Lean Six Sigma uses data to reduce waste, lower variation, and improve process performance. The basic idea is that instead of guessing what is wrong with a process, we collect data, look for patterns, and use those patterns to decide what should be improved.

Lean focuses on reducing waste and improving flow. Six Sigma focuses on reducing defects and variation. Together, they help organizations make processes faster, more consistent, and more reliable.

DMAIC stands for Define, Measure, Analyze, Improve, and Control. This gives improvement teams a structure to follow so they can define the problem, measure the current process, analyze the causes, improve the process, and then control the process so the improvement does not disappear later.

One basic statistical tool is the sample mean. The mean gives us a simple way to describe the center of the data.

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

In Lean Six Sigma, this can help us understand the average cycle time, average defect rate, or average output volume of a process.

The sample standard deviation helps measure how spread out the data is. This matters because two processes can have the same average, but one process may be much more consistent than the other.

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

A lower standard deviation usually means the process is more predictable.

For this example, I created simulated data for a warehouse process with five steps: Receiving, Storage, Picking, Packing, and Shipping. This dataset includes cycle time, defect rate, output volume, and whether the data point came from before or after an improvement.

ggplot(process_data, aes(x = improvement_stage, y = cycle_time)) +
  geom_boxplot() +
  labs(title = "Cycle Time Before and After Improvement", x = "Improvement Stage", y = "Cycle Time")

A basic defect rate compares the number of defects to the total number of opportunities for defects.

\[ Defect\ Rate = \frac{Number\ of\ Defects}{Total\ Opportunities} \]

Process capability compares the allowed specification range to the amount of process variation.

\[ C_p = \frac{USL - LSL}{6\sigma} \]

## # A tibble: 2 × 4
##   improvement_stage average_cycle_time average_defect_rate average_output_volume
##   <chr>                          <dbl>               <dbl>                 <dbl>
## 1 After Improvement               36.5              0.0447                  503.
## 2 Before Improveme…               42.2              0.0698                  517.

The Control phase makes sure the improvements are sustained after the project ends. Common control tools include standard operating procedures, dashboards, audits, training, and control charts. In Lean Six Sigma, statistics help turn process improvement into evidence-based decision making.