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CUSUM Control Chart Check (Tabular Upper and Lower Sums, Decision Interval, First Signal)

Has my process mean shifted away from target? Using the tabular cumulative sum (CUSUM) method on these readings, do the upper or lower sums pass the decision interval, and when?

Tabular CUSUM control chart from up to twenty readings or subgroup means in time order: the reference value from the shift you want to detect, the upper and lower cumulative sums, the decision interval you set, and the first reading at which either sum exceeds it, with the direction of the shift.

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How this tool works

Inputs, units and where to find them

InputUnitTypical source
Target (in-control mean)Mean the process should run at when in control.
Standard deviation of the plotted valuesStandard deviation of the values being plotted (of subgroup means if you plot means).
Shift to detectstandard deviationsSize of shift you want to detect quickly, in standard deviations of the plotted values.
Decision interval hDecision interval in the same units as the readings; a sum above it signals.
Number of readings usedreadingsHow many of the values below to use, in time order.
Readings or subgroup means in time order
Value 1Reading or subgroup mean number 1 in time order.
Value 2Reading or subgroup mean number 2 in time order.
Value 3Reading or subgroup mean number 3 in time order.
Value 4Reading or subgroup mean number 4 in time order.
Value 5Reading or subgroup mean number 5 in time order.
Value 6Reading or subgroup mean number 6 in time order.
Value 7Reading or subgroup mean number 7 in time order.
Value 8Reading or subgroup mean number 8 in time order.
Value 9Reading or subgroup mean number 9 in time order.
Value 10Reading or subgroup mean number 10 in time order.
Value 11Reading or subgroup mean number 11 in time order.
Value 12Reading or subgroup mean number 12 in time order.
Value 13Reading or subgroup mean number 13 in time order.
Value 14Reading or subgroup mean number 14 in time order.
Value 15Reading or subgroup mean number 15 in time order.
Value 16Reading or subgroup mean number 16 in time order.
Value 17Reading or subgroup mean number 17 in time order.
Value 18Reading or subgroup mean number 18 in time order.
Value 19Reading or subgroup mean number 19 in time order.
Value 20Reading or subgroup mean number 20 in time order.

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 (in-control mean)
Illustration: Target (in-control mean)A number line with a marker at the value of Target (in-control mean), between its lowest and highest accepted values.−10⁹10⁹μ₀Example: 325Target (in-control mean)
What it is
The mean the process should run at when in control, the value the cumulative sums measure departures from.
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 325 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Readings above it build the upper sum and readings below it build the lower sum.
Get it right
Use the target or the mean of a stable historical period.
What this means: Standard deviation of the plotted values
Illustration: Standard deviation of the plotted valuesA nominal size line with an upper and a lower limit around it; parts inside the band are acceptable. The figure illustrates Standard deviation of the plotted values.upperlowerσExample: 0.635Standard deviation of the plotted values
What it is
The standard deviation of the values being plotted: of single readings, or of subgroup means if you plot means.
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 0.635 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Sets the reference value k with the shift to detect; a larger value lets bigger departures pass without adding to the sums.
Get it right
For subgroup means it is the process standard deviation divided by the square root of the subgroup size.
What this means: Shift to detect
Illustration: Shift to detectTwo bars: a base value and the same value multiplied by Shift to detect, the factor.base valuewith factorδExample: 1 standard deviationsShift to detect
What it is
The size of the shift in the mean you want to detect quickly, in standard deviations of the plotted values.
Units and format
Enter a number in standard deviations: standard deviations. The form accepts 0.1 standard deviations to 10 standard deviations. Required: the tool does not calculate until every input is valid.
Example
The form opens with 1 standard deviations as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Raises the reference value k, so the sums grow only for larger departures.
Get it right
One standard deviation is a common choice; CUSUM charts are best at shifts of about 0.5 to 2.
What this means: Decision interval h
Illustration: Decision interval hA nominal size line with an upper and a lower limit around it; parts inside the band are acceptable. The figure illustrates Decision interval h.upperlowerhExample: 4.1959Decision interval h
What it is
The decision interval: a sum above this value signals that the process mean has shifted, in the same units as the readings.
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 4.1959 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A larger interval gives fewer false alarms but later signals.
Get it right
Choose it with the shift to detect from your chart design or the average run length tables, not by trial on the data.
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 values listed below are used, counting from the first 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
Values beyond this count are ignored.
Get it right
Enter values from the oldest; unused fields at the end are ignored.

