Precision Ontology
Measurement Uncertainty
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 16 September 2026
Audio Overview
# Precision Ontology — Flow Algebra
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/measurement-uncertainty/precision-ontology.html
FLOW PrecisionOntology {
DOMAIN:
Mathematical Ontologies
-> Measurement & Units Ontologies
-> Measurement Uncertainty
-> Precision Ontology
PURPOSE:
Represent measurement precision as the closeness of agreement among indications or measured quantity values obtained by replicate measurements on the same or similar objects under specified conditions, including repeatability, intermediate precision, reproducibility, dispersion, variance, standard deviation, coefficient of variation, pooled precision, heteroscedasticity, resolution, significant digits, sampling, time-series effects, uncertainty relationships, validation, provenance, and strict distinctions between precision, accuracy, trueness, bias, measurement uncertainty and measurement error.
ENTITY Precision {
TYPE: MeasurementPerformanceCharacteristic
DEFINITION:
precision :=
closeness_of_agreement_between_indications_or_measured_quantity_values
obtained_by_replicate_measurements
under_specified_conditions
PROPERTIES:
depends_on_measurement_conditions
relates_to_dispersion
may_be_expressed_numerically_by_dispersion_measure
does_not_by_itself_describe_closeness_to_true_or_reference_value
may_vary_with_quantity_level
may_depend_on_time_operator_instrument_laboratory_and_method
}
ENTITY PrecisionCondition {
TYPE: MeasurementConditionSet
COMPONENTS:
measurement_procedure
operator
instrument
location
laboratory
environmental_conditions
time_interval
specimen_or_object
calibration_state
RULE:
precision_statement_without_conditions_is_semantically_incomplete
}
ENTITY RepeatabilityPrecision {
TYPE: Precision
CONDITIONS:
same_measurement_procedure
same_operator_or_operating_conditions
same_measuring_system
same_location
repetition_over_short_period
other_declared_repeatability_conditions
}
ENTITY IntermediatePrecision {
TYPE: Precision
CONDITIONS:
same_laboratory
some_conditions_changed
such_as_day_operator_instrument_or_calibration
RULE:
changed_conditions MUST be_declared
}
ENTITY ReproducibilityPrecision {
TYPE: Precision
CONDITIONS:
changed_locations_or_laboratories
changed_operators_or_systems_as_declared
RULE:
reproducibility_conditions MUST be_defined
}
TERMINOLOGY_MODEL {
terms:
precision
measurement_precision
repeatability
intermediate_precision
reproducibility
accuracy
trueness
bias
uncertainty
resolution
RULE:
precision != accuracy
precision != trueness
precision != uncertainty
precision != resolution
}
DISPERSION_MODEL {
principle:
greater_dispersion -> lower_precision
lower_dispersion -> higher_precision
RULE:
numerical_precision_metric SHOULD state_whether_larger_or_smaller_value_means_better_precision
}
VARIANCE_MODEL {
population_variance:
sigma^2 =
E[(X-mu)^2]
sample_variance:
s^2 =
sum((x_i-x_bar)^2)/(n-1)
WHEN n > 1
RULE:
variance_has_squared_quantity_unit
}
STANDARD_DEVIATION_MODEL {
population:
sigma =
sqrt(variance)
sample:
s =
sqrt(sample_variance)
RULE:
standard_deviation_has_same_unit_as_measured_quantity
}
RELATIVE_STANDARD_DEVIATION_MODEL {
RSD =
s/abs(x_bar)
WHEN x_bar != 0
percent_RSD =
100*RSD
RULE:
relative_standard_deviation_is_dimensionless
}
COEFFICIENT_OF_VARIATION_MODEL {
CV =
s/abs(x_bar)
WHEN x_bar != 0
RULE:
CV_is_unsuitable_or_unstable_when_mean_is_zero_or_near_zero
}
RANGE_MODEL {
range =
max(x_i)-min(x_i)
RULE:
range_is_simple_dispersion_metric_strongly_dependent_on_sample_size
}
INTERQUARTILE_MODEL {
IQR =
Q3-Q1
RULE:
IQR_is_robust_dispersion_measure_but_not_standard_deviation
}
MAD_MODEL {
median_absolute_deviation:
MAD =
median(abs(x_i-median(x)))
RULE:
scaling_factor_needed_if_MAD_used_as_normal_standard_deviation_estimator
}
