 = 0$, $D_4 = 2.114$.

1. $R$-Chart Limits: $$\text{UCL}_R = D_4 \bar{R} = 2.114 \times 0.080 = 0.1691\text{ mm}$$ $$\text{LCL}_R = D_3 \bar{R} = 0 \times 0.080 = 0.000\text{ mm}$$
2. $\bar{X}$-Chart Limits: $$\text{UCL}_{\bar{X}} = \bar{\bar{x}} + A_2 \bar{R} = 25.040 + (0.577 \times 0.080) = 25.040 + 0.04616 = 25.0862\text{ mm}$$ $$\text{LCL}_{\bar{X}} = \bar{\bar{x}} - A_2 \bar{R} = 25.040 - 0.04616 = 24.9938\text{ mm}$$

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