Readings or subgroup means in time order

What this means: Value 1
Illustration: Value 1A number line with a marker at the value of Value 1, between its lowest and highest accepted values.−10⁹10⁹x1Example: 324.93Value 1
What it is
Reading or subgroup mean number 1 in time order, in the same units as the target.
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 324.93 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Enter the first value taken; the order matters because each new value builds on the ones before.
What this means: Value 2
Illustration: Value 2A number line with a marker at the value of Value 2, between its lowest and highest accepted values.−10⁹10⁹x2Example: 324.68Value 2
What it is
Reading or subgroup mean number 2 in time order, in the same units as the target.
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 324.68 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 2 must be the one taken after value 1.
What this means: Value 3
Illustration: Value 3A number line with a marker at the value of Value 3, between its lowest and highest accepted values.−10⁹10⁹x3Example: 324.73Value 3
What it is
Reading or subgroup mean number 3 in time order, in the same units as the target.
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 324.73 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 3 must be the one taken after value 2.
What this means: Value 4
Illustration: Value 4A number line with a marker at the value of Value 4, between its lowest and highest accepted values.−10⁹10⁹x4Example: 324.35Value 4
What it is
Reading or subgroup mean number 4 in time order, in the same units as the target.
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 324.35 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 4 must be the one taken after value 3.
What this means: Value 5
Illustration: Value 5A number line with a marker at the value of Value 5, between its lowest and highest accepted values.−10⁹10⁹x5Example: 325.35Value 5
What it is
Reading or subgroup mean number 5 in time order, in the same units as the target.
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 325.35 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 5 must be the one taken after value 4.
What this means: Value 6
Illustration: Value 6A number line with a marker at the value of Value 6, between its lowest and highest accepted values.−10⁹10⁹x6Example: 325.23Value 6
What it is
Reading or subgroup mean number 6 in time order, in the same units as the target.
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 325.23 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 6 must be the one taken after value 5.
What this means: Value 7
Illustration: Value 7A number line with a marker at the value of Value 7, between its lowest and highest accepted values.−10⁹10⁹x7Example: 324.13Value 7
What it is
Reading or subgroup mean number 7 in time order, in the same units as the target.
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 324.13 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 7 must be the one taken after value 6.
What this means: Value 8
Illustration: Value 8A number line with a marker at the value of Value 8, between its lowest and highest accepted values.−10⁹10⁹x8Example: 324.53Value 8
What it is
Reading or subgroup mean number 8 in time order, in the same units as the target.
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 324.53 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 8 must be the one taken after value 7.
What this means: Value 9
Illustration: Value 9A number line with a marker at the value of Value 9, between its lowest and highest accepted values.−10⁹10⁹x9Example: 325.23Value 9
What it is
Reading or subgroup mean number 9 in time order, in the same units as the target.
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 325.23 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 9 must be the one taken after value 8.
What this means: Value 10
Illustration: Value 10A number line with a marker at the value of Value 10, between its lowest and highest accepted values.−10⁹10⁹x10Example: 324.6Value 10
What it is
Reading or subgroup mean number 10 in time order, in the same units as the target.
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 324.6 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 10 must be the one taken after value 9.
What this means: Value 11
Illustration: Value 11A number line with a marker at the value of Value 11, between its lowest and highest accepted values.−10⁹10⁹x11Example: 324.63Value 11
What it is
Reading or subgroup mean number 11 in time order, in the same units as the target.
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 324.63 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 11 must be the one taken after value 10.
What this means: Value 12
Illustration: Value 12A number line with a marker at the value of Value 12, between its lowest and highest accepted values.−10⁹10⁹x12Example: 325.15Value 12
What it is
Reading or subgroup mean number 12 in time order, in the same units as the target.
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 325.15 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 12 must be the one taken after value 11.
What this means: Value 13
Illustration: Value 13A number line with a marker at the value of Value 13, between its lowest and highest accepted values.−10⁹10⁹x13Example: 328.33Value 13
What it is
Reading or subgroup mean number 13 in time order, in the same units as the target.
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 328.33 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 13 must be the one taken after value 12.
What this means: Value 14
Illustration: Value 14A number line with a marker at the value of Value 14, between its lowest and highest accepted values.−10⁹10⁹x14Example: 327.25Value 14
What it is
Reading or subgroup mean number 14 in time order, in the same units as the target.
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 327.25 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 14 must be the one taken after value 13.
What this means: Value 15
Illustration: Value 15A number line with a marker at the value of Value 15, between its lowest and highest accepted values.−10⁹10⁹x15Example: 327.83Value 15
What it is
Reading or subgroup mean number 15 in time order, in the same units as the target.
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 327.83 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 15 must be the one taken after value 14.
What this means: Value 16
Illustration: Value 16A number line with a marker at the value of Value 16, between its lowest and highest accepted values.−10⁹10⁹x16Example: 328.5Value 16
What it is
Reading or subgroup mean number 16 in time order, in the same units as the target.
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 328.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 16 must be the one taken after value 15.
What this means: Value 17
Illustration: Value 17A number line with a marker at the value of Value 17, between its lowest and highest accepted values.−10⁹10⁹x17Example: 326.68Value 17
What it is
Reading or subgroup mean number 17 in time order, in the same units as the target.
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 326.68 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 17 must be the one taken after value 16.
What this means: Value 18
Illustration: Value 18A number line with a marker at the value of Value 18, between its lowest and highest accepted values.−10⁹10⁹x18Example: 327.78Value 18
What it is
Reading or subgroup mean number 18 in time order, in the same units as the target.
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 327.78 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 18 must be the one taken after value 17.
What this means: Value 19
Illustration: Value 19A number line with a marker at the value of Value 19, between its lowest and highest accepted values.−10⁹10⁹x19Example: 326.88Value 19
What it is
Reading or subgroup mean number 19 in time order, in the same units as the target.
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 326.88 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 19 must be the one taken after value 18.
What this means: Value 20
Illustration: Value 20A number line with a marker at the value of Value 20, between its lowest and highest accepted values.−10⁹10⁹x20Example: 328.35Value 20
What it is
Reading or subgroup mean number 20 in time order, in the same units as the target.
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 328.35 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A value above the target plus k adds to the upper sum; one below the target minus k adds to the lower sum.
Get it right
Keep the time order: value 20 must be the one taken after value 19.

