Repeatability and Reproducibility Calculations in Turbine Oil Analysis

Turbine engineers sometimes take one oil sample from the same turbine reservoir, split it into two or three bottles, and send the bottles to different laboratories. The objective is simple:

“Let us compare the laboratories and see which one is accurate.”

Then the reports come back, and everyone is shocked.

Test	Lab A	Lab B	Lab C
TAN, ASTM D664	0.15 mg KOH/g	0.21 mg KOH/g	0.18 mg KOH/g
MPC, ASTM D7843	ΔE 24	ΔE 31	ΔE 28
LSV/RULER, ASTM D6971	62% amine remaining	48% amine remaining	55% amine remaining
RPVOT, ASTM D2272	520 min	630 min	575 min

The first reaction is normally emotional:

“How can this happen? It was the same oil.”

But technically, the better question is:

“Is the difference larger than the repeatability or reproducibility limit of the ASTM test method?”

This is the missing discipline in many turbine oil programs. A laboratory number is not an absolute truth. It is a measured result with a defined precision. If we ignore the precision statement of the method, we may accuse a good laboratory unfairly, or worse, make a wrong maintenance decision.

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1. Repeatability and Reproducibility

Repeatability, r

Repeatability is the expected agreement between two test results obtained in the same laboratory, by the same operator, using the same equipment, on the same material, over a short time interval.

In practical turbine oil language:

“If the same lab tests the same turbine oil twice, how different can the two results be before the difference becomes statistically suspicious?”

The repeatability limit is normally written as:

r

The comparison rule is:

Absolute difference = |X1 – X2|

Same-lab duplicate results are acceptable when:

|X1 – X2| <= r

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Reproducibility, R

Reproducibility is the expected agreement between two test results obtained in different laboratories, by different operators, using different equipment, on the same material.

In practical turbine oil language:

“If two different laboratories test the same turbine oil, how different can their results be before the difference becomes statistically suspicious?”

The reproducibility limit is normally written as:

R

The comparison rule is:

Absolute difference = |X1 – X2|

Different-lab results are acceptable when:

|X1 – X2| <= R

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2. General Calculation Method

For two results:

X1 = result from test 1

X2 = result from test 2

Absolute difference:

D = |X1 – X2|

Average:

X = (X1 + X2) / 2

For same-lab repeatability:

D <= r

For different-lab reproducibility:

D <= R

If D > r for same-lab testing, the two results should be investigated.

If D > R for different-lab testing, the two laboratory results should be investigated.

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3. TAN by ASTM D664

TAN, or acid number, is one of the most common turbine oil tests. It is usually reported in mg KOH/g.

For turbine oils, TAN is typically low, so even small absolute differences can look large when expressed as a percentage.

Example:

0.15 mg KOH/g to 0.22 mg KOH/g

Absolute difference:

D = 0.22 – 0.15 = 0.07 mg KOH/g

Percentage difference relative to 0.15:

Percentage difference = (0.07 / 0.15) x 100 = 46.7%

This looks alarming, but statistically it may still be within ASTM D664 reproducibility.

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3.1 ASTM D664 Precision Calculation

For D664 acid number by inflection endpoint, the commonly referenced precision equations are:

Repeatability:

r = 0.1938 x X^0.8199

Reproducibility:

R = 0.4022 x X^0.8199

Where:

X = (X1 + X2) / 2

For buffer endpoint around pH 10, the commonly referenced precision equations are:

Repeatability:

r = 0.2681 x X^0.9748

Reproducibility:

R = 0.4506 x X^0.9748

The engineer must confirm which endpoint was used by the laboratory.

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3.2 TAN Example: Same Laboratory

Same laboratory duplicate results:

X1 = 0.15 mg KOH/g

X2 = 0.18 mg KOH/g

Average:

X = (0.15 + 0.18) / 2 = 0.165 mg KOH/g

Difference:

D = |0.18 – 0.15| = 0.03 mg KOH/g

Using the inflection endpoint repeatability equation:

r = 0.1938 x 0.165^0.8199

r ≈ 0.044 mg KOH/g

Decision:

D <= r

0.03 <= 0.044

The same-lab duplicate TAN results are acceptable.

