Dart Impact Test Calculation: How to Find F50 from Your Staircase Data
Learn the ASTM D1709 dart impact test calculation formula step by step. Find F50 from staircase data, compute standard deviation, and avoid common errors.
The staircase method gives you twenty pass/fail results but no raw force measurements. Turning that string of passes and failures into a single F50 number requires a specific formula, and getting one variable wrong shifts your reported impact resistance by a full weight increment.
A dart impact test calculation is the statistical procedure used to convert staircase test data from ASTM D1709 or ISO 7765-1 into F50 — the dart mass, in grams, at which 50% of film specimens fail under the specified drop conditions. Unlike tests that output a direct reading, the falling dart staircase method produces a binary pass/fail sequence that must be processed through a formula to yield a reportable result.
This guide walks through the ASTM D1709 formula variable by variable, provides a worked calculation example, compares the staircase and group methods, and flags the mistakes that most often corrupt F50 results.
What the Calculation Actually Produces
The dart impact test calculation produces a statistically estimated mass — not a directly measured one. F50 (also called M50 or impact failure weight) is that estimate. No single specimen is tested at the exact 50% failure threshold. Instead, the staircase procedure brackets that threshold by adjusting the dart weight up after a pass and down after a failure, using a fixed weight increment each time.
The formula then estimates where the 50% failure point falls within the bracket. That estimate is what gets reported on a test certificate, compared against a specification limit, or used to qualify a film lot.
| Term | Meaning |
|---|---|
| F50 / M50 / WF | Impact failure weight — the dart mass at which 50% of specimens would fail |
| DW | The uniform weight increment added or removed between specimens (e.g., 15 g, 25 g, 50 g) |
| Wo | The lowest dart weight at which the less frequent event occurred |
| N | Total count of the less frequent event across all weight levels |
| A | Weighted sum: each weight level's index (i) multiplied by its count (ni), then summed |
| B | Sum of each weight level's index squared (i squared) multiplied by its count (ni) |
| Less frequent event | Whichever outcome — pass or failure — occurred fewer times in the data set |
Before running the formula, the F50 result only becomes useful when paired with specification limits. For guidance on translating F50 into accept/reject criteria, see setting specification limits for dart impact M50.
The ASTM D1709 Staircase Formula
The formula from ASTM D1709 Section 10:
WF = Wo + DW x (A/N +/- 1/2)
Use + 1/2 when the less frequent event is failure (more specimens passed than failed overall).
Use - 1/2 when the less frequent event is pass (more specimens failed than passed overall).
Identifying the Less Frequent Event
After running all specimens, count total passes and total failures separately. The smaller count determines the less frequent event. All subsequent counting — building the weight-level table, computing N, A, and B — is done using only the less frequent event's data.
Why does the formula need this step? The staircase converges around the 50% threshold, so the split between passes and failures will be close to even. Using the less frequent event reduces the number of data points going into the calculation, which makes the estimate more stable when the sample size is small.
Variable Definitions
| Variable | Definition | How to find it |
|---|---|---|
| WF | Impact failure weight (F50), in grams | The output of the formula |
| Wo | Lowest weight level where the less frequent event occurred | Assign this weight level i = 0 |
| DW | Uniform weight increment between levels, in grams | Fixed before testing begins; must stay constant throughout the test |
| ni | Number of times the less frequent event occurred at weight level i | Count from your data |
| A | Sum of (i x ni) across all weight levels | Compute from the counting table |
| N | Sum of ni across all weight levels | Equals the total less frequent event count |
| B | Sum of (i-squared x ni) across all weight levels | Needed for standard deviation only |
Computing the Standard Deviation
The standard deviation formula for the staircase method:
s = 1.62 x DW x [(N x B - A-squared) / N-squared + 0.029]
The standard flags the result as suspect if the quantity (N x B - A-squared) / N-squared is less than 0.3. A value below that threshold suggests the weight increment DW may be too large relative to the film's actual variability, and the test should be repeated with a smaller increment.
Worked Example: Staircase Calculation Step by Step
A complete staircase calculation starts with raw pass/fail data. Suppose you are testing a polyethylene film using Method A (0.66 m drop height, 38.1 mm dart). Your weight increment DW is 25 g. After conditioning and testing 20 specimens, the sequence looks like this (P = pass, F = fail):
Weights tested (grams): 200, 225, 200, 225, 250, 225, 250, 275, 250, 275, 250, 225, 250, 275, 250, 275, 300, 275, 300, 275
Results: P, F, P, P, F, P, F, F, P, F, P, P, F, F, P, F, F, P, F, P
Step 1: Count passes and failures.
Passes: 10. Failures: 10. When counts are equal, use failures as the less frequent event (ASTM D1709 convention).
Step 2: Build the counting table for failures only.
| Weight (g) | Level index i | Failures at this level (ni) | i x ni | i-squared x ni |
|---|---|---|---|---|
| 225 | 0 | 1 | 0 | 0 |
| 250 | 1 | 2 | 2 | 2 |
| 275 | 2 | 4 | 8 | 16 |
| 300 | 3 | 1 | 3 | 9 |
Wo = 225 g (lowest weight where a failure occurred, assigned i = 0).