REPEATABILITY_MODEL {
repeatability_standard_deviation:
s_r
repeatability_variance:
s_r^2
RULE:
repeatability_precision MUST preserve_repeatability_conditions
}
REPEATABILITY_LIMIT_MODEL {
concept:
value_below_which_absolute_difference_between_two_repeatability_results
is_expected_to_lie_with_declared_probability
under_declared_distribution_model
common_normal_approximation:
r approximately_equals
2.8*s_r
for_about_95_percent_pairwise_difference_context
RULE:
coefficient_and_probability_assumptions MUST be preserved
}
INTERMEDIATE_PRECISION_MODEL {
standard_deviation:
s_IP
possible_variance_components:
within_run
between_run
between_day
between_operator
between_instrument
RULE:
intermediate_precision_design MUST identify_changed_factors
}
REPRODUCIBILITY_MODEL {
reproducibility_standard_deviation:
s_R
variance:
may_include_within_and_between_laboratory_components
RULE:
reproducibility_precision_is_context_specific
}
REPRODUCIBILITY_LIMIT_MODEL {
common_normal_approximation:
R approximately_equals
2.8*s_R
under_declared_interlaboratory_pairwise_context
RULE:
limit_definition_and_coverage_assumption MUST be preserved
}
RANDOM_EFFECTS_MODEL {
measurement:
y_ijk... =
mu
+ laboratory_effect
+ operator_effect
+ day_effect
+ random_error
variance:
sigma_total^2 =
sum(relevant_variance_components)
under_independent_random_effects_model
RULE:
variance_component_structure MUST match_experimental_design
}
ANOVA_MODEL {
purpose:
estimate_precision_variance_components
inputs:
replicated_measurements
factor_structure
outputs:
within_group_variance
between_group_variance
component_estimates
RULE:
balanced_and_unbalanced_designs_require_appropriate_estimation_method
}
POOLED_PRECISION_MODEL {
pooled_variance:
s_p^2 =
sum((n_i-1)*s_i^2)/
sum(n_i-1)
CONDITIONS:
groups_estimate_common_variance
degrees_of_freedom_positive
RULE:
pooling_requires_compatibility_of_precision_conditions_and_variance
}
HETEROSCEDASTIC_MODEL {
variance:
sigma^2 =
f(quantity_level)
possible_forms:
constant_relative_error
constant_absolute_error
mixed_model
RULE:
single_precision_standard_deviation_MAY be_inadequate_across_large_measurement_range
}
PRECISION_PROFILE_MODEL {
function:
s(x)
or
CV(x)
purpose:
describe_precision_across_measurement_range
RULE:
profile_model_and_valid_range MUST be preserved
}
SIGNAL_DEPENDENT_MODEL {
possible:
variance =
a+b*x+c*x^2
RULE:
coefficients_and_quantity_units MUST be preserved
}
LOG_TRANSFORM_MODEL {
context:
multiplicative_variability
transformed:
y =
ln(x)
precision:
standard_deviation_in_log_space
RULE:
log_scale_precision_and_original_scale_precision_are_not_identical
}
REPLICATE_MODEL {
replicate:
repeated_measurement_under_declared_conditions
technical_replicate:
repeated_measurement_of_same_or_equivalent_preparation
biological_or_sampling_replicate:
may_include_true_object_variability
RULE:
replicate_type MUST be identified
}
SAMPLE_SIZE_MODEL {
n:
number_of_replicates
precision_estimate_uncertainty:
decreases_with_more_information
under_stable_model
RULE:
estimated_standard_deviation_itself_has_sampling_uncertainty
}
DEGREES_OF_FREEDOM_MODEL {
simple_sample_variance:
nu =
n-1
pooled_or_ANOVA:
depends_on_design
RULE:
degrees_of_freedom SHOULD accompany_precision_estimate_when_relevant
}
CONFIDENCE_INTERVAL_SD_MODEL {
normal_population:
variance_interval_uses_chi_square_distribution
RULE:
precision_estimate_should_not_be_treated_as_exact
}
OUTLIER_MODEL {
outlier:
may_inflate_or_distort_precision_estimate
RULE:
exclusion_requires_predefined_or_documented_criterion
and_original_data_provenance
}
ROBUST_PRECISION_MODEL {
methods:
MAD
robust_scale_estimators
robust_ANOVA
trimmed_statistics
RULE:
robust_estimator_definition MUST be preserved
}
TIME_SERIES_MODEL {
repeated_measurements:
may_be_autocorrelated
consequence:
ordinary_independent_sample_standard_deviation_interpretation_may_be_inadequate