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3.3 TAN Example: Different Laboratories

Lab A:

X1 = 0.15 mg KOH/g

Lab B:

X2 = 0.22 mg KOH/g

Average:

X = (0.15 + 0.22) / 2 = 0.185 mg KOH/g

Difference:

D = |0.22 – 0.15| = 0.07 mg KOH/g

Reproducibility:

R = 0.4022 x 0.185^0.8199

R ≈ 0.101 mg KOH/g

Decision:

D <= R

0.07 <= 0.101

The two laboratories are statistically acceptable under this precision calculation.

Practical message:

0.15 and 0.22 mg KOH/g may look very different emotionally, but they may not be different statistically.

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4. MPC by ASTM D7843-25 — Corrected Calculation

ASTM D7843-25 measures lubricant-generated insoluble color bodies in in-service turbine oils using membrane patch colorimetry. The result is reported as ΔE within the CIELAB scale.

The critical correction is this:

ASTM D7843-25 does not use one universal r and R equation for all MPC results.

The precision treatment depends on the solvent used.

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4.1 MPC Sampling and Preparation Sensitivity

MPC can vary significantly because it is sensitive to:

• bottle type;
• UV exposure;
• heating time;
• incubation time;
• sample mixing;
• adherent material on the bottle wall;
• membrane type;
• vacuum setting;
• solvent used;
• drying time;
• spectrophotometer geometry.

A small deviation in sample handling can change the final ΔE result.

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4.2 MPC ΔE Calculation

ASTM D7843 uses the CIE 1976 Lab* color space.

The total color difference is:

Delta Eab = [ (Delta L)^2 + (Delta a*)^2 + (Delta b*)^2 ]^0.5

Or in plain text:

Delta E = square root of [(Delta L)^2 + (Delta a)^2 + (Delta b)^2]

Where:

Delta L* = Lsample patch – Lblank patch

Delta a* = asample patch – ablank patch

Delta b* = bsample patch – bblank patch

Example:

Delta L* = 18

Delta a* = 6

Delta b* = 10

Delta E = [(18)^2 + (6)^2 + (10)^2]^0.5

Delta E = [324 + 36 + 100]^0.5

Delta E = 460^0.5

Delta E = 21.4

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4.3 MPC r and R Approach in ASTM D7843-25

ASTM D7843-25 gives two different precision approaches.

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A. Petroleum Ether / Petroleum Spirit

For petroleum ether, repeatability and reproducibility limits are table-based.

Use:

Table 1 for 72 h / 3 day incubation.

Table 2 for 120 h / 5 day incubation.

Therefore, for petroleum ether:

r is not calculated from a universal equation.

R is not calculated from a universal equation.

Instead, use the r and R values from the applicable table, using the closest relevant material average and correct incubation time.

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B. Heptane

For heptane, ASTM D7843-25 gives equation-based precision.

Repeatability:

r = 1.1499 x (X + 0.0001)^0.3155

Reproducibility:

R = 5.6250 x (X + 0.0001)^0.3155

Where:

X = (X1 + X2) / 2

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4.4 Petroleum Ether: Table-Based MPC Examples

Table 1 — 72 h / 3 day incubation

ASTM D7843-25 Table 1 gives petroleum ether precision values for 72 h / 3 day incubation.

Material	Average ΔE	r	R
Sample 1	15.332	2.410	8.874
Sample 2	7.409	2.547	7.925
Sample 3	6.970	2.349	8.637
Sample 4	3.472	1.309	5.732
Sample 5	6.993	2.386	8.309
Sample 6	34.258	10.236	32.357
Sample 7	3.475	0.494	4.611

Important point:

For petroleum ether, do not calculate r and R with the heptane equations. Use the table values.

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Petroleum Ether Example 1: Same Laboratory, 72 h Incubation

Same lab duplicate MPC results:

X1 = 31.0 ΔE

X2 = 38.0 ΔE

Average:

X = (31.0 + 38.0) / 2 = 34.5 ΔE

Difference:

D = |38.0 – 31.0| = 7.0 ΔE

Nearest D7843-25 Table 1 material average:

Sample 6 average = 34.258 ΔE

For Sample 6:

r = 10.236 ΔE

Decision:

D <= r

7.0 <= 10.236

Therefore, the same-lab MPC difference is acceptable for petroleum ether at a similar ΔE level under the 72 h table.

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Petroleum Ether Example 2: Different Laboratories, 72 h Incubation

Lab A:

X1 = 24.0 ΔE

Lab B:

X2 = 31.0 ΔE

Average:

X = (24.0 + 31.0) / 2 = 27.5 ΔE

Difference:

D = |31.0 – 24.0| = 7.0 ΔE

There is no exact table average at 27.5 ΔE. The engineer should not invent an equation for petroleum ether.

