Forty readings from a process that moves 2.2 units at point 26 — more than two sigma. Not one point crosses the three-sigma limits, because the shift is inside the data the limits were computed from: it raises the centre to 100.9807 and would have raised the sigma too if the moving range could see it. Twenty-one points are flagged by rules 2, 5 and 6 — runs on one side of the centre, which is exactly what a sustained shift looks like when the limits have been widened by the shift itself.
40 readings · centre 100.9807
Limits 97.4079 to 104.5536
Sigma from the average moving range is 1.1910; the standard deviation of the whole series is 1.4983 — 25.8% larger, which means something changed between readings rather than within them. 0 readings cross the limits and 21 break at least one of the eight rules.
Centre
100.9807
limits ±3.5729
Sigma (moving range)
1.1910
MR-bar 1.3438 ÷ 1.128379
Sigma (whole series)
1.4983
25.8% larger
Readings flagged
21 of 40
0 beyond the limits
Which rules fired
How many readings each Nelson rule flagged
Rule
Readings
What it detects
2
16
nine points in a row on the same side of the centre
5
2
two of three points beyond two sigma on the same side
6
11
four of five points beyond one sigma on the same side
Rule 1 did not fire: nothing is outside the limits, and everything flagged here was caught by a pattern rather than by a limit.
Every reading
Each reading with its moving range and any rules it breaks
#
Value
Moving range
Sigmas from centre
Rules
1
99.3800
—
-1.34
—
2
100.7900
1.4100
-0.16
—
3
100.4100
0.3800
-0.48
—
4
101.9000
1.4900
+0.77
—
5
99.0100
2.8900
-1.65
—
6
101.6400
2.6300
+0.55
—
7
99.8200
1.8200
-0.97
—
8
99.4200
0.4000
-1.31
6
9
99.1100
0.3100
-1.57
6
10
101.2500
2.1400
+0.23
—
11
99.4100
1.8400
-1.32
6
12
99.2500
0.1600
-1.45
6
13
99.7600
0.5100
-1.03
6
14
101.5500
1.7900
+0.48
—
15
99.9400
1.6100
-0.87
—
16
98.2200
1.7200
-2.32
—
17
100.3300
2.1100
-0.55
—
18
101.5400
1.2100
+0.47
—
19
99.4300
2.1100
-1.30
—
20
101.3200
1.8900
+0.28
—
21
100.4800
0.8400
-0.42
—
22
100.0200
0.4600
-0.81
—
23
98.5600
1.4600
-2.03
—
24
99.7800
1.2200
-1.01
—
25
101.5800
1.8000
+0.50
2
26
101.2900
0.2900
+0.26
2
27
101.4000
0.1100
+0.35
2
28
101.1800
0.2200
+0.17
2
29
103.7900
2.6100
+2.36
2, 5
30
101.3200
2.4700
+0.28
2
31
103.5500
2.2300
+2.16
2, 5, 6
32
102.1400
1.4100
+0.97
2
33
103.0200
0.8800
+1.71
2, 6
34
103.7700
0.7500
+2.34
2, 6
35
103.1400
0.6300
+1.81
2, 6
36
101.3900
1.7500
+0.34
2
37
102.6900
1.3000
+1.44
2, 6
38
103.6200
0.9300
+2.22
2, 6
39
102.0400
1.5800
+0.89
2
40
100.9900
1.0500
+0.01
2
The moving range chart has its own limit at 4.3897, and 0 of 39 moving ranges exceed it. A moving range out of control means the sigma estimate underneath the whole chart is not trustworthy, so it is worth reading first.
No subgroups needed Sigma from the moving range All eight rules Slow to see a small shift
What this tool shows
The shipped preset shifts by 2.2 units — more than two sigma — and not one reading crosses the three-sigma limits. The shift is inside the data those limits were computed from, so it raised the centre and would have widened the limits too if the moving range could see it. Twenty-one of forty readings are flagged by the run rules instead, which is what a sustained shift looks like on a chart it has already contaminated.
