6.10 Descriptive Statistics: Central Tendency, Dispersion, and Shape

In the Measure phase of a Six Sigma project, numerical data serves as the foundation for evaluating process performance, identifying sources of variation, and establishing baseline capability. A primary responsibility of a Black Belt is to classify data correctly according to its statistical characteristics and select appropriate descriptive statistics to summarize process performance without introducing distortion.

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Measures of Central Tendency (Location)

Central tendency describes the central point or typical value around which process observations cluster:

1. Sample Mean (Arithmetic Average)

The sample mean $\bar{x}$ is the sum of all individual observations divided by the sample size $n$:

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

• Properties: Includes every data point in its calculation; mathematically tractable. Highly sensitive to extreme outliers or skewed distributions, which pull the mean toward the long tail.

2. Sample Median ($\tilde{x}$ or $Q_2$)

The median is the middle value of a dataset arranged in ascending numerical order ($50^{\text{th}}$ percentile).

• For an odd sample size $n$, the median is the exact center observation at position $\frac{n+1}{2}$.
• For an even sample size $n$, the median is the arithmetic mean of the two middle observations at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$.
• Properties: Non-parametric metric. Highly robust against outliers and distribution skewness.

3. Mode

The mode is the most frequently occurring value in a dataset.

• Properties: Can be used with nominal categorical data. A dataset may be unimodal, bimodal, multimodal, or have no mode.

Distribution Skewness and Central Tendency Relationships

• Symmetric Distribution: $\text{Mean} = \text{Median} = \text{Mode}$
• Right-Skewed (Positive Skew): Tail extends toward larger values $\rightarrow \text{Mode} < \text{Median} < \text{Mean}$
• Left-Skewed (Negative Skew): Tail extends toward smaller values $\rightarrow \text{Mean} < \text{Median} < \text{Mode}$

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Measures of Dispersion (Variability / Spread)

Dispersion quantifies the magnitude of variation or scatter within a process dataset:

1. Range ($R$)

The difference between the maximum and minimum observed values: $R = x_{\max} - x_{\min}$. Range is simple to calculate but volatile, as it depends entirely on the two extreme observations.

2. Sample Variance ($s^2$) & Bessel's Correction

Sample variance measures the average squared deviation of observations from the sample mean. The sample variance $s^2$ is defined as:

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

Why Divide by $(n - 1)$ Instead of $n$?

When estimating population variance $\sigma^2$ from a sample, using $n$ in the denominator results in a biased estimator that systematically underestimates population variance because deviations are calculated around the sample mean $\bar{x}$ rather than the true population mean $\mu$. Dividing by $n - 1$ (known as Bessel's Correction) adjusts for the loss of one degree of freedom, yielding an unbiased estimator such that the expected value $E[s^2] = \sigma^2$.

3. Sample Standard Deviation ($s$)

The positive square root of sample variance, returning the dispersion metric to the original units of measurement:

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

4. Interquartile Range (IQR)

The interquartile range measures the spread of the middle $50%$ of the data, calculated as the difference between the third quartile ($Q_3$, $75^{\text{th}}$ percentile) and first quartile ($Q_1$, $25^{\text{th}}$ percentile):

$\text{IQR} = Q_3 - Q_1$

Like the median, IQR is a non-parametric measure robust against extreme outliers.

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Distribution Shape: Skewness and Kurtosis

Central tendency and dispersion describe where a distribution sits and how wide it is. Shape describes how it is distributed within that width, and it determines whether normal-theory methods apply at all.

Skewness

Skewness measures asymmetry:

$g_1 = \frac{\frac{1}{n}\sum (x_i - \bar{x})^3}{s^3}$

A rule of thumb: $|g_1| < 0.5$ is approximately symmetric; $0.5$ to $1.0$ is moderately skewed; above $1.0$ is highly skewed and normal-theory capability calculations should not be applied without transformation.

Right skew is by far the most common in transactional processes, because time-based measures are bounded below by zero and unbounded above. That is also why the lognormal distribution fits them so often.

Kurtosis

Kurtosis measures tail weight relative to a normal distribution. Excess kurtosis is reported so that a normal distribution scores zero:

$g_2 = \frac{\frac{1}{n}\sum (x_i - \bar{x})^4}{s^4} - 3$

Leptokurtosis matters practically: heavy tails mean the process produces far more extreme values than a normal model predicts, so a capability index computed under the normality assumption understates the true defect rate, sometimes by an order of magnitude.

Neither statistic replaces a plot. A histogram plus a normal probability plot shows skew, tail weight, bimodality, and truncation at once, and bimodality -- which usually means two processes mixed together -- is invisible in the skewness and kurtosis values.

Relative Dispersion: the Coefficient of Variation

Standard deviation is expressed in the units of the measurement, so it cannot be compared across characteristics with different units or very different magnitudes. The coefficient of variation normalizes it:

$CV = \frac{s}{\bar{x}} \times 100\%$

Two uses in Six Sigma work:

• Comparing variability across dissimilar characteristics. A fill weight with $s = 1.2$ g on a mean of 500 g ($CV = 0.24%$) is far better controlled than a cycle time with $s = 1.2$ minutes on a mean of 8 minutes ($CV = 15%$), even though the standard deviations are numerically identical.
• Comparing the same characteristic across processes running at different levels, such as two lines producing different sizes of the same part.

Two cautions. CV is meaningful only on a ratio scale with a true zero -- it is undefined in any useful sense for Celsius temperature -- and it becomes unstable when the mean is near zero, because a small denominator inflates the ratio without limit.