The nearest higher table level is Sample 6:

Sample 6 average = 34.258 ΔE

For Sample 6:

R = 32.357 ΔE

Decision using nearest higher table level:

D <= R

7.0 <= 32.357

The difference between Lab A and Lab B is statistically acceptable using this practical comparison reference.

Professional wording:

“The two results are not necessarily contradictory. They are within the broad interlaboratory precision behavior shown in the D7843-25 petroleum ether precision data. However, because the measured average falls between table values, exact interpretation should consider the closest applicable ASTM table level, laboratory validation data, and the full sampling and test procedure.”

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Petroleum Ether Example 3: Low MPC Values, 72 h Incubation

Lab A:

X1 = 3.0 ΔE

Lab B:

X2 = 7.5 ΔE

Average:

X = (3.0 + 7.5) / 2 = 5.25 ΔE

Difference:

D = |7.5 – 3.0| = 4.5 ΔE

Nearby Table 1 values:

Sample 4 average = 3.472, R = 5.732

Sample 7 average = 3.475, R = 4.611

Sample 3 average = 6.970, R = 8.637

Sample 5 average = 6.993, R = 8.309

Decision:

Using Sample 7:

D <= R

4.5 <= 4.611

Using Sample 4:

D <= R

4.5 <= 5.732

The difference may be acceptable, even though the percentage difference looks large.

Important point:

At low ΔE values, percentage difference can look dramatic, but the absolute reproducibility limit may still allow the difference.

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4.5 Petroleum Ether: 120 h / 5 Day Incubation

ASTM D7843-25 Table 2 gives petroleum ether precision values for 120 h / 5 day incubation.

Material	Average ΔE	r	R
Sample 1	17.125	4.284	12.004
Sample 2	7.618	2.003	8.174
Sample 3	7.473	1.989	8.401
Sample 4	3.584	1.452	5.242
Sample 5	7.883	1.736	8.770
Sample 6	38.615	13.690	34.439
Sample 7	3.867	1.223	4.392

The incubation time must be known.

Do not compare a 72 h result with the 120 h table unless the reported incubation condition matches.

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Petroleum Ether Example 4: Different Laboratories, 120 h Incubation

Lab A:

X1 = 35.0 ΔE

Lab B:

X2 = 45.0 ΔE

Average:

X = (35.0 + 45.0) / 2 = 40.0 ΔE

Difference:

D = |45.0 – 35.0| = 10.0 ΔE

Nearest Table 2 high-level material:

Sample 6 average = 38.615 ΔE

For Sample 6:

R = 34.439 ΔE

Decision:

D <= R

10.0 <= 34.439

The between-lab difference is acceptable under the high-MPC petroleum ether 120 h precision table.

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4.6 Heptane: D7843-25 Equation-Based Examples

For heptane, D7843-25 gives equations.

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Heptane Example 1: Same Laboratory

Same lab duplicate results:

X1 = 24.0 ΔE

X2 = 31.0 ΔE

Average:

X = (24.0 + 31.0) / 2 = 27.5 ΔE

Difference:

D = |31.0 – 24.0| = 7.0 ΔE

Repeatability:

r = 1.1499 x (27.5 + 0.0001)^0.3155

r ≈ 3.27 ΔE

Decision:

D <= r ?

7.0 <= 3.27 ? No.

Therefore, for heptane, these same-lab results do not meet the repeatability limit.

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Heptane Example 2: Different Laboratories

Lab A:

X1 = 24.0 ΔE

Lab B:

X2 = 31.0 ΔE

Average:

X = (24.0 + 31.0) / 2 = 27.5 ΔE

Difference:

D = |31.0 – 24.0| = 7.0 ΔE

Reproducibility:

R = 5.6250 x (27.5 + 0.0001)^0.3155

R ≈ 16.01 ΔE

Decision:

D <= R

7.0 <= 16.01

For heptane, the between-lab difference is acceptable.

Important point:

The same numerical difference can fail repeatability but pass reproducibility.

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Heptane Example 3: High MPC

Lab A:

X1 = 35.0 ΔE

Lab B:

X2 = 45.0 ΔE

Average:

X = (35.0 + 45.0) / 2 = 40.0 ΔE

Difference:

D = |45.0 – 35.0| = 10.0 ΔE

Repeatability:

r = 1.1499 x (40.0 + 0.0001)^0.3155

r ≈ 3.68 ΔE

Reproducibility:

R = 5.6250 x (40.0 + 0.0001)^0.3155

R ≈ 18.00 ΔE

Same-lab decision:

D <= r ?

