# Individuals and Moving Range (I-MR) Chart Limits Calculator

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

## Short answer

Calculate X chart and moving range limits when you can only measure one value at a time, with the 2.66 and 3.267 constants applied.

## Category

quality

## Industry

Discrete Manufacturing

## Audience

- Quality engineer
- Process engineer
- Six Sigma Green Belt

## Tool

**Kind:** calculator

**Inputs:**

#### Mean of individual values

**Key:** xbar

**Type:** number

**Default:** 25.4

**Step:** 0.001

#### Average moving range (MR-bar)

**Key:** mrbar

**Type:** number

**Default:** 0.18

**Min:** 0

**Step:** 0.001

**Help:** Average of absolute differences between consecutive readings.

#### Sigma limit multiplier

**Key:** k

**Type:** select

**Default:** 3

**Options:**

##### 3 sigma (standard)

**Value:** 3

##### 2 sigma (warning)

**Value:** 2

**Outputs:**

#### Estimated sigma (MR-bar / 1.128)

**Key:** sigma

**Expr:** mrbar / 1.128

**Format:** decimal

**Description:** d2 for a moving range of 2 is 1.128.

#### Individuals UCL

**Key:** ucl

**Expr:** xbar + k * sigma

**Format:** decimal

**Primary:** true

#### Individuals LCL

**Key:** lcl

**Expr:** xbar - k * sigma

**Format:** decimal

**Primary:** true

#### Moving range UCL (3.267 x MR-bar)

**Key:** ucl_mr

**Expr:** 3.267 * mrbar

**Format:** decimal

#### Zone A upper boundary (+2 sigma)

**Key:** zone_a

**Expr:** xbar + 2 * sigma

**Format:** decimal

#### Zone B upper boundary (+1 sigma)

**Key:** zone_b

**Expr:** xbar + sigma

**Format:** decimal

**Chart:**

#### Individuals limits as MR-bar grows

**Type:** line

**X:** mrbar

**From:** 0.05

**To:** 0.5

**Steps:** 20

**Series:**

- ucl
- lcl

**Disclaimer:** Assumes moving ranges of 2 consecutive points and a stable process. Highly autocorrelated or very non-normal data may need other methods.

## Content

**Intro:** When each reading is a single value, such as a daily batch result, a tank assay or one part per hour, you cannot form subgroups. The individuals and moving range chart is the standard answer. Enter the mean and average moving range to get the limits and zone boundaries.

**Sections:**

#### How to use it

**Body:** List at least 20 to 25 consecutive individual readings in time order. Compute the mean. Then compute the moving range for each pair of consecutive readings, the absolute difference, and average them to get MR-bar. Enter both numbers. The calculator gives limits for the individuals chart and the moving range chart, plus the plus-one and plus-two sigma boundaries you need for run rules.

**Bullets:**

- Keep data in time order
- Do not sort or group the readings
- Remove only special-cause points you have explained

#### Method and constants

**Body:** Sigma is estimated as MR-bar divided by 1.128 (the d2 constant for n = 2). The individuals limits are the mean plus or minus 3 x MR-bar / 1.128, which is the familiar 2.66 x MR-bar. The moving range chart has an upper limit of 3.267 x MR-bar and no lower limit. Estimating sigma from short-term moving ranges rather than the overall standard deviation keeps the limits sensitive to real shifts.

#### Worked example

**Body:** With a mean of 25.400 and MR-bar of 0.180, sigma is about 0.160, the individuals limits are about 24.92 and 25.88, and the upper moving range limit is 0.588. A reading of 26.0 is a signal, as is a run of eight points above 25.400. The chart sweeps MR-bar to show how noisier measurement widens the limits and hides real shifts, which is a good argument for improving gauge repeatability.

#### Common mistakes

**Body:** Using the overall standard deviation of all readings to set the limits inflates them when the mean has drifted, so shifts go undetected. Another mistake is applying the chart to data that is naturally cyclic or autocorrelated, such as temperature drifting through a shift, which causes false alarms. A moving range chart of non-normal data will also have an uneven shape. If the process is slow and the data skewed, consider transforming the data.

**Cta:** If you capture single readings in an ERP or historian, we can build a private chart view with these rules applied and alerts for signals.

**Faq:**

#### Why is the constant 2.66?

**Answer:** It is 3 divided by 1.128. Three-sigma limits use sigma estimated as MR-bar / 1.128, so the limits are the mean plus or minus 2.66 times the average moving range.

#### How many points do I need?

**Answer:** Aim for at least 20 to 25 points from a period when the process was operating normally. Fewer points give unstable limits, and they should be recalculated as data accumulates.

#### Does the moving range chart need a lower limit?

**Answer:** No. With a moving range of 2 the lower limit is zero, so only the upper limit of 3.267 times MR-bar is drawn.

#### Is the I-MR chart robust to non-normal data?

**Answer:** Reasonably, but very skewed data produces misleading limits and false signals. Check a histogram, and consider a transformation or a method suited to the distribution.

## Search terms

i-mr chart calculator, individuals chart control limits, moving range chart calculator, 2.66 moving range constant, xmr chart limits, control chart for individual measurements, mrbar 1.128 sigma

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