2. Descriptive statistics for quality data

Before any control chart is drawn or any capability index is computed, the quality engineer must first understand what the data actually look like. Descriptive statistics provide that first, honest picture. They answer four practical questions: where is the process centered, how much does it spread, is it symmetric, and does it have unusual tails or outliers? This chapter builds those four views and then shows how to summarize them graphically with histograms, box plots, and run charts.

Measures of Central Tendency

The mean, median, and mode describe the "typical" value of a quality characteristic. For a sample of n measurements x₁, x₂, …, xₙ, the sample mean is:

x̄ = (Σ xᵢ) / n

The mean is the natural estimator of process center and is used in X̄–R and X̄–s charts, but it is sensitive to outliers. The median is the middle value when data are ordered; it resists extreme readings and is preferred when the distribution is heavily skewed, for example surface finish or waiting-time data. The mode is the most frequent value and is mainly useful for attribute or discretized data. A quick rule: if mean and median differ noticeably, the distribution is probably skewed, and that is a signal to look at the histogram before trusting any normality-based calculation.

Measures of Dispersion

Center alone is dangerous. Two processes can share the same mean while one produces scrap and the other does not. Dispersion measures quantify that risk.

• Range (R): maximum minus minimum. Simple, used in R charts, but driven entirely by the two extreme points.
• Sample variance (s²): s² = Σ(xᵢ − x̄)² / (n − 1). The n − 1 divisor corrects bias when estimating the population variance.
• Sample standard deviation (s): the square root of s², expressed in the same units as the data. This is the workhorse for capability analysis.
• Coefficient of variation (CV = s / x̄): a unitless relative spread, useful when comparing variability across characteristics with different scales.

For a stable process, the within-subgroup standard deviation estimate σ̂ = R̄ / d₂ is often used instead of the overall s, because it isolates common-cause variation. The gap between s and σ̂ is itself diagnostic: a large gap suggests between-subgroup shifts.

Skewness and Kurtosis

Skewness measures asymmetry. A positive skew means a long right tail (common for cycle time and defect counts); a negative skew means a long left tail. The sample skewness is approximately:

g₁ = [n / ((n−1)(n−2))] · Σ((xᵢ − x̄)/s)³

Kurtosis measures tail heaviness relative to a normal distribution. The excess kurtosis g₂ is zero for a normal distribution; positive values indicate heavy tails and more outliers than expected, negative values indicate light tails. In practice, |g₁| > 1 or |g₂| > 2 is a reasonable flag to check normality before applying X̄–R charts or Cp/Cpk, since those tools assume an approximately normal, stable process.

Graphical Summaries

Numbers compress; graphs reveal. Three plots cover most day-to-day needs.

Histogram. Bin the data and plot frequency per bin. Choose the number of bins with Sturges' rule, k ≈ 1 + 3.322·log₁₀(n), or the Freedman–Diaconis rule for skewed data. Overlay specification limits (LSL, USL) to see whether the distribution sits inside tolerance and whether it is centered. A histogram that is bimodal or has a shoulder usually means two streams, two shifts, or two suppliers are mixed in one dataset.

Box plot. Shows median, quartiles Q1 and Q3, the interquartile range IQR = Q3 − Q1, and whiskers extending to the most extreme points within 1.5·IQR. Points beyond the whiskers are plotted individually as suspected outliers. Box plots are ideal for comparing several machines, lines, or shifts side by side.

Run chart. Plots measurements in time order with a center line at the median. It detects non-random patterns — runs above or below the center, trends, or too few/many crossings — that a histogram cannot show because it discards time order. A run chart is the correct first look at any sequential quality data before formal control limits are applied.

Worked Example: Shaft Diameter, n = 20

Suppose 20 shaft diameters (mm) are measured in production order. The values below are illustrative sample data used for teaching the calculations.

Order:  1     2     3     4     5     6     7     8     9    10
Value: 10.02 10.05 10.01 10.04 10.03 10.06 10.02 10.04 10.05 10.03

Order: 11    12    13    14    15    16    17    18    19    20
Value: 10.07 10.04 10.02 10.05 10.03 10.06 10.04 10.03 10.05 10.04

Compute the mean: x̄ = 10.041 mm. The ordered median is the average of the 10th and 11th values, median = 10.04 mm. The range is R = 10.07 − 10.01 = 0.06 mm. The sample standard deviation is approximately s ≈ 0.016 mm. Skewness is near zero and excess kurtosis is slightly negative, consistent with a roughly symmetric, light-tailed distribution. With LSL = 9.95 and USL = 10.10, all points fall well inside tolerance, and the histogram is centered near the midpoint 10.025 — a small but visible offset worth noting before capability is computed.

Reading the Three Plots Together

The three graphical tools answer different questions and should be used as a set. The run chart answers "is the process stable over time?" The histogram answers "what shape and spread does the process have?" The box plot answers "how do groups compare, and are there outliers?" A useful workflow is: run chart first for stability, then histogram for shape and centering, then box plot for subgroup comparison. Only after all three look consistent should you proceed to capability indices.

Practical Checklist Before Capability Analysis

Descriptive statistics are not a preliminary formality — they are the diagnostic layer that determines whether the more advanced tools in this course are even valid to apply. Master the mean, standard deviation, skewness, kurtosis, and the three plots, and every subsequent control chart and capability study will rest on a defensible foundation.