Method and formulas

  1. Reference value k = δ σ ÷ 2 (half the shift to detect, in data units).
  2. S_hi(i) = max(0, S_hi(i − 1) + xᵢ − μ₀ − k); S_lo(i) = max(0, S_lo(i − 1) + μ₀ − k − xᵢ), both starting at 0 (NIST e-Handbook 6.3.2.3, tabular form).
  3. The process is out of control when either sum exceeds the decision interval h.
  4. Example: k = 0.3175; first signal at reading 14; final upper sum 19.06.

Limitations

  • The decision interval h sets the false alarm rate and the speed of detection: take it from your chart design or the average run length tables (NIST 6.3.2.3.1).
  • Readings must be independent; autocorrelated data, common in continuous processes, give false signals.
  • Example values are illustrations. Replace them with values for your actual task.

Typical uses

  • Detect a small, sustained shift in a process mean
  • Find roughly when a process changed
  • Watch a filling weight or dimension against its target

Who it is for

  • Quality engineers and technicians
  • Process engineers
  • Production supervisors

What you get

  • Reference value k
  • Upper and lower cumulative sums after each reading
  • First signal and its direction

Interpreting the result

  • After a signal, the reading where the sum last left zero shows roughly when the shift began.

References

  • NIST/SEMATECH e-Handbook of Statistical Methods, 6.3.2.3 Cusum Control Charts — https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc323.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 CUSUM Control Chart Check (Tabular Upper and Lower Sums, Decision Interval, First Signal) calculator

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