Attribute data arise when each inspected item is classified as conforming or non-conforming (go / no-go), or when we count the number of defects on each item. Unlike variable data (length, weight), attribute data are discrete counts or proportions.
| Flavour | Question Answered | Chart Type |
|---|---|---|
| Defective items (binomial) | How many items are non-conforming? | p chart, np chart |
| Defects per unit (Poisson) | How many defects are on each item? | c chart, u chart |
A defective item is one that fails to meet specifications (e.g., a cracked tile). A defect is an imperfection — an item can have multiple defects and still be usable (e.g., 3 scratches on a car panel).
The p chart monitors the proportion (fraction) of defective items in each subgroup. It works for both fixed and variable subgroup sizes.
For \(k\) subgroups, let subgroup \(j\) have size \(n_j\) and contain \(d_j\) defectives. The sample fraction defective is:
The average fraction defective:
\[ \bar{p} = \frac{\sum_{j=1}^{k} d_j}{\sum_{j=1}^{k} n_j}. \]Control limits (for subgroup of size \(n_j\)):
If LCL computes negative, set LCL = 0.
25 subgroups of \(n = 100\) items each. Total defectives = 75. \(\bar{p} = 75/2500 = 0.030\).
Standard error: \(\sqrt{0.030 \times 0.970/100} = \sqrt{0.000291} = 0.01706\).
UCL = 0.030 + 3 × 0.01706 = 0.0812; CL = 0.030; LCL = 0.030 − 0.0512 = −0.0212 → set LCL = 0 (cannot be negative).
If a subgroup shows 11 defectives out of 100 (\(p = 0.11\)), it exceeds UCL → out of control.
Subgroup sizes vary: \(n_1 = 80\), \(n_2 = 120\), \(n_3 = 100\), etc. With \(\bar{p} = 0.05\):
For \(n_1 = 80\): UCL\(_1\) = 0.05 + 3\(\sqrt{0.05 \times 0.95/80}\) = 0.05 + 0.0731 = 0.1231.
For \(n_2 = 120\): UCL\(_2\) = 0.05 + 3\(\sqrt{0.05 \times 0.95/120}\) = 0.05 + 0.0597 = 0.1097.
Each subgroup gets its own limits — wider for smaller samples, narrower for larger.
If the target fraction defective \(p_0\) is given:
The np chart is used instead of the p chart when the subgroup size \(n\) is constant across all subgroups. It plots the number of defective items \(d\) directly, which is easier for shop-floor operators to interpret.
20 subgroups, each \(n = 50\). Total defectives = 40. \(\bar{p} = 40/1000 = 0.04\). \(\bar{d} = 50 \times 0.04 = 2.0\).
UCL = 2.0 + 3\(\sqrt{50 \times 0.04 \times 0.96}\) = 2.0 + 3\(\sqrt{1.92}\) = 2.0 + 3 × 1.386 = 6.16.
LCL = 2.0 − 4.16 = −2.16 → set LCL = 0 (cannot be negative). CL = 2.0.
If a subgroup shows 8 defectives, it exceeds UCL = 6.16 → out of control.
Target \(p_0 = 0.02\), \(n = 200\). \(np_0 = 4.0\).
UCL = 4 + 3\(\sqrt{200 \times 0.02 \times 0.98}\) = 4 + 3\(\sqrt{3.92}\) = 4 + 3 × 1.980 = 9.94.
LCL = 4 − 5.94 = −1.94 → set LCL = 0 (cannot be negative). CL = 4.
The c chart monitors the total number of defects (not defectives) in each inspection unit. It assumes the number of defects follows a Poisson distribution with parameter \(c\). The inspection unit must be constant (e.g., one television set, one roll of cloth, one square metre of painted surface).
25 rolls of fabric inspected. Number of weaving defects per roll: 4, 6, 3, 8, 5, 7, 2, 9, 4, 5, 6, 3, 10, 5, 4, 6, 7, 3, 5, 4, 8, 6, 5, 3, 7. The total is 135, so \(\bar{c} = 135/25 = 5.40\).
UCL = 5.40 + 3\(\sqrt{5.40}\) = 5.40 + 6.97 = 12.37; LCL = 5.40 − 6.97 = −1.57 → set LCL = 0 (cannot be negative). CL = 5.40.
All observed values are between 2 and 10 — within limits. Process is in control.
Target: \(c_0 = 3.0\) defects per television set. UCL = 3 + 3\(\sqrt{3}\) = 3 + 5.196 = 8.20; LCL = 0. CL = 3. A set with 9 defects exceeds UCL — assignable cause suspected.
The u chart is used when the inspection unit varies in size — e.g., cloth rolls of different lengths, or sheets of different areas. We monitor the rate of defects per unit rather than the total count.
Let \(c_j\) be the number of defects in inspection unit \(j\) which has area/size \(n_j\) (in standard units).
Inspection of painted panels. Roll 1: 6 defects over 2 m² (\(u_1 = 3.0\)); Roll 2: 8 defects over 3 m² (\(u_2 = 2.67\)); Roll 3: 4 defects over 1 m² (\(u_3 = 4.0\)); … Total defects = 55 over 20 m². \(\bar{u} = 55/20 = 2.75\) defects per m².
For Roll 1 (\(n_1 = 2\) m²): UCL = 2.75 + 3\(\sqrt{2.75/2}\) = 2.75 + 3 × 1.173 = 6.27; LCL = 0. \(u_1 = 3.0\) is within limits.
For Roll 3 (\(n_3 = 1\) m²): UCL = 2.75 + 3\(\sqrt{2.75/1}\) = 2.75 + 4.974 = 7.72. Wider limits for smaller inspection units.
Target: \(u_0 = 1.5\) defects per m². For a panel of 4 m²: UCL = 1.5 + 3\(\sqrt{1.5/4}\) = 1.5 + 3 × 0.612 = 3.34; LCL = 0. If 16 defects are found on this 4 m² panel, \(u = 16/4 = 4.0 > 3.34\) → out of control.