Step 3: Sum the columns.
N = 1 + 2 + 4 + 1 = 8
A = 0 + 2 + 8 + 3 = 13
B = 0 + 2 + 16 + 9 = 27
Step 4: Apply the formula.
Since the less frequent event is failure, use + 1/2:
WF = 225 + 25 x (13/8 + 1/2)
WF = 225 + 25 x (1.625 + 0.5)
WF = 225 + 25 x 2.125
WF = 278 g
Step 5: Compute standard deviation.
s = 1.62 x 25 x [(8 x 27 - 169) / 64 + 0.029]
s = 40.5 x [(216 - 169) / 64 + 0.029]
s = 40.5 x [0.734 + 0.029]
s = 30.9 g
Check: (N x B - A-squared) / N-squared = 47/64 = 0.734. This exceeds 0.3, so the result is valid.
Your reported result: F50 = 278 g with a standard deviation of 30.9 g.
Action item: after completing your own calculation, verify the (N x B - A-squared) / N-squared check before reporting. If it falls below 0.3, reduce DW and retest.
Staircase Method vs Group Method
Two data-reduction approaches exist in the standard. The staircase method is far more common in production QC because it uses fewer specimens.
| Factor | Staircase method | Group method |
|---|---|---|
| Specimen count | 20 minimum (typically 25-30) | 30 minimum |
| Test flow | Sequential — each specimen's weight depends on the previous result | Parallel — test batches of 10 at fixed weights |
| Calculation | WF = Wo + DW x (A/N +/- 1/2) | Plot percent failure vs weight; interpolate to 50% |
| Precision | Good when DW is well-matched to film variability | Higher precision with enough specimens, but needs more material |
| Best for | Routine QC, incoming inspection, single-operator labs | R&D characterization, referee testing, resolving disputes |
Which one to pick? If your lab tests the same film grades regularly, the staircase method is the practical choice: fewer specimens, shorter test time, and a formula that can be built into a spreadsheet template once and reused. The group method becomes worthwhile when two labs disagree on a result and both need to demonstrate their data independently.
Common Mistakes That Shift Your F50
Errors in the dart impact test calculation usually trace back to five sources. Each one shifts the reported F50 without any warning from the instrument itself — the math runs cleanly with wrong inputs.
| Mistake | Effect on F50 | How to catch it |
|---|---|---|
| Wrong sign (+/- 1/2) | Shifts F50 by one full DW (e.g., 25 g off) | Verify: did you use the less frequent event? Failures = +1/2, passes = -1/2 |
| Wrong Wo (assigned i = 0 to the wrong weight level) | Offsets the entire result | Wo must be the lowest weight where the less frequent event actually happened |
| Non-uniform DW during testing | Formula assumes constant increment; varying DW invalidates the result | Review your weight sequence before calculating — every step must be the same |
| Too few specimens (under 20) | Wider confidence interval, F50 unreliable | ASTM D1709 requires a minimum of 20; aim for 25-30 |
| Counting both events instead of only the less frequent one | N and A are wrong; F50 is meaningless | Double-check: N should equal only the count of the less frequent event |
Quick Self-Audit Checklist
Use this self-audit checklist before signing off on a dart impact test calculation:
- Passes and failures counted separately; less frequent event identified
- Wo assigned to the correct (lowest) weight level of the less frequent event
- Weight levels indexed starting at i = 0
- All three sums (A, N, B) computed from the less frequent event only
- Correct sign applied (+1/2 for failures, -1/2 for passes)
- Standard deviation computed and (N x B - A-squared) / N-squared checked against 0.3
- DW was constant throughout the entire test run
Frequently Asked Questions
Q: What does F50 mean in a dart impact test?
F50 is the dart mass, in grams, at which 50% of film specimens are statistically expected to fail when dropped from the height specified by the test method. It is the primary reported result from both ASTM D1709 and ISO 7765-1.
Q: How many specimens does the staircase method need?
ASTM D1709 requires a minimum of 20 specimens for the staircase method. Running 25 to 30 specimens improves the reliability of the F50 estimate, especially when the film shows high variability.
Q: What is the difference between F50 and M50?
Same measurement, different abbreviations. F50 stands for failure mass at 50%, M50 for mass at 50% failure, and the standard itself uses the notation WF.
Q: Can I use the ASTM D1709 staircase formula for ISO 7765-1?
Yes. ISO 7765-1 uses the same staircase statistical method, the same formula structure, and test parameters that match Method A. A spreadsheet built for D1709 staircase data works for ISO 7765-1 without modification.
Q: What does a high standard deviation mean in dart impact results?
Wide scatter in impact resistance across the specimen set. Check three things: uneven film thickness across the web, inconsistent specimen conditioning (temperature or humidity out of range), and whether specimens came from different production lots. Any of these inflates the standard deviation beyond what the film's inherent variability would produce.
Q: When should the test be repeated instead of reporting the result?
When (N x B - A-squared) / N-squared falls below 0.3. That threshold, specified in ASTM D1709, signals that the weight increment was too coarse to resolve the film's failure point. Reduce DW and run a fresh set.
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KHT Instruments is a Jinan, China-based maker of packaging test equipment, founded in 2013.
