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EWMA Control Chart Check (Weighted Moving Average, Control Limits, Small Shifts)

Is my process drifting by a small amount that an ordinary control chart would miss? Where are the EWMA control limits for these readings, and does the weighted average cross them?

Exponentially weighted moving average (EWMA) control chart from up to twenty readings in time order: the EWMA statistic starting from the target, its standard deviation for your weighting factor, the upper and lower control limits from your historical standard deviation, and the first reading at which the EWMA leaves the limits, so small sustained shifts show sooner than on a Shewhart chart.

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How this tool works

Inputs, units and where to find them

InputUnitTypical source
Target or historical meanMean of the historical in-control data, used as the starting EWMA and the centre line.
Historical standard deviationStandard deviation of the historical in-control data.
Weighting factor λWeight given to the newest reading (0 to 1); NIST notes 0.2 to 0.3 are usual.
Limit width kstandard deviationsHow many standard deviations of the EWMA the limits sit from the centre (3 is common).
Number of readings usedreadingsHow many of the readings below to use, in time order.
Readings in time order
Reading 1Measurement number 1 in the order it was taken.
Reading 2Measurement number 2 in the order it was taken.
Reading 3Measurement number 3 in the order it was taken.
Reading 4Measurement number 4 in the order it was taken.
Reading 5Measurement number 5 in the order it was taken.
Reading 6Measurement number 6 in the order it was taken.
Reading 7Measurement number 7 in the order it was taken.
Reading 8Measurement number 8 in the order it was taken.
Reading 9Measurement number 9 in the order it was taken.
Reading 10Measurement number 10 in the order it was taken.
Reading 11Measurement number 11 in the order it was taken.
Reading 12Measurement number 12 in the order it was taken.
Reading 13Measurement number 13 in the order it was taken.
Reading 14Measurement number 14 in the order it was taken.
Reading 15Measurement number 15 in the order it was taken.
Reading 16Measurement number 16 in the order it was taken.
Reading 17Measurement number 17 in the order it was taken.
Reading 18Measurement number 18 in the order it was taken.
Reading 19Measurement number 19 in the order it was taken.
Reading 20Measurement number 20 in the order it was taken.

What each input means

Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.

What this means: Target or historical mean
Illustration: Target or historical meanA number line with a marker at the value of Target or historical mean, between its lowest and highest accepted values.−10⁹10⁹EWMA₀Example: 50Target or historical mean
What it is
The mean of the process when it was in control, used as the starting EWMA value and the centre line of the chart.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 50 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the centre line and both limits up or down by the same amount.
Get it right
Take it from at least 20 to 25 historical readings from a stable period, or use the nominal if the process is centred on it.
What this means: Historical standard deviation
Illustration: Historical standard deviationA nominal size line with an upper and a lower limit around it; parts inside the band are acceptable. The figure illustrates Historical standard deviation.upperlowersExample: 2.0539Historical standard deviation
What it is
The standard deviation of the historical in-control readings, which sets how far apart the limits are.
Units and format
Enter a plain number with no unit. The form accepts 0.000001 or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 2.0539 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
The limits widen in proportion; a larger value makes signals rarer.
Get it right
Calculate it from the same stable period as the target; data containing a shift give limits that are too wide.
What this means: Weighting factor λ
Illustration: Weighting factor λTwo bars: a base value and the same value multiplied by Weighting factor λ, the factor.base valuewith factorλExample: 0.3Weighting factor λ
What it is
The weight given to the newest reading in the moving average, between 0 and 1; older readings get geometrically less.
Units and format
Enter a plain number with no unit. The form accepts 0.01 to 1. Required: the tool does not calculate until every input is valid.
Example
The form opens with 0.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A smaller weight narrows the limits and smooths more, finding small shifts but reacting slowly to large ones.
Get it right
NIST notes 0.2 to 0.3 are usual; use a smaller value to find smaller shifts.
What this means: Limit width k
Illustration: Limit width kTwo bars: a base value and the same value multiplied by Limit width k, the factor.base valuewith factorkExample: 3 standard deviationsLimit width k
What it is
How many standard deviations of the EWMA the control limits sit above and below the centre line.
Units and format
Enter a number in standard deviations: standard deviations. The form accepts 0.5 standard deviations to 6 standard deviations. Required: the tool does not calculate until every input is valid.
Example
The form opens with 3 standard deviations as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
The limits move out in proportion.
Get it right
Three is common; narrower limits give earlier signals and more false alarms.
What this means: Number of readings used
Illustration: Number of readings usedA row of identical items being counted; Number of readings used is how many there are.nExample: 20 readingsNumber of readings used
What it is
How many of the readings listed below are charted, counting from the first reading in time order.
Units and format
Enter a number in readings: readings. The form accepts 3 readings to 20 readings. Use a whole number. Required: the tool does not calculate until every input is valid.
Example
The form opens with 20 readings as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Readings beyond this count are ignored.
Get it right
Enter readings from the oldest; unused fields at the end are ignored.