10.0 <= 3.68 ? No.

Different-lab decision:

D <= R ?

10.0 <= 18.00 ? Yes.

So, for heptane, a 10 ΔE difference at this level is not acceptable as same-lab repeatability, but it is acceptable as between-lab reproducibility.

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4.7 What This Means for Turbine Engineers Using MPC

For petroleum ether:

Do not use a universal equation.

Use:

Table 1 for 72 h / 3 day incubation.

Table 2 for 120 h / 5 day incubation.

Use the closest applicable material average, and apply engineering caution if the measured average falls between table values.

For heptane:

Use:

r = 1.1499 x (X + 0.0001)^0.3155

R = 5.6250 x (X + 0.0001)^0.3155

Where:

X = (X1 + X2) / 2

Never compare MPC results without knowing:

• solvent used: petroleum ether or heptane;
• incubation time: 72 h or 120 h;
• bottle type;
• UV protection;
• whether the bottle had adherent deposits;
• whether patch image, Delta L, Delta a, and Delta b were reported;
• whether the lab deviated from ASTM D7843-25.

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5. LSV / RULER by ASTM D6971

RULER is the trade name of one instrument used for linear sweep voltammetry, or LSV. ASTM D6971 refers to the method as linear sweep voltammetry.

The test is used to measure antioxidant depletion, typically:

• hindered phenolic antioxidants;
• aromatic amine antioxidants.

Results are often reported as percentage remaining compared with a fresh oil reference.

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5.1 LSV / RULER Precision Calculation

A commonly referenced ASTM D6971 precision expression is:

Repeatability standard deviation expression:

sr = 1.5094 x (x + 8.6662)^0.46390

Reproducibility standard deviation expression:

sR = 3.0067 x (x + 8.6662)^0.46390

Where:

x = (X1 + X2) / 2

Important note:

Because these expressions are often presented as standard deviation expressions rather than direct r and R limit language, the current ASTM D6971 edition should be checked before using them for contractual acceptance.

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5.2 LSV / RULER Example

Lab A:

X1 = 62% amine remaining

Lab B:

X2 = 48% amine remaining

Average:

x = (62 + 48) / 2 = 55%

Difference:

D = |62 – 48| = 14%

Reproducibility expression:

sR = 3.0067 x (55 + 8.6662)^0.46390

sR ≈ 20.8%

The 14% difference may be explainable by method precision and laboratory differences, depending on the exact acceptance language used in the current standard.

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5.3 Practical LSV / RULER Warnings

LSV / RULER results can vary because of:

• wrong fresh oil reference;
• changed oil formulation;
• mixed turbine oils;
• phenol and amine peak overlap;
• oxidation by-product interference;
• electrode cleanliness;
• extraction technique;
• peak integration differences.

For turbine oil reliability, do not ask only:

“What is the RULER percentage?”

Ask:

“Which antioxidant peak was measured, against which fresh oil reference, and how does it trend against TAN, MPC, and RPVOT?”

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6. RPVOT by ASTM D2272

RPVOT measures oxidation stability under accelerated test conditions and is normally reported in minutes.

It is important, but it is not a direct varnish test. A turbine oil can show varnish tendency while still having a reasonable RPVOT value. Similarly, a falling RPVOT does not identify the exact degradation mechanism.

Commonly referenced D2272 precision guidance for many turbine oil discussions is:

Repeatability:

r ≈ 12% of the mean

Or:

r = 0.12 x X

Reproducibility:

R ≈ 22% of the mean

Or:

R = 0.22 x X

Where:

X = (X1 + X2) / 2

For formal acceptance, the current D2272 precision statement and applicable result range must be checked.

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6.1 RPVOT Example: Different Laboratories

Lab A:

X1 = 520 min

Lab B:

X2 = 630 min

Average:

X = (520 + 630) / 2 = 575 min

Difference:

D = |630 – 520| = 110 min

Reproducibility:

R = 0.22 x 575

R = 126.5 min

Decision:

D <= R

110 <= 126.5

The two laboratories may be statistically acceptable.

