Standard deviation, and the other standard deviation

The usual summary statistics, and the thing a spreadsheet will not tell you: the within-subgroup sigma a control chart uses, beside the overall sigma a textbook means. The gap between them is why Cpk and Ppk disagree.

Nothing you paste leaves this calculation. It is not stored, not logged and not sent anywhere else — the numbers are computed and returned, and that is all.

Why your standard deviation disagrees with somebody else's

Two reasons, and they are different problems.

1. n or n−1

The sample standard deviation divides by n−1; the population one divides by n. Excel offers both — STDEV.S and STDEV.P — and which one a calculator picked by default is the usual reason two people get different numbers from identical data.

Use the sample version, essentially always. Your readings are a sample from a process that will carry on producing parts, not the entire population of interest, and dividing by n systematically understates the spread. The gap matters most exactly where it is easiest to overlook: at n = 10 it is about 5%, at n = 100 it is 0.5%.

2. Overall or within-subgroup

This one is bigger, and it is the reason two capability numbers from the same data can disagree substantially.

Overall sigma is the plain standard deviation of every reading, treating them as one pot. It includes everything that happened while you were measuring — every drift, every setup change, every shift handover.
Within-subgroup sigma is estimated from short-term variation only: R̄/d₂ across subgroups, or MR̄/d₂ from consecutive readings when you have individuals. It deliberately excludes movement between subgroups.

A control chart uses the within-subgroup estimate, on purpose — it is asking whether the process centre has moved relative to its own short-term noise, and computing the limits from the long-term spread would fold the very shift you are hunting into the limits that are supposed to detect it.

Cp and Cpk use within-subgroup sigma; Pp and Ppk use overall. So when the two sigmas above differ, so will the two capability figures, and the size of the gap measures how much the process moved during your study. There is a whole guide on reading that gap, and the capability calculator reports both side by side for the same reason.

The mean and the median together

When they sit on top of each other, the data is roughly symmetric. When the mean is pulled away from the median, something is dragging it — either genuine skew, or one or two extreme readings.

That difference is worth acting on, because capability indices are built on the mean. A distribution skewed enough to separate the two will not give a Cpk that means what a Cpk normally means, and the resulting parts-per-million figure will be wrong on the skewed side. Look at the shape before quoting either — the histogram is the honest check.

The confidence interval on the mean

Twenty readings do not pin a mean down as precisely as their decimals suggest, and the interval above says by how much. This matters when you are comparing two processes, two machines or two suppliers: if their confidence intervals overlap comfortably, the difference between the sample means is not yet evidence of a difference between the processes.

And a caution that applies to every number on this page: none of them means anything about the future unless the process was stable while you measured. Summary statistics of an out-of-control process describe a blend of several processes and predict nothing. Chart it first.

Questions people ask about this

Should I divide by n or n minus 1?

n−1, essentially always. Your readings are a sample from a process that will carry on producing parts, not the whole population, and dividing by n systematically understates the spread. Excel calls them STDEV.S and STDEV.P, and which one a calculator picked by default is the usual reason two people get different answers from identical data.

Why does my control chart use a different standard deviation?

Because it estimates sigma from short-term variation only — R-bar over d2 across subgroups, or the mean moving range over d2 for individuals — deliberately excluding movement between subgroups. Computing limits from the long-term spread would fold the very shift you are hunting into the limits meant to detect it.

Which sigma does Cpk use?

The within-subgroup one. Pp and Ppk use the overall standard deviation instead, which is why the two disagree whenever the process moved during your study — and the size of that gap is a measure of how much it moved.

What does the coefficient of variation tell me?

The standard deviation as a percentage of the mean, which makes spread comparable across characteristics measured in different units or at very different magnitudes. It is meaningless when the mean is near zero, and misleading on a scale with an arbitrary origin such as degrees Celsius.

Want this to keep itself up to date?

A calculator answers for the data you pasted. A control chart answers for the data your line produced this morning — limits frozen at a baseline you locked, rules evaluated on every new reading, an alert when one trips.

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More analysis tools

Grubbs outlier test — The G statistic against its critical value, the p-value, and the point implicated — plus a straight answer about what to do next, which is almost never to delete the reading.
Scatter diagram and correlation — Paired readings plotted, with Pearson r, Spearman rho, r², the least-squares line and its p-value — and the two reported together, because when they disagree the disagreement is the finding.

Everything else

Cp / Cpk calculator — Paste measurements — or type a mean and a sigma — with your tolerance, and get Cp, Cpk, Pp, Ppk, the sigma level and the expected parts per million out of spec.
Control limit calculator — Give it subgroups or individual readings and it returns the control limits for the chart, the range or sigma chart beneath it, and every constant it used to get there.
Control chart generator — Paste a column of numbers, or rows of subgroups, and get a real control chart: limits from the data, Nelson rules 1–4 evaluated, out-of-control points marked.
Nelson & Western Electric rules checker — Every run rule evaluated on your data, each violation named in plain English with what it usually indicates — a shift, a trend, tool wear, two machines mixed.
Cpk confidence interval & sample size — A Cpk of 1.33 from 30 pieces has a 95% interval of roughly 0.97 to 1.69. See the uncertainty in your own number, and how many parts would settle it.
Cpk ↔ PPM ↔ sigma level converter — What PPM is a Cpk of 1.33? What Cpk does 3.4 PPM need? Both conventions shown side by side, because the 1.5 sigma shift is why two sources disagree by a factor of ten.
Control chart constants — The whole table, n = 2 to 25, with the formula each constant belongs to. The same values the charts on this site are computed from.
X̄-R chart calculator — Subgroups in, X̄ and R charts out — limits from A₂, D₃ and D₄, run rules evaluated, and the range chart shown first because it decides whether the averages chart can be trusted.
X̄-s chart calculator — For subgroups big enough that the range wastes them. Limits from A₃, B₃ and B₄, sigma recovered with the c₄ correction, and the s chart read first because it decides whether the averages chart can be trusted.
I-MR chart calculator — For processes that give you one number at a time — a batch, an oven, a destructive test. Limits from the mean moving range, run rules evaluated, and an honest note about what an individuals chart cannot see.
Levey-Jennings chart with Westgard rules — Paste your QC results with the assigned mean and SD from the package insert, and get the chart with every sigma band drawn and all six Westgard rules evaluated — each one named, and labelled random or systematic error.
Bland-Altman plot — Paired readings from two methods, plotted as difference against average — with the bias, the limits of agreement, their confidence intervals, and a test for whether the disagreement depends on the magnitude.
Pareto chart generator — Categories and counts in, ranked bars and the cumulative line out — with the vital few named, an "Other" bucket that always sorts last, and an honest warning when the distribution is flat and there is no dominant cause to attack.
Box and whisker plot generator — One column per group, boxes side by side. Quartiles, the 1.5×IQR fences, whiskers that stop at real readings and outliers drawn individually — with the quartile method stated, because that is why your plot and Excel's disagree.
Histogram and normality test — A histogram with the fitted normal curve, both standard bin rules with a reasoned recommendation, and an Anderson-Darling test that refuses to tell you your data is normal — because no test can.
Gage R&R calculator (ANOVA) — Paste a crossed study — part, operator, reading — and get the full ANOVA: repeatability and reproducibility separated, the part-by-operator interaction tested rather than assumed away, %GRR, %Tolerance and ndc against AIAG's bands.
p, np, u and c chart calculator — Counted data rather than measured: proportion defective, number defective, defects per unit. Limits that step with the sample size instead of pretending it never changed, and an honest warning when the counts are too low for three-sigma limits to mean anything.