Graphical Methods

Graphical Methods

Quick Answer: Graphical methods turn raw measurements into visual patterns. For CSSGB BoK III.D.4 (Create), you must build and read scatter diagrams, normal probability plots, histograms, stem-and-leaf plots, and box-and-whisker plots so you can see shape, association, outliers, and stratification effects before you trust averages, capability indices, or hypothesis tests.

Why Graphics Come Before Formulas

Numbers without pictures hide structure. A stable-looking mean can sit on a bimodal mixture; a “strong correlation” can be two clusters with no linear relationship inside either group. In the Measure and early Analyze phases, Green Belts use graphs to:

• Check distribution shape (symmetric, skewed, multimodal, truncated)
• Spot outliers and data-entry errors
• See relationships between inputs (X) and outputs (Y)
• Compare strata (shifts, lines, suppliers) side by side
• Decide whether tools that assume normality or linearity are safe

ASQ expects Create-level skill: choose the right plot, construct it correctly (including scales and labels), and state what the picture implies for the project Y and measurement plan.

Scatter Diagrams

A scatter diagram (scatter plot) displays paired observations $(x_i, y_i)$ as points in a plane. Use it when you suspect that changing an input X (temperature, speed, supplier score) moves an output Y (thickness, cycle time, defect rate).

How to construct:

1. Collect paired data under a clear operational definition for both X and Y.
2. Plot X on the horizontal axis and Y on the vertical axis with readable scales covering the observed range.
3. Mark each pair as one point (do not connect points unless time order is the story).
4. Optionally code points by stratum (color or symbol for shift/machine).

How to interpret:

CSSGB tip: Scatter plots show association, not proof of causation. Use them to prioritize Xs for multi-vari, regression, or DOE—not to claim “X causes Y” from a single plot alone.

Worked sketch: Twenty runs of oven temperature (°C) vs. bond strength (N). Points form a rising band with moderate scatter and no extreme outliers → temperature is a candidate X worth quantifying with correlation/regression later.

Histograms

A histogram groups continuous data into bins (intervals) and draws bar heights equal to counts or relative frequencies in each bin. It is the primary picture of a univariate process distribution.

How to construct:

1. Collect a reasonable sample (often n ≥ 50–100 for a clear shape; smaller samples still help screening).
2. Choose bin width so you get roughly $\sqrt{n}$ to $2\sqrt{n}$ bars (rules of thumb—not sacred).
3. Align bin edges with measurement resolution when possible.
4. Label axes: measurement units on X, frequency (or density) on Y; title the CTQ and sample context.

How to interpret:

• Symmetric unimodal — often compatible with normal-based capability and z thinking (still verify with a normal plot).
• Right-skewed — common for cycle time and wait time; mean > median; capability formulas that assume normality can understate risk in the long tail.
• Left-skewed — less common; mean pulled left.
• Bimodal / multimodal — mixture of streams (two machines, day vs. night); stratify before one σ or one Cpk.
• Cliffs / truncation — inspection sorting, physical limits at zero, or censoring.
• Gaps and spikes — measurement resolution, preferred numbers, or data issues.

Always mark LSL/USL on the histogram when specs exist so the voice of the process sits next to the voice of the customer.

Normal Probability Plots

A normal probability plot (normal quantile–quantile plot) orders the data and plots each ordered value against the expected normal score (z-quantile) for its rank. If data are approximately normal, points fall close to a straight line.

How to construct (conceptually):

1. Sort observations $x_{(1)} \le \cdots \le x_{(n)}$.
2. Assign plotting positions (software uses standard formulas such as $(i-0.5)/n$).
3. Convert each position to a normal z-score.
4. Plot $x_{(i)}$ vs. z (or vs. theoretical quantile); add a reference line through the bulk of the data.

How to interpret:

Green Belts use normal plots before trusting % out of spec from z-tables, many capability narratives, and some residual checks. If the plot is badly curved, prefer transformation, nonparametric views, or stratified analysis—not blind z-math.

Stem-and-Leaf Plots

A stem-and-leaf plot splits each number into a stem (leading digits) and a leaf (trailing digit). Unlike a histogram, it preserves the actual data values while ranking them.

How to construct:

1. Choose stem unit (e.g., tens place for values 10–99).
2. Write stems in a column; attach leaves in order for each stem.
3. Include a key (e.g., “3 | 2 = 32 mm”).

How to interpret: Shape mirrors a sideways histogram; you can still read min, max, rough median, and clusters. Useful for small-to-moderate n in workshops when software is unavailable. For large n or multi-decimal data, histograms and box plots scale better.

Box-and-Whisker Plots

A box-and-whisker plot (box plot) summarizes a distribution with five-number logic: minimum (or lower whisker), first quartile $Q_1$, median, third quartile $Q_3$, and maximum (or upper whisker), plus points beyond fences as outliers.

Recall:

$\mathrm{IQR} = Q_3 - Q_1$

Common fence rule: points below $Q_1 - 1.5,\mathrm{IQR}$ or above $Q_3 + 1.5,\mathrm{IQR}$ plot as individual outliers.

How to construct:

1. Order the data; compute median, $Q_1$, $Q_3$, and IQR.
2. Draw the box from $Q_1$ to $Q_3$ with a line at the median.
3. Extend whiskers to the most extreme non-outlier points (or to min/max if your standard does not flag outliers).
4. Plot outliers individually.
5. For comparisons, place side-by-side box plots by stratum.

How to interpret:

• Box height = middle 50% spread (IQR)—robust to extremes.
• Median position inside the box shows skew (median near $Q_1$ → right skew).
• Long whiskers or many outliers → heavy tails or special causes.
• Side-by-side boxes quickly show which shift or supplier has higher center or wider spread.

CSSGB tip: Box plots hide multimodality inside a single box. If a histogram is bimodal, do not rely on one box for the mixed stream.

Choosing and Combining Graphs

Measure-phase sequence that exam items reward: operational definition → clean data → histogram/box for shape → normal plot if needed → scatter for candidate Xs → only then formal correlation, capability, or MSA numbers.

Common Pitfalls on the CSSGB Exam

• Treating a scatter plot cloud as causation or as “DOE complete.”
• Drawing a histogram with too few bins (everything looks unimodal) or too many (noise looks like multimodality).
• Ignoring stratification when the histogram is multimodal.
• Claiming normality from a “pretty” histogram alone without a normal probability plot for borderline cases.
• Using a box plot alone when the decision needs the full shape (defects in a long tail).

Master construction and interpretation of these five tools and you meet BoK III.D.4 at the Create level—and you protect every later calculation that depends on seeing the data correctly first.