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6.2 RPVOT Example: Same Laboratory

Same laboratory duplicate results:

X1 = 520 min

X2 = 600 min

Average:

X = (520 + 600) / 2 = 560 min

Difference:

D = |600 – 520| = 80 min

Repeatability:

r = 0.12 x 560

r = 67.2 min

Decision:

D <= r ?

80 <= 67.2 ? No.

This difference may be too large for same-lab repeatability and should be investigated.

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7. Summary Table

Test	ASTM Method	Result	Same-Lab Check	Between-Lab Check	Major Warning
TAN	ASTM D664	mg KOH/g	r = 0.1938 x X^0.8199, if inflection endpoint	R = 0.4022 x X^0.8199, if inflection endpoint	Small absolute differences look large at low TAN
MPC, petroleum ether	ASTM D7843-25	ΔE	Use Table 1 or Table 2	Use Table 1 or Table 2	No universal equation
MPC, heptane	ASTM D7843-25	ΔE	r = 1.1499 x (X + 0.0001)^0.3155	R = 5.6250 x (X + 0.0001)^0.3155	Confirm solvent
LSV/RULER	ASTM D6971	% antioxidant remaining	Use current ASTM D6971 precision expression	Use current ASTM D6971 precision expression	Fresh oil reference is critical
RPVOT	ASTM D2272	minutes	r ≈ 0.12 x X	R ≈ 0.22 x X	Not a direct varnish test

Where:

X = average of two results = (X1 + X2) / 2

D = absolute difference = |X1 – X2|

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8. Practical Laboratory Comparison Protocol

If you want to compare laboratories properly, use this procedure.

Step 1: Sample from the correct point

Take the sample from a live, circulating, representative sampling point. Avoid drains, dead legs, dirty sample tubing, and stagnant oil.

Step 2: Sample hot and circulating

For turbine oil varnish potential, sampling hot is critical because soluble degradation products can precipitate during cooling.

Step 3: Use correct bottles

For MPC, UV exposure is a serious issue. Use suitable amber HDPE bottles or immediately protect clear bottles from light.

Step 4: Split carefully

Do not simply fill three bottles randomly. Mix the bulk sample properly and split under clean, controlled conditions.

Step 5: Run blind duplicates

Send two duplicate samples to the same lab under different IDs. This checks repeatability.

Step 6: Send split samples to different labs

This checks reproducibility.

Step 7: Demand method details

Ask every lab to report:

• ASTM method edition;
• solvent used for MPC;
• incubation time for MPC;
• whether any method deviation occurred;
• TAN endpoint type;
• LSV/RULER fresh oil reference;
• phenol and amine values separately;
• RPVOT instrument and endpoint criteria;
• uncertainty or internal QC data.

Step 8: Compare using r and R

Do not compare emotionally. Compare mathematically.

Absolute difference:

D = |X1 – X2|

Same-lab comparison:

D <= r

Different-lab comparison:

D <= R

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9. Final Engineering Message

When the same turbine oil sample is sent to different laboratories and the results vary, the correct conclusion is not immediately:

“One lab is wrong.”

The correct conclusion is:

“Let us check whether the difference exceeds the ASTM repeatability or reproducibility limit.”

For TAN, use the ASTM D664 precision equation appropriate to the endpoint.

For MPC, be very careful:

• petroleum ether uses D7843-25 table-based r and R limits;
• heptane uses D7843-25 equations;
• incubation time must be known;
• UV protection and sample handling can strongly affect results.

For LSV/RULER, confirm the fresh oil reference and antioxidant peak interpretation.

For RPVOT, understand that a 100-minute difference may be statistically acceptable depending on the average value.

The mature turbine oil reliability mindset is not:

“The number changed, so the oil changed.”

It is:

“The number changed. Now we must determine whether the change is real, statistically significant, and consistent with the full oil condition picture.”

That is how turbine oil analysis moves from laboratory reporting to reliability engineering.

Last Khash Words

To compare different lab results for the same oil sample, first calculate the average of the two results: X = (Result 1 + Result 2) / 2. Then calculate the absolute difference: D = |Result 1 – Result 2|. After that, compare D with the ASTM repeatability limit, r, if both results came from the same lab, or with the ASTM reproducibility limit, R, if the results came from different labs. If D <= r or D <= R, the difference is statistically acceptable and should not be overreacted to. If D > r or D > R, the difference is larger than expected and should be investigated by checking sampling point, sample handling, bottle type, test method, instrument, operator, incubation/pre-treatment, and laboratory procedure before making any maintenance decision.