Descriptive Statistics

Descriptive Statistics

Quick Answer: Descriptive statistics summarize a dataset’s center and spread without yet testing hypotheses. Use mean, median, and mode for central tendency; range, variance, standard deviation, and IQR for dispersion; and frequency / cumulative frequency tables to see distribution shape. Green Belts calculate these on baseline CTQs before capability, SPC, or Analyze tests.

Role in DMAIC Measure

Before you claim “the process is broken” or “the pilot worked,” you need a clear numerical portrait of Y: Where is the center? How wide is the spread? Is the shape skewed? CSSGB BoK III.D.3 expects you to evaluate descriptive measures—select the right statistic and compute/interpret it correctly.

Descriptive methods describe the sample (or population dump) you have. Inferential methods (t-tests, confidence intervals) go further to statements about a larger population—covered more deeply with hypothesis testing, but always built on clean descriptive understanding.

Measures of Central Tendency

Mean (arithmetic average)

For a sample of n observations $x_1, \ldots, x_n$:

$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

Population mean μ uses all N population values. The mean uses every point—powerful but sensitive to outliers and skew.

Median

The median is the middle value when data are ordered. For odd n, it is the center point; for even n, the average of the two central values. The median is robust to extreme values.

Mode

The mode is the most frequent value (or bin). Datasets may be unimodal, bimodal, or have no unique mode. Useful for categorical/nominal data and for spotting multi-process mixtures (two peaks).

When to prefer which

Worked example — center: Five cycle times (minutes): 10, 12, 11, 13, 34.

• Ordered: 10, 11, 12, 13, 34
• Mean: $(10+12+11+13+34)/5 = 80/5 = \mathbf{16.0}$
• Median: 12
• Mode: none unique (all appear once)

The mean is pulled high by 34; the median better represents a “typical” ticket if 34 was a one-off system outage. Investigate the outlier—do not silently drop it without rules—but choose the summary that matches the question.

Measures of Dispersion

Range

$R = x_{\max} - x_{\min}$

Simple and used in X-bar/R charts, but based on only two points—volatile for large n.

Variance

Population variance:

$\sigma^2 = \frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}$

Sample variance (unbiased estimator used in almost all project work):

$s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}$

The $n-1$ denominator (Bessel’s correction) accounts for using $\bar{x}$ instead of μ.

Standard deviation

$\sigma = \sqrt{\sigma^2}, \quad s = \sqrt{s^2}$

Same units as the data—easier to interpret than variance. In a rough normal model, mean ± s covers about two-thirds of points; control charts and capability indices are built from σ estimates.

Interquartile range (IQR)

$\mathrm{IQR} = Q_3 - Q_1$

Spread of the middle 50%; robust to outliers. Box plots display IQR visually.

Coefficient of variation (awareness)

$\mathrm{CV} = \frac{s}{\bar{x}}$ (often ×100%)

Compares relative spread across metrics with different units or means—use only with ratio-scale data and $\bar{x} > 0$.

Worked example — dispersion: Use 10, 12, 11, 13 (omit the outlier to show clean arithmetic; n = 4).

• $\bar{x} = 11.5$
• Deviations: −1.5, 0.5, −0.5, 1.5
• Squared: 2.25, 0.25, 0.25, 2.25; sum = 5.0
• $s^2 = 5.0 / (4-1) = \mathbf{1.667}$
• $s = \sqrt{1.667} \approx \mathbf{1.29}$
• Range = 13 − 10 = 3

If the 34-minute point returns to the set (n = 5, mean 16), s jumps dramatically—dispersion measures expose instability the mean alone hides.

Full five-point check (including 34):

• Mean = 16
• Squared deviations from 16: 36, 16, 25, 9, 324; sum = 410
• $s^2 = 410/(5-1) = 102.5$; $s \approx \mathbf{10.12}$

Report both center and spread: “mean 16 min with s ≈ 10 min” warns the process is not a tight 16.

Frequency and Cumulative Frequency Distributions

A frequency distribution lists each value or class interval with its count (frequency). A relative frequency is count/n; a cumulative frequency adds counts from the lowest class upward (or downward for “more than” tables).

Why Green Belts use them:

• Build histograms and see shape (symmetric, skewed, bimodal)
• Read approximate percentiles and % below a spec from cumulative relative frequency
• Communicate baseline without dumping raw lists to leadership

Worked example — discrete frequencies: Defect counts per batch over 20 batches:
0,1,0,2,1,0,1,3,1,0, 2,1,0,1,2,0,1,1,0,2

Interpretation: 75% of batches have ≤1 defect; only 5% hit 3. Mode = 1; mean defect count = $(0·7+1·8+2·4+3·1)/20 = 18/20 = 0.90$.

Continuous data: form class intervals (bins)—e.g., cycle time 0–5, 5–10, 10–15, …—with consistent width when possible. Too few bins hide shape; too many create noisy spikes. Software defaults are a start; adjust with process knowledge.

Cumulative use case: From a cumulative relative frequency chart, estimate the fraction of product below USL even before formal capability indices—useful in early Measure storytelling (still verify normality assumptions later for % predictions).

Connecting Descriptives to Other Measure Tools

Descriptives also feed control chart constants and baseline DPMO calculations. Incomplete description leads to wrong chart scaling and false “improvements.”

Calculation Discipline for the Open-Book Exam

1. Sort data before median and for easy min/max.
2. State whether you use n−1 (sample) or N (population).
3. Keep units on s and range; variance is squared units (minutes²).
4. Show sum of squares steps when options are close numerically.
5. Pair every mean with a spread measure in written conclusions.

Green Belt Mini-Workflow

1. QC the data (prior section).
2. Plot histogram or stem-and-leaf / ordered list for small n.
3. Compute mean, median, mode as appropriate.
4. Compute range, s (and IQR if skewed).
5. Build frequency / cumulative tables for communication and percentile reads.
6. Only then proceed to capability, control limits, or hypothesis tests.

Descriptive statistics turn a pile of Measure-phase numbers into a defensible baseline story: where the process sits, how much it wanders, and how mass piles up relative to requirements.