Lesson 6: Basic statistics, the normal distribution, and graphical analysis

Once a process is mapped, the Measure phase turns to actually quantifying how it performs. This lesson covers the statistical foundation every later lesson in this course builds on: the basic descriptive statistics, the normal distribution, and the graphical tools used to see a process's behavior rather than just calculate it.

Basic and descriptive statistics

Descriptive statistics summarize a data set using a small number of values instead of forcing a reader to look at every individual data point. They split into two families: measures of central tendency, which describe where the middle of the data sits, and measures of dispersion, which describe how spread out the data is.

The three common measures of central tendency are the mean (the arithmetic average, the sum of all values divided by the count), the median (the middle value when all data points are sorted, less sensitive than the mean to extreme outliers), and the mode (the value that occurs most frequently).

The common measures of dispersion are the range (the highest value minus the lowest value, simple but very sensitive to a single outlier), the variance (roughly the average of the squared differences between each data point and the mean, where the sample version divides by n minus 1 rather than n, which weights larger deviations more heavily), and the standard deviation (the square root of the variance, which returns the spread to the data's original units and is the most commonly reported measure of dispersion in Six Sigma work, usually denoted with the Greek letter sigma). Two data sets can have the identical mean and tell completely different stories once their standard deviations are compared: one tightly clustered around the average, the other wildly inconsistent.

Normal distributions and normality

The normal distribution is a specific, symmetric, bell-shaped probability distribution that shows up constantly in natural and process data, where the mean, median, and mode all fall at the same central point and data becomes progressively less common the further it sits from that center in either direction. A process whose output follows a normal distribution is predictable in a specific, useful way, known as the empirical rule: roughly 68 percent of output falls within one standard deviation of the mean, roughly 95 percent within two standard deviations, and roughly 99.7 percent within three standard deviations. This predictability is the foundation underneath process capability calculations, covered in Lesson 8, and underneath the hypothesis tests Green Belts use, covered later in this program. Not every process is normally distributed, and checking whether data is approximately normal, called a normality check, is a routine early step before applying tools that assume it.

Graphical analysis

Before running any statistical test, an experienced practitioner looks at the data. A handful of simple charts reveal patterns that a table of numbers alone hides:

• Histogram: bars showing how many data points fall into each of a series of value ranges, the standard way to see a distribution's overall shape, including whether it looks roughly normal, skewed, or has more than one peak.
• Box plot: a compact summary showing the median, the middle 50 percent of the data (the box itself), and the overall spread, including individual outlier points plotted separately, useful for comparing several groups side by side at a glance.
• Scatter plot: individual points plotted against two variables at once, the standard way to visually check whether a relationship exists between an X and a Y before running a formal regression, covered later at the Green Belt level.
• Run chart: individual data points plotted in the order they occurred over time, the simplest way to spot a trend, a shift, or a cycle that a simple average would completely hide.

A team that jumps straight to a statistical test without first looking at a histogram or a run chart risks missing something a five-second glance would have caught, such as an obvious shift partway through the data or a handful of extreme outliers distorting the average.

Measures of shape: skewness and kurtosis

Central tendency and dispersion describe the middle and the spread of a data set, but they say nothing about its shape. A third family of descriptive statistics, measures of shape, describes how symmetric or lopsided a distribution is. Skewness measures that asymmetry. A distribution with a long tail stretching toward higher values is positively skewed, or right-skewed; its mean is pulled above its median by the small number of unusually high values in that tail. A distribution with a long tail stretching toward lower values is negatively skewed, or left-skewed, with the mean pulled below the median instead. A perfectly symmetric distribution has zero skew, and for the normal distribution the mean, median, and mode all coincide. Housing rehab data is often right-skewed in practice: most work orders close out in a predictable handful of days, but a small number run far longer because of a permit delay or a supply backorder, and that long tail pulls the average completion time higher than the typical, median job.

Kurtosis measures how heavy a distribution's tails are compared with the normal distribution, whose excess kurtosis is defined as zero. A distribution with excess kurtosis above zero, called leptokurtic, has heavier tails than normal, meaning more frequent extreme values than the normal distribution would predict, and is often drawn with a sharper peak as well. A distribution with excess kurtosis below zero, called platykurtic, has thinner tails, with fewer extreme values, and is often drawn with a flatter top. Both skewness and kurtosis are typically checked, along with a histogram, as part of the normality check described above, before applying a tool that assumes normally distributed data.