# X-bar and S Control Chart Limits Calculator

> Source: https://netray.co/tools/xbar-s-control-chart-limits-calculator
> Publisher: Netray (netray.co), AI and integration engineering for ERP and manufacturing systems.
> Section: Free Tools

## Short answer

Compute X-bar and S chart control limits from the grand mean, average subgroup standard deviation and subgroup size 2 to 10.

## Category

quality

## Industry

Discrete Manufacturing

## Audience

- Quality engineer
- Process engineer
- Six Sigma Green Belt

## Tool

**Kind:** calculator

**Inputs:**

#### Grand mean (X-double-bar)

**Key:** xbb

**Type:** number

**Default:** 50.02

**Step:** 0.001

**Help:** Average of the subgroup means.

#### Average subgroup standard deviation (S-bar)

**Key:** sbar

**Type:** number

**Default:** 0.42

**Min:** 0

**Step:** 0.001

**Help:** Average of the subgroup sample standard deviations.

#### Subgroup size

**Key:** n

**Type:** select

**Default:** 8

**Options:**

##### 2

**Value:** 2

##### 3

**Value:** 3

##### 4

**Value:** 4

##### 5

**Value:** 5

##### 6

**Value:** 6

##### 7

**Value:** 7

##### 8

**Value:** 8

##### 9

**Value:** 9

##### 10

**Value:** 10

**Outputs:**

#### c4

**Key:** c4

**Expr:** if(eq(n,2),0.7979,if(eq(n,3),0.8862,if(eq(n,4),0.9213,if(eq(n,5),0.9400,if(eq(n,6),0.9515,if(eq(n,7),0.9594,if(eq(n,8),0.9650,if(eq(n,9),0.9693,0.9727))))))))

**Format:** decimal

**Description:** Bias correction for the sample standard deviation.

#### Estimated process sigma (S-bar / c4)

**Key:** sigma

**Expr:** sbar / c4

**Format:** decimal

#### X-bar chart UCL

**Key:** ucl_x

**Expr:** xbb + 3 * sigma / sqrt(n)

**Format:** decimal

**Primary:** true

#### X-bar chart LCL

**Key:** lcl_x

**Expr:** xbb - 3 * sigma / sqrt(n)

**Format:** decimal

**Primary:** true

#### B4

**Key:** b4

**Expr:** 1 + 3 * sqrt(1 - c4 * c4) / c4

**Format:** decimal

#### B3

**Key:** b3

**Expr:** max(0, 1 - 3 * sqrt(1 - c4 * c4) / c4)

**Format:** decimal

#### S chart UCL

**Key:** ucl_s

**Expr:** b4 * sbar

**Format:** decimal

#### S chart LCL

**Key:** lcl_s

**Expr:** b3 * sbar

**Format:** decimal

**Chart:**

#### X-bar limits vs subgroup size

**Type:** line

**X:** n

**From:** 2

**To:** 10

**Steps:** 9

**Series:**

- ucl_x
- lcl_x

**Disclaimer:** Uses exact c4 values for n = 2 to 10. Confirm against your control chart software and your customer's SPC manual.

## Content

**Intro:** When subgroups are larger than about 8, or when you have the standard deviation recorded anyway, the X-bar and S chart is the better choice over X-bar and R. This calculator turns your grand mean and average S into both sets of limits, and shows the c4, B3 and B4 constants it used.

**Sections:**

#### How to use it

**Body:** Take at least 20 to 25 subgroups from a stable period. Calculate each subgroup mean and sample standard deviation, then average the means to get X-double-bar and average the standard deviations to get S-bar. Enter both with the subgroup size. The calculator returns the X-bar chart and S chart limits and the estimated process sigma. Plot your subgroup means and standard deviations against these limits.

**Bullets:**

- Use sample standard deviation with n-1, not population
- Keep subgroup size constant
- Recalculate limits only after a verified process change

#### Formulas and constants

**Body:** X-bar limits are X-double-bar plus or minus A3 times S-bar, where A3 = 3 / (c4 times the square root of n). S chart limits are B3 times S-bar and B4 times S-bar, with B4 = 1 + 3 sqrt(1 - c4^2) / c4 and B3 = the larger of 0 and 1 - 3 sqrt(1 - c4^2) / c4. For subgroups of 5 you should see A3 of about 1.427 and B4 of about 2.089, which matches published tables. B3 is zero for subgroups of 5 or fewer.

#### Worked example

**Body:** With the defaults, subgroup size 8 gives c4 of 0.9650, so estimated sigma is 0.42 / 0.965, about 0.435. The X-bar limits are 50.02 plus or minus 3 x 0.435 / sqrt(8), roughly 49.56 and 50.48. The S chart upper limit is about 0.42 x 1.815, near 0.76. A subgroup standard deviation above that, or a run of 8 points on one side of the centre line, is a signal to investigate.

#### Common mistakes

**Body:** Putting specification limits on a control chart is the classic error; control limits come from the process, not the drawing. Others mix subgroups from different machines, which inflates S-bar and widens the limits so nothing ever signals. Using R-bar constants with S-bar values gives wrong limits. If you do not have S recorded, the X-bar and R chart is simpler for subgroups below 6.

**Cta:** If your measurement data sits in an ERP quality module or a database, we can build a private SPC view that applies these limits automatically.

**Faq:**

#### When should I use X-bar and S instead of X-bar and R?

**Answer:** Use S when subgroups are larger than about 8 to 10, because the range uses only two of the readings and loses efficiency. For small subgroups the range chart is almost as good and easier to compute by hand.

#### Why is B3 zero for small subgroups?

**Answer:** The calculated lower limit goes negative for subgroups of 5 or fewer, and a standard deviation cannot be negative, so the lower limit is set to zero.

#### Can I use this with varying subgroup sizes?

**Answer:** Not directly. Variable subgroup sizes need limits recalculated for each subgroup, or a standardised chart. This tool assumes a constant n.

#### What should I do when a point is outside the limits?

**Answer:** Treat it as a signal of special cause. Check the measurement, the setup and recent changes, record the cause, and exclude the point from limit calculation only if you have found and corrected the cause.

## Search terms

xbar s chart calculator, x-bar s control chart limits, control limits calculator subgroup standard deviation, c4 A3 B3 B4 constants, xbar s chart formula, when to use xbar s instead of xbar r, UCL LCL calculator

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