ΣSumCalculator
CONTROL CHART TOOL

R Chart UCL Calculator

Turn subgroup ranges into a clear upper control limit.

Three-sigma R chartCustom subgroup sizeRuns in your browser

01 / INPUT

Your subgroups

Use the same number of observations in each subgroup. Enter maximum/minimum pairs or ranges, or paste a written problem and let AI suggest the data.

Input type

Enter any whole-number n from 2 to 1,000,000. For n above 10, factors are calculated numerically; an s chart is usually preferable for larger subgroups.

One nonnegative range per line. Up to 100 subgroups.

02 / RESULT

Range chart limits

LIVE RESULT
UPPER CONTROL LIMIT · UCL
0.614 decimal places: 0.6058
Average range · R̄0.325
Lower limit · LCL0.044 decimal places: 0.0442
D₄ factor1.864
Subgroups4

UCL = R̄ × D₄ = 0.325 × 1.864 = 0.6058 ≈ 0.61

Regular calculation stays in this browser. AI extraction is optional.

YOUR CALCULATION

How this answer was calculated

  1. Find each subgroup range

    Use the 4 ranges you entered, one per subgroup.

    Show every subgroup range
    1. R1 = 0.3
    2. R2 = 0.4
    3. R3 = 0.2
    4. R4 = 0.4

  2. Find the average range

    R̄ = (0.3 + 0.4 + 0.2 + 0.4) ÷ 4 = 1.3 ÷ 4 = 0.325

  3. Choose the three-sigma factors

    For 8 observations per subgroup and z = 3, D₄ = 1.864 and D₃ = 0.136.

  4. Calculate the upper control limit

    UCL = R̄ × D₄ = 0.325 × 1.864 = 0.6058

  5. Calculate the lower control limit

    LCL = R̄ × D₃ = 0.325 × 0.136 = 0.0442

SEE EACH SUBGROUP

Ranges and control lines

The chart updates after you enter data. Red points lie outside the three-sigma limits.

Calculated range for each subgroup
SubgroupMaxMinRangeStatus
1——0.3Within limits
2——0.4Within limits
3——0.2Within limits
4——0.4Within limits

WORKED CASES

Two versions of the range-chart question

Both cases use subgroup ranges 0.3, 0.4, 0.2, and 0.4. The wording changes; the calculation and its subgroup-size assumption are the same.

CASE 01

What is the UCL of the range chart when z = 3?

Sample 3: max 4.6, min 4.4, range 0.2. Sample 4: max 4.9, min 4.5, range 0.4. The four subgroup ranges are 0.3, 0.4, 0.2, and 0.4.

ANSWERUCL = 0.61

Assuming 8 observations per subgroup. The original question gives z = 3 but does not state n.

R̄ = (0.3 + 0.4 + 0.2 + 0.4) ÷ 4 = 0.325. For n = 8, D₄ = 1.864, so UCL = 0.325 × 1.864 = 0.6058 ≈ 0.61.

Show original case text

"sample 3: \\["max = 4.6, min = 4.4 → range = 4.6 - 4.4 = 0.2"\\] sample 4: \\["max = 4.9, min = 4.5 → range = 4.9 - 4.5 = 0.4"\\] compute average range rˉ rˉ=(0.3+0.4+0.2+0.4)/4=1.3/4=0.325\mathrm{none} ucl r" or "chart when \\[z=3\\] is \\[0.61\\] final answer: \\[ucl = 0.61\\]" or "0.61 \mathrm{none} the upper control limit (ucl) of the range"

CASE 02

Is 0.61 the upper control limit (UCL) of the range?

Using the same four ranges, check the stated final answer for a three-sigma R chart.

ANSWERYes, if n = 8.

The unrounded UCL is 0.6058, which rounds to 0.61. A different subgroup size changes D₄ and the answer.

R̄ = 1.3 ÷ 4 = 0.325. For n = 8, UCL = D₄ × R̄ = 1.864 × 0.325 = 0.6058 ≈ 0.61.

Show original case text

"sample 3: \\["max = 4.6, min = 4.4 → range = 4.6 - 4.4 = 0.2"\\] sample 4: \\["max = 4.9, min = 4.5 → range = 4.9 - 4.5 = 0.4"\\] compute average range rˉ rˉ=(0.3+0.4+0.2+0.4)/4=1.3/4=0.325\mathrm{none} ucl r" or "0.61 \mathrm{none} the upper control limit (ucl) of the range" or "chart when \\[z=3\\] is \\[0.61\\] final answer: \\[ucl = 0.61\\]"

HOW IT WORKS

R chart UCL questions

What is the upper control limit of the range?

For a standard three-sigma R chart, UCL = D₄ × R̄. Each subgroup's range is its maximum minus its minimum; R̄ is the average of those ranges. The center line is R̄ and the lower control limit is D₃ × R̄.

Is z = 3 enough to calculate UCL?

No. You also need the average range and the number of observations in each subgroup. The subgroup size determines D₄. The number of subgroups used to compute R̄ is a separate count.

Why does another answer differ from 0.61?

0.61 assumes R̄ = 0.325, subgroup size n = 8, the tabulated D₄ = 1.864, and rounding to two decimals. For n = 4, for example, D₄ = 2.282 and UCL = 0.74165, or 0.74 to two decimals.

Can n be greater than 10?

Yes. Enter a whole-number subgroup size from 2 to 1,000,000. The calculator computes the three-sigma range-chart factors numerically above n = 10. For larger subgroups, an s chart is usually more suitable.

Can I enter ranges directly?

Yes. Change Input type to “Ranges already calculated,” then enter one nonnegative range per line. Enter values from comparable subgroups of the selected size.

Can AI read a written R-chart question?

Yes. Paste the question into the input box and select “Extract subgroup data with AI.” Review the proposed ranges or max/min pairs before applying them. AI does not use a stated UCL as an input range. If the text gives z = 3 but not the number of observations per subgroup, choose n yourself before calculating.

Where do the constants come from?

The D₃ and D₄ values for subgroup sizes 2–10 come from the NIST Engineering Statistics Handbook. Above n = 10, this tool calculates the range-distribution moments under the normality assumption, then uses D₃ = max(0, 1 − 3d₃/d₂) and D₄ = 1 + 3d₃/d₂. This tool uses z = 3 throughout.