How this calculation works
An exponentially weighted moving average retains information from prior process observations. Its startup variance is smaller than its long-run variance when the initial statistic is fixed at the target. Using only asymptotic limits can hide that distinction. This page calculates the sequence and its finite-startup limits from a separately entered baseline. It is a monitoring chart, different from a forecasting smoother that estimates the next observation without control limits.
zt=lambda xt+(1−lambda)zt−1; variance_t=sigma² lambda/(2−lambda)[1−(1−lambda)^(2t)]A worked example
For target zero, sigma one, lambda 0.5 and observations two and two, the first EWMA is one and the second is 1.5. First-step standard deviation is 0.5, so three-sigma limits are ±1.5. At the second step the variance is 0.3125 and limits are about ±1.67705. The finite-startup limits widen toward their long-run values even though the target stays fixed.
Good to know
Independent observations, constant baseline variance and an appropriate baseline mean are assumed. Autocorrelation can change chart performance substantially. No baseline estimation, change-point diagnosis or automatic reset is included. A point inside the displayed limits does not certify control, and a signal needs investigation rather than an inferred cause.
What this calculation answers
An exponentially weighted moving average retains information from prior process observations. Its startup variance is smaller than its long-run variance when the initial statistic is fixed at the target. Using only asymptotic limits can hide that distinction.
This page calculates the sequence and its finite-startup limits from a separately entered baseline. It is a monitoring chart, different from a forecasting smoother that estimates the next observation without control limits.
The One sided standardized CUSUM control chart calculator produces a forecasting level. This monitoring calculation additionally needs an established target, sigma and finite-startup control limits.
Use the right input conventions
Preserve time order and use individual observations in common units. Initial EWMA is set to the entered target. Lambda ranges above zero through one and controls how heavily the newest observation is weighted.
Sigma is the established standard deviation of individual observations, not the standard deviation of the EWMA sequence. The limit multiplier applies to the evolving EWMA standard deviation.
Interpret the result
Smaller lambda creates a longer memory, but the chart’s alarm behavior also depends on its limit multiplier. Neither parameter is selected automatically to meet a desired average run length.
At lambda one, the statistic equals the current observation and startup limits reduce to target plus or minus multiplier times sigma. That boundary supplies an independent check on the general formula.
Work through the example
For target zero, sigma one, lambda 0.5 and observations two and two, the first EWMA is one and the second is 1.5. First-step standard deviation is 0.5, so three-sigma limits are ±1.5.
At the second step the variance is 0.3125 and limits are about ±1.67705. The finite-startup limits widen toward their long-run values even though the target stays fixed.
Where the model stops
Independent observations, constant baseline variance and an appropriate baseline mean are assumed. Autocorrelation can change chart performance substantially.
No baseline estimation, change-point diagnosis or automatic reset is included. A point inside the displayed limits does not certify control, and a signal needs investigation rather than an inferred cause.
A few questions, answered
Why do limits change early in the sequence?
The EWMA starts from a fixed target and accumulates observation variance gradually.
What changes in the comparison?
Use lambda one and compare with a chart of the raw observations and constant limits.