Individuals and moving range charts from a single stream of readings
Sigma from the average moving range over d₂(2), never from the standard deviation of the series
Both sigma estimates printed, because their gap is the diagnosis
All eight Nelson rules, with a count of which ones fired
The moving range chart with its own limit, which validates the sigma underneath everything
A preset where a single spike widens the limits by 36%
No subgroups needed Moving-range sigma Eight rules counted Both sigmas shown
Sigma from the moving range. Using the series standard deviation draws limits the problem widened.
Updated 13 September 2026 · Works in any browser, no installation
An individuals chart is a control chart for processes that produce one measurement at a time. There are no subgroups, so the yardstick has to come from somewhere else: the moving range between consecutive readings, which captures short-term variation and nothing else. That is the whole trick, and it is why sigma here is the average moving range divided by 1.128379 rather than the standard deviation of the series — a series that has shifted has a large standard deviation and a small moving range.
At a glance
Formula shown
Sigma = MR̄ / d₂(2) with d₂(2) = 2/√π = 1.128379, the expected range of two standard normals. Limits are x̄ ± 3σ, and the moving range chart runs from 0 to D₄·MR̄ with D₄ = 3.2665 at n = 2. The choice of MR̄ over the sample standard deviation is not a convention: a sustained shift inflates the second and leaves the first alone, which is exactly the property that lets the limits detect it.
Scenario support
Batch processes where each batch yields one measurement, chemical assays, monthly financial or operational figures, destructive testing, environmental readings on a single instrument, and any process where subgrouping is impossible or meaningless.
Educational estimate
Planning support from the values you enter — not professional advice.
A two-sigma shift that crosses nothing
The first preset is built so the chart fails at the thing it is usually trusted for, and the numbers say why.
The process moves 2.2 units at reading 26, in a process whose sigma is about 1.19. That is nearly a two-sigma step.
Not one reading crosses a limit. The shift is in the data the limits were built from: it pulled the centre up to 100.9807, so the post-shift readings sit near the middle of the chart rather than at the top of it.
Twenty-one readings are flagged by rules 2, 5 and 6. Nine in a row on one side, four of five beyond one sigma — the patterns that exist precisely because the limits cannot see this.
Which is the case for using the run rules on an individuals chart in particular. With no subgrouping to average over, a single reading is noisy and a limit is a blunt detector; the run rules are most of the sensitivity this chart has.
Why not the standard deviation
The most common way to get an individuals chart wrong is to compute sigma the obvious way, and it fails in the direction that hides problems.
On the shifted preset, sigma from the moving range is 1.1910 and the standard deviation of the series is 1.4983. 25.8% larger.
The standard deviation contains the shift. The moving range does not. A moving range only ever compares two consecutive readings, so a step that happens once contributes one large moving range and nothing else.
Limits built on the larger figure would be 25.8% wider, which is the process using its own instability to excuse itself.
On the stable preset the two agree to within 1.8%, which is what a process with nothing happening looks like — and is why both are printed here rather than only the one used.
One spike widens the limits by 36%
The moving-range estimate is robust to a sustained shift and not robust to a single outlier, which is the opposite failure and worth knowing separately.
The third preset is the stable series with one reading changed from 50.00 to 59.00. One value.
It creates two large moving ranges, not one, since the spike differs from both its neighbours. The average moving range goes from 1.6355 to 2.2210 — up 35.8%.
The limits widen from ±4.3483 to ±5.9050. The spike itself is still caught, at 4.67 sigma, but the chart is now 36% less sensitive to everything else.
The standard fix is a median moving range. On this data it gives sigma 1.3732 against 1.9683 from the mean, and it is the right default when a chart is being set up on historical data that has not been cleaned.
Read the moving range chart first
The moving range chart is usually treated as an afterthought. It validates the number the individuals chart is built on, so it comes first.
Its centre is MR̄ and its upper limit is 3.2665 times that. A moving range above the limit means two consecutive readings were further apart than the process should allow.
If the moving range chart is out of control, the sigma estimate is not trustworthy, and neither are the individuals limits derived from it.
There is no lower limit at n = 2. D₃ is zero, so a run of identical readings produces no signal here — which is a real blind spot on rounded or low-resolution data.
