Methodology Brief

Per-meal error versus per-day error: why validation reports should say which one they mean

A methodology brief

Daniel Okafor, PhD, MS; Lars Henriksen, PhD
Published October 1, 2026

Per-meal error and per-day error are different measurements, and the same records give a different number under each. Per-meal error takes the absolute value of every eating occasion’s error before averaging; per-day error adds the signed errors across a day first, so errors in opposite directions offset one another. On any single day, the daily percentage error cannot exceed the intake-weighted per-meal error (total absolute error divided by total reference energy) from the same data, and it can be far smaller, but only when the errors point in different directions. A bias that runs the same way through every meal does not offset at all. A validation report that quotes one percentage without saying which unit it describes cannot be reliably compared with any other report.

What is the difference between per-meal error and per-day error?

Take one day with several eating occasions. Each has a weighed reference energy and an instrument estimate, and its signed error is the estimate minus the reference.

• Per-meal error for one occasion is 100 × |signed error| ÷ reference energy. Averaged over every occasion in a study, it gives the mean absolute percentage error (MAPE) in its usual form.
• Per-day error for one day is 100 × |sum of that day’s signed errors| ÷ sum of that day’s reference energies. Averaged over days, it gives a daily MAPE.

The only structural difference is where the absolute value is taken: before the errors are combined in the first case, after in the second. The same logic applies one level down. Errors on individual foods partly offset within a plate, so a figure given per food item, per eating occasion or per day describes three different things.

Why is the per-day figure usually smaller?

There are two reasons, one exact and one conditional.

The exact reason is arithmetic. The absolute value of a sum is never larger than the sum of the absolute values. Divide both by the day’s total reference energy and the daily percentage error is at most the intake-weighted per-meal error for that day’s meals. The two are equal only when no errors on that day point in opposite directions.

The conditional reason concerns size. If the meals are similar in size, each meal’s percentage error has the same spread (σ), and the errors are uncorrelated, the spread of the day’s percentage error is σ divided by the square root of the number of meals: half of σ for four meals. With unequal meals the reduction is smaller, because the largest meal carries most of the day’s total.

Illustrative arithmetic. The table uses invented numbers for two unnamed systems, chosen so the sums are easy to check. They are not data from any study, instrument or product.

Summarised three ways:

• Per meal, unweighted mean of the four percentages: System A 12.5%, System B 10.0%.
• Per meal, intake-weighted (total absolute error ÷ total reference energy): System A 11.0% (220 ÷ 2,000), System B 10.0% (200 ÷ 2,000).
• Per day: System A 2.0%, System B 10.0%.

System B has the lower per-meal figure and the higher per-day figure. System A’s errors are larger on average but run in opposite directions, so most of them offset; System B’s are smaller but all run the same way, so none do. Neither summary is wrong. They answer different questions, and a reader shown only one of them would reach the opposite conclusion about the two systems from a reader shown only the other.

When does the daily figure not shrink?

When errors share a cause. Three situations are worth separating.

A systematic bias. The National Cancer Institute’s Dietary Assessment Primer defines systematic error as “a type of error that results in measurements that consistently depart from the true value in the same direction” and says it “cannot be reduced or eliminated by taking repeat measures.” A uniform 10% underestimate of every meal is a 10% underestimate of the day, as System B shows.

A person-specific bias. The same primer separates bias tied to true intake from person-specific bias, which it says may be related to individual characteristics such as social or cultural desirability. Kipnis and colleagues examined eight validation studies that used urinary nitrogen as a reference biomarker for protein intake. They reported that, for protein, conventional dietary-report instruments (series of 24-hour recalls, 4-day weighed food records and 7-day diaries) showed systematic bias that was partly person-specific. When people log their own meals, one person’s habits run through every meal of the day.

Errors that move together. For k equal-sized meals whose percentage errors have spread σ and a common pairwise correlation ρ, the spread of the daily percentage error is σ × √((1 + (k − 1)ρ) ÷ k). With four meals that is 0.5σ at ρ = 0, about 0.79σ at ρ = 0.5, and the full σ at ρ = 1. The correlation between a day’s errors is not visible in either headline figure, which is why neither figure can be derived from the other.

Can a per-meal figure be converted into a per-day figure?

Not from the headline number alone. Three things get in the way.

First, the size of the gap depends on the sign pattern and the within-day correlation described above, and a single percentage carries neither.

Second, the aggregation matters as much as the unit. The intake-weighted figure and the unweighted mean of per-meal percentages are different statistics, and the daily figure can fall on either side of the second. Take an illustrative day of two eating occasions with reference energies of 100 kcal and 900 kcal. The instrument is exact on the first and overestimates the second by 90 kcal, or 10%. The mean of the two per-meal percentages is 5%; the daily error is 90 ÷ 1,000, or 9%.