Readings in time order

What this means: Reading 1
Illustration: Reading 1A number line with a marker at the value of Reading 1, between its lowest and highest accepted values.−10⁹10⁹Y1Example: 52Reading 1
What it is
Measurement number 1 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 52 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Enter the first reading taken; the order matters because each new value builds on the ones before.
What this means: Reading 2
Illustration: Reading 2A number line with a marker at the value of Reading 2, between its lowest and highest accepted values.−10⁹10⁹Y2Example: 47Reading 2
What it is
Measurement number 2 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 47 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 2 must be the one taken after reading 1.
What this means: Reading 3
Illustration: Reading 3A number line with a marker at the value of Reading 3, between its lowest and highest accepted values.−10⁹10⁹Y3Example: 53Reading 3
What it is
Measurement number 3 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 53 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 3 must be the one taken after reading 2.
What this means: Reading 4
Illustration: Reading 4A number line with a marker at the value of Reading 4, between its lowest and highest accepted values.−10⁹10⁹Y4Example: 49.3Reading 4
What it is
Measurement number 4 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 49.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 4 must be the one taken after reading 3.
What this means: Reading 5
Illustration: Reading 5A number line with a marker at the value of Reading 5, between its lowest and highest accepted values.−10⁹10⁹Y5Example: 50.1Reading 5
What it is
Measurement number 5 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 50.1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 5 must be the one taken after reading 4.
What this means: Reading 6
Illustration: Reading 6A number line with a marker at the value of Reading 6, between its lowest and highest accepted values.−10⁹10⁹Y6Example: 47Reading 6
What it is
Measurement number 6 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 47 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 6 must be the one taken after reading 5.
What this means: Reading 7
Illustration: Reading 7A number line with a marker at the value of Reading 7, between its lowest and highest accepted values.−10⁹10⁹Y7Example: 51Reading 7
What it is
Measurement number 7 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 51 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 7 must be the one taken after reading 6.
What this means: Reading 8
Illustration: Reading 8A number line with a marker at the value of Reading 8, between its lowest and highest accepted values.−10⁹10⁹Y8Example: 50.1Reading 8
What it is
Measurement number 8 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 50.1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 8 must be the one taken after reading 7.
What this means: Reading 9
Illustration: Reading 9A number line with a marker at the value of Reading 9, between its lowest and highest accepted values.−10⁹10⁹Y9Example: 51.2Reading 9
What it is
Measurement number 9 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 51.2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 9 must be the one taken after reading 8.
What this means: Reading 10
Illustration: Reading 10A number line with a marker at the value of Reading 10, between its lowest and highest accepted values.−10⁹10⁹Y10Example: 50.5Reading 10
What it is
Measurement number 10 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 50.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 10 must be the one taken after reading 9.
What this means: Reading 11
Illustration: Reading 11A number line with a marker at the value of Reading 11, between its lowest and highest accepted values.−10⁹10⁹Y11Example: 49.6Reading 11
What it is
Measurement number 11 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 49.6 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 11 must be the one taken after reading 10.
What this means: Reading 12
Illustration: Reading 12A number line with a marker at the value of Reading 12, between its lowest and highest accepted values.−10⁹10⁹Y12Example: 47.6Reading 12
What it is
Measurement number 12 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 47.6 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 12 must be the one taken after reading 11.
What this means: Reading 13
Illustration: Reading 13A number line with a marker at the value of Reading 13, between its lowest and highest accepted values.−10⁹10⁹Y13Example: 49.9Reading 13
What it is
Measurement number 13 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 49.9 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 13 must be the one taken after reading 12.
What this means: Reading 14
Illustration: Reading 14A number line with a marker at the value of Reading 14, between its lowest and highest accepted values.−10⁹10⁹Y14Example: 51.3Reading 14
What it is
Measurement number 14 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 51.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 14 must be the one taken after reading 13.
What this means: Reading 15
Illustration: Reading 15A number line with a marker at the value of Reading 15, between its lowest and highest accepted values.−10⁹10⁹Y15Example: 47.8Reading 15
What it is
Measurement number 15 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 47.8 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 15 must be the one taken after reading 14.
What this means: Reading 16
Illustration: Reading 16A number line with a marker at the value of Reading 16, between its lowest and highest accepted values.−10⁹10⁹Y16Example: 51.2Reading 16
What it is
Measurement number 16 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 51.2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 16 must be the one taken after reading 15.
What this means: Reading 17
Illustration: Reading 17A number line with a marker at the value of Reading 17, between its lowest and highest accepted values.−10⁹10⁹Y17Example: 52.6Reading 17
What it is
Measurement number 17 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 52.6 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 17 must be the one taken after reading 16.
What this means: Reading 18
Illustration: Reading 18A number line with a marker at the value of Reading 18, between its lowest and highest accepted values.−10⁹10⁹Y18Example: 52.4Reading 18
What it is
Measurement number 18 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 52.4 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 18 must be the one taken after reading 17.
What this means: Reading 19
Illustration: Reading 19A number line with a marker at the value of Reading 19, between its lowest and highest accepted values.−10⁹10⁹Y19Example: 53.6Reading 19
What it is
Measurement number 19 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 53.6 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 19 must be the one taken after reading 18.
What this means: Reading 20
Illustration: Reading 20A number line with a marker at the value of Reading 20, between its lowest and highest accepted values.−10⁹10⁹Y20Example: 52.1Reading 20
What it is
Measurement number 20 in the order the readings were taken, one value of the characteristic being charted.
Units and format
Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
Example
The form opens with 52.1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Moves the EWMA towards this value by the weighting factor; a run of high or low values carries it across a limit.
Get it right
Keep the time order: reading 20 must be the one taken after reading 19.