Consecutive moving ranges share a reading, so they are correlated. That is why the run rules are not applied to the moving range chart on this page: a run there can be an artefact of the overlap rather than a signal.
Normality matters more here than anywhere else
An X-bar chart averages within subgroups, and averages are close to normal whatever the underlying data does. This chart has no such protection.
Each point is a single reading, so the three-sigma limits inherit the shape of the process distribution directly.
On a skewed process the false-alarm rate is asymmetric: far more signals above the upper limit than below the lower one, and neither at the nominal rate.
Bounded or heavily skewed data — times, counts, concentrations — usually wants a transform. Charting the logarithm is the common route and is often enough.
Check the distribution before charting, with a normality test or simply a histogram; the chart itself will not tell you that its limits are the wrong shape.
It is slow, and the alternatives are not
An individuals chart is the least sensitive Shewhart chart there is, and for small sustained shifts there are better tools.
With no subgrouping there is no averaging, so each point carries the full process noise and a one-sigma shift is invisible against it.
An EWMA chart carries the past forward and detects a small shift in a handful of points where a Shewhart chart takes hundreds.
A CUSUM does the same by accumulating deviations from a target, and is the better choice when the shift size you care about is known in advance.
What the individuals chart keeps is interpretability. Every point is a reading someone took, in the units they took it in — which matters on a shop floor in a way an exponentially weighted statistic does not.
Reporting an individuals chart
Four items, and the first is the one that decides whether the limits mean anything.
Say how sigma was estimated. Average moving range, median moving range or sample standard deviation give materially different limits on the same data.
Give the number of readings the limits came from. Below about twenty-five they are unstable; below fifteen they are barely worth drawing.
Say which rules were applied. On this chart the run rules do most of the detecting, so leaving them out changes the result far more than it would on an X-bar chart.
And say whether the data was transformed. Charting logarithms is normal practice on skewed data and has to be stated, because the limits are then in log units.
Sources and methodology
References for individuals charts and their limits.
Method. Sigma comes from the average moving range divided by d₂(2), and that constant is computed rather than hard-coded: it is the expected range of two standard normals, which has the closed form 2/√π = 1.128379 and which the suite checks against both the closed form and the published table. The standard deviation of the whole series is computed alongside and printed, because the gap between the two is the diagnosis rather than a curiosity — the suite asserts on sixty generated series that a sustained shift always inflates the second above the first. The eight Nelson rules are applied to the individuals chart and only rule 1 to the moving range chart, since consecutive moving ranges share a reading and a run among them can be an artefact of that overlap. A series with no movement at all returns no result rather than a chart with zero-width limits that could never signal. The suite also asserts that sigma is exactly MR̄/d₂(2), that the limits sit exactly three sigma either side, and that the moving range chart carries one fewer point than the series. That engine is verified on every change against 139 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
X-bar and R ChartX-bar with R or S charts, every Shewhart constant computed from its definition rather than looked up, and all eight Nelson rules.
EWMA ChartEWMA and CUSUM computed on the same readings with a Shewhart chart beside them, a lambda sensitivity table and both statistics per reading.
p Chartp and np charts with limits computed per point when the sample sizes differ, and the cost of flat limits measured on your own chart.
Normality TestShapiro-Wilk, Anderson-Darling and Jarque-Bera with a Q-Q plot, plus a resampled sweep answering the question the tests cannot: was your sample size big enough to detect anything?
CpkCp, Cpk, Pp and Ppk with the defect rates they predict and the rate actually observed — including the built-in case where Cp is 2.05, Cpk is 0.57 and a quarter of the sample is already out of spec.
Moving AverageSimple, exponential, weighted and centred moving averages with the lag each one carries — exactly (n−1)/2 periods for a simple average, which is also the reason α = 2/(n+1) is in every EMA formula.
An educational tool. An individuals chart has no subgrouping to average over, so its three-sigma limits inherit the shape of the process distribution directly and behave badly on skewed data. Its sigma estimate is robust to a sustained shift and not to a single outlier — one spike widens the limits by 36% on the shipped preset — and it is the least sensitive Shewhart chart to a small sustained change.