Third, the day has to be defined. Meals prepared separately and grouped into notional days afterwards give a per-day figure that depends on the grouping rule, and real days may carry errors that are more alike than those of randomly grouped meals; such a figure should be labelled as constructed, with the rule stated. A per-day figure from complete free-living days also contains anything eaten but not logged, or logged but not eaten, which a test of prepared meals excludes by design. And a daily total is a daily total only if the reference covers everything eaten that day; otherwise call it a partial-day sum.

Which unit should a validation report treat as primary?

The one that matches the decision the figure is meant to inform, named in the protocol before data collection. Decisions taken at the table, such as choosing a portion or counting carbohydrate for a single meal, depend on per-meal error, because an error at one meal is not offset by an opposite error at another. A budget for the day depends on the daily total, where offsetting errors do help. Averages over many days depend increasingly on bias, because random error keeps shrinking while systematic error does not. Our position paper on accuracy thresholds for self-monitoring ties thresholds to decisions in this way; some of its thresholds are set per day and others per meal.

Where the data allow, report both figures, each labelled, and designate one as primary in advance. Our brief on reporting MAPE covers rounding, confidence intervals and companion metrics.

This is a research-reporting convention, not dietary advice. Anyone who uses tracked intake for a medical decision, such as insulin dosing, should do so with their clinician or a registered dietitian.

What should a report say so readers know which one they are reading?

Seven short statements are enough.

• The unit. Food item, eating occasion or person-day, and what counted as an eating occasion (snacks and beverages included or not).
• The order of operations. Whether the absolute value was taken before or after summing within the day.
• The normalisation. Mean of per-occasion percentages, or total error divided by total reference energy.
• The reference for any daily total. Whether it covers everything eaten that day or only some meals.
• Signed error as well as absolute error, in kilocalories and in percent. Only signed errors show whether anything offset, and the difference-based approach Bland and Altman described in 1986 starts from them.
• The counts. Participants, days, eating occasions per day, and any days excluded as incomplete.
• Uncertainty at the right level. Meals from one person or one day are clustered, whereas the basic limits-of-agreement calculation assumes each pair of observations comes from a different subject. Bland and Altman (2007) describe methods for repeated pairs on the same subject; resampling by day or by participant addresses the same problem in bootstrap intervals.

The sixth item of our reporting checklist already asks authors to say whether error is reported per meal or per day. This brief adds the arithmetic behind that item and the conditions under which its shorthand, that a per-day figure will be lower, holds. It makes no claim about how large the gap is for any real instrument: that depends on the instrument’s error structure and the study design, and no conversion factor is implied.

Frequently asked questions

What is the difference between random error and systematic error in dietary assessment?

Random error varies from one measurement to the next and is as likely to run high as low; systematic error pushes measurements the same way each time. The NCI primer notes that data affected by within-person random error only “are not biased but may be imprecise.” Summing meals into a day offsets part of the random error, not the systematic error.

Does averaging reduce error in dietary intake data?

It reduces random error but not systematic error. The NCI primer says that when within-person random error is the only kind in the data and repeat measures exist, averaging across days “will provide a better approximation of usual dietary intakes than will a single day’s measure.” Systematic error, in contrast, cannot be removed by repeat measures. The same arithmetic operates when meals are summed into a day.

How is mean absolute percentage error calculated?

For each observation, take the absolute difference between estimate and reference, divide by the reference, multiply by 100, and average over observations. The observation is the unit: eating occasions give a per-meal MAPE, and day totals against day references give a per-day MAPE.

Can Bland–Altman analysis handle several meals from the same person?

Yes, with an adjustment. The basic limits-of-agreement calculation assumes each pair of observations comes from a different subject. Bland and Altman (2007) describe methods for repeated pairs on the same subject, including the case where the underlying value changes from one observation to the next, as it does between meals.

How accurate is calorie counting over a whole day compared with a single meal?

No single figure applies, and one cannot be derived from the other by formula. Errors that point in different directions partly offset over a day, while a consistent bias is the same size at both levels, and research on self-reported intake describes systematic bias in conventional instruments for protein intake, part of it person-specific. Before comparing any quoted figure with another, ask which unit it describes.

References

1. Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurement. Lancet. 1986;1(8476):307-310. PubMed 2868172
2. Bland JM, Altman DG. Agreement between methods of measurement with multiple observations per individual. J Biopharm Stat. 2007;17(4):571-582. doi:10.1080/10543400701329422. PubMed 17613642
3. Kipnis V, Midthune D, Freedman L, Bingham S, Day NE, Riboli E, Ferrari P, Carroll RJ. Bias in dietary-report instruments and its implications for nutritional epidemiology. Public Health Nutr. 2002;5(6A):915-923. doi:10.1079/PHN2002383. PubMed 12633516
4. National Cancer Institute, Division of Cancer Control and Population Sciences, Epidemiology and Genomics Research Program. Dietary Assessment Primer: Types of Measurement Error. Last updated 29 December 2025. epi.grants.cancer.gov. Accessed 1 October 2026.