Method and formulas

  1. EWMAₜ = λ Yₜ + (1 − λ) EWMAₜ₋₁, starting from EWMA₀ = the target (NIST e-Handbook 6.3.2.4).
  2. Standard deviation of the EWMA s_ewma = s √(λ ÷ (2 − λ)); control limits = target ± k s_ewma (the approximation NIST gives for t not small).
  3. An EWMA above the upper or below the lower limit signals a shift in the mean.
  4. Example: limits 47.412 to 52.588; final EWMA 51.988; no signal.

Limitations

  • Uses the steady-state limits NIST gives for t not small; the first few EWMA values have a smaller spread, so the chart is slightly slow to signal right at the start.
  • The historical mean and standard deviation must come from in-control data; limits from data that contain a shift hide it.
  • Example values are illustrations. Replace them with values for your actual task.

Typical uses

  • Watch a slowly drifting dimension or temperature
  • Detect a small shift in a process mean early
  • Chart individual readings where single points are noisy

Who it is for

  • Quality engineers and technicians
  • Process engineers
  • Laboratory and calibration staff

What you get

  • EWMA after each reading
  • Upper and lower control limits
  • First reading beyond the limits

Interpreting the result

  • A small λ averages many readings and finds small shifts but reacts slowly to large ones; λ = 1 turns the chart into an ordinary individuals chart.

References

  • NIST/SEMATECH e-Handbook of Statistical Methods, 6.3.2.4 EWMA Control Charts — https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc324.htm

Learn the Concepts Behind This Tool

These KEVOS® Library readings explain the method this tool uses, with worked examples.

KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.

Open the EWMA Control Chart Check (Weighted Moving Average, Control Limits, Small Shifts) calculator

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