EWMA Chart (Exponentially Weighted Moving Average)
The Exponentially Weighted Moving Average (EWMA) chart is a sophisticated tool within the Lean Six Sigma framework, particularly under the umbrella of Six Sigma Control Plans. As part of the broader category of control charts, the EWMA chart stands out for its ability to detect small shifts in the process mean more efficiently than traditional Shewhart control charts. This article delves into the theory behind the EWMA chart, its construction, and its application in monitoring and improving process performance.
What is an EWMA Chart?
The EWMA chart is a type of control chart used to monitor processes where it is crucial to detect small and persistent shifts in the process mean. Unlike simple moving averages or Shewhart control charts that primarily detect large shifts, the EWMA chart is sensitive to smaller changes due to its unique weighting mechanism. It applies exponentially decreasing weights to past observations, with more recent data receiving more weight. This method enables the chart to smooth out noise and detect subtle trends in the process.
Theory Behind EWMA Chart
The theoretical foundation of the EWMA chart lies in its formula, where the EWMA statistic Zi is calculated using the formula:
In this formula, Xi represents the individual value at point i, Zi−1 is the EWMA statistic from the previous point, and λ is the weighting factor, which ranges between 0 and 1. The choice of λ affects the sensitivity of the chart: a smaller λ makes the chart more sensitive to small shifts.
Construction of an EWMA Chart
Constructing an EWMA chart involves several steps:
Selection of Lambda (λ): The value of λ is critical and is usually chosen based on the desired sensitivity to shifts in the process mean. A common choice is λ=0.2.
Calculation of Initial EWMA Statistic (Z0): This is often set equal to the process mean or the first data point.
Sequential Calculation of EWMA Statistics: Using the EWMA formula, calculate the statistics for each data point in the series.
Determination of Control Limits: The control limits for the EWMA chart are calculated based on the standard deviation of the process and the selected λ. These limits are adjusted to account for the weighting applied to the data points.
Plotting: Plot the EWMA statistics on the chart with the control limits. Any point outside these limits indicates a potential process shift that may require investigation.
Applications of EWMA Chart
The EWMA chart is versatile and can be applied in various industries and processes, including manufacturing, healthcare, and financial services. It is particularly useful in processes where the quality characteristic is autocorrelated, or in situations where the process exhibits a slow drift. By detecting these small shifts early, organizations can take corrective actions sooner, potentially saving costs and improving quality.
Conclusion
The EWMA chart is a powerful tool in the Lean Six Sigma toolkit for monitoring process performance. Its ability to detect small shifts in the process mean makes it an invaluable asset for quality control and process improvement. Understanding the theory behind its construction and application allows practitioners to effectively implement this chart in their control plans, leading to enhanced process stability and performance.
EWMA Chart example
For this example, let's consider a Lean Six Sigma project focused on improving the quality of coffee in a café. The project team aims to reduce the variability in the temperature of the served coffee, ensuring it is consistently within the customer's preferred range. The team decides to use an Exponentially Weighted Moving Average (EWMA) chart to monitor the coffee temperature over time, identifying any shifts or trends away from the target temperature. The EWMA chart is chosen because it gives more weight to recent observations, which is helpful in quickly detecting small shifts in the process.
Project Background
Objective: Ensure the coffee temperature is consistently within the preferred range (ideally between 65°C to 70°C).
Data Collection: The team measures the temperature of the first coffee served every hour for 10 hours.
Data
Here's the temperature data collected (in °C):
Hour | Temperature (°C) |
1 | 67 |
2 | 68 |
3 | 66 |
4 | 69 |
5 | 65 |
6 | 67 |
7 | 70 |
8 | 68 |
9 | 66 |
10 | 64 |
EWMA Chart Creation
The EWMA chart calculation involves several steps. We'll use a lambda (λ) value of 0.2 for the weighting factor, which is a common choice. The formula to calculate the EWMA is:
where:
The control limits are calculated using the following formulas:
where:
μ is the target or mean temperature,
L is the distance of the control limits from the mean in terms of standard deviation (usually set to 3 for control charts),
σEWMA is the standard deviation of the EWMA values.
The standard deviation of the EWMA (σEWMA) is calculated based on the standard deviation of the original observations and the weighting factor.
Let's calculate the EWMA values for our data, along with the UCL and LCL, assuming the target temperature is the mean of the collected data.
Calculate the EWMA's
Hour | Temperature (°C) | EWMA (°C) |
1 | 67 | 67.00* |
2 | 68 | 67.20** |
3 | 66 | 66.96*** |
4 | 69 | 67.37 |
5 | 65 | 66.89 |
6 | 67 | 66.92 |
7 | 70 | 67.53 |
8 | 68 | 67.63 |
9 | 66 | 67.30 |
10 | 64 | 66.64 |
EWMA Hour 1*
For the first hour, the temperature measured was 67°C. Since there's no previous EWMA value, we set the EWMA for Hour 1 directly to the temperature observed:
EWMA1=X1=67°C
EWMA Hour 2**
For the second hour, the temperature measured was 68°C. Now, we apply the EWMA formula using the measurement from Hour 2 and the EWMA value from
EWMA2=λ×X2+(1−λ)×EWMA1
EWMA2=0.2×68+(1−0.2)×67
EWMA2=13.6+53.6=67.2°C
EWMA Hour 3***
EWMA3=λ×X3+(1−λ)×EWMA2
EWMA3=0.2×66+(1−0.2)×67.2
EWMA3=13.2+53.76=66.96°C
Calculate UCL and LCL:
The mean temperature (μ) calculated from the data is 67°C (I do not detail the math here to be quick)
Then we calculate σ with:
For our data, the standard deviation (σ) was approximately:
σ=1.83
the standard deviation of the EWMA values (σEWMA) is not directly measured but can be approximated using the standard deviation of the original observations (σ) and the weighting factor (λ).
he UCL and LCL are calculated by adding and subtracting, respectively, a specified number of standard deviations (σ EWMA) from the mean (μ). Typically,=3 L=3 standard deviations are used.
Substituting the values:
Create the EWMA chart
Note, The actual temperatures measured (in blue), is not required on the chart.
Interpretation
The EWMA chart indicates that the coffee temperature control process is stable and within predefined control limits, showing no signs of undesirable variability or trends. This suggests effective temperature management, with no immediate actions required. Continuous monitoring is recommended to maintain this level of process control.
This analysis helps the café team in their Lean Six Sigma project by providing a method to continuously monitor and control the coffee temperature, ensuring a consistent quality experience for customers.