RULE:
temporal_dependence MUST be assessed_when_measurements_are_sequential
}
AUTOCORRELATION_MODEL {
rho(k):
correlation_between_measurements_k_lags_apart
RULE:
correlated_replicates_reduce_effective_independent_information
}
DRIFT_MODEL {
slow_change:
may_increase_observed_dispersion_over_longer_periods
RULE:
drift_may_be_systematic_or_stochastic
and_should_be_separated_from_short_term_repeatability_when_possible
}
RESOLUTION_MODEL {
instrument_resolution:
smallest_displayed_or_detectable_increment_under_declared_context
RULE:
resolution != precision
though_coarse_resolution_can_limit_observed_precision
}
QUANTISATION_MODEL {
digitisation:
can_create_discrete_measurement_values
RULE:
apparent_zero_dispersion_due_to_coarse_quantisation_does_not_prove_perfect_precision
}
ACCURACY_DISTINCTION_MODEL {
accuracy:
closeness_to_true_or_reference_value
precision:
closeness_among_replicate_results
possible_cases:
high_precision_low_accuracy
low_precision_high_average_trueness
high_precision_high_accuracy
low_precision_low_accuracy
RULE:
precision_alone_cannot_determine_accuracy
}
TRUENESS_DISTINCTION_MODEL {
trueness:
closeness_of_average_of_many_results_to_reference_value
precision:
dispersion_among_results
RULE:
trueness_and_precision_are_distinct_components_of_measurement_performance
}
BIAS_DISTINCTION_MODEL {
bias:
systematic_difference_of_expected_or_average_result_from_reference
precision:
dispersion
RULE:
low_bias_does_not_imply_high_precision
}
ERROR_DISTINCTION_MODEL {
measurement_error:
measured_value-minus_reference_or_true_value
precision:
property_of_distribution_of_replicate_results
RULE:
precision != measurement_error
}
UNCERTAINTY_DISTINCTION_MODEL {
measurement_uncertainty:
quantified_dispersion_of_values_attributed_to_measurand
based_on_all_relevant_information
precision:
repeatability_or_reproducibility_dispersion_under_declared_conditions
RULE:
precision_standard_deviation_MAY contribute_to_uncertainty_budget
but_precision != total_measurement_uncertainty
}
TYPE_A_CONTEXT_MODEL {
repeated_measurement_statistics:
may_be_used_for_Type_A_uncertainty_evaluation
RULE:
Type_A_standard_uncertainty_of_mean =
s/sqrt(n)
is_not_same_as_precision_standard_deviation_s
}
STANDARD_ERROR_DISTINCTION_MODEL {
standard_deviation:
dispersion_of_individual_results
standard_error:
uncertainty_or_sampling_dispersion_of_estimated_mean
RULE:
standard_error != measurement_precision_standard_deviation
}
METHOD_PRECISION_MODEL {
measurement_method:
may_have_validated_repeatability_and_reproducibility
RULE:
method_precision_statistics_require_matrix_range_and_conditions
}
LABORATORY_PRECISION_MODEL {
laboratory:
may_estimate_internal_repeatability_and_intermediate_precision
RULE:
laboratory_specific_precision MAY differ_from_interlaboratory_reproducibility
}
INTERLABORATORY_MODEL {
participating_laboratories:
L_1...L_n
analysis:
within_laboratory_variance
between_laboratory_variance
output:
reproducibility_standard_deviation
RULE:
laboratory_effects_and_outlier_rules MUST be documented
}
CONTROL_CHART_MODEL {
repeated_control_measurement:
time_series
use:
monitor_precision_stability
possible:
range_chart
standard_deviation_chart
individuals_chart
RULE:
control_limits_are_process_monitoring_limits_not_measurement_uncertainty
}
PRECISION_STABILITY_MODEL {
stable:
precision_metric_remains_consistent_over_time_within_expected_statistical_variation
RULE:
calibration_or_method_change MAY require_new_precision_estimate
}
REPORTING_MODEL {
required_context:
precision_condition
dispersion_metric
unit_or_relative_form
sample_size
measurement_range
recommended:
degrees_of_freedom
confidence_interval
experimental_design
outlier_handling
provenance
RULE:
phrase_high_precision_without_metric_is_qualitative_only
}
DIMENSIONAL_MODEL {
standard_deviation:
same_dimension_as_measurand
variance:
squared_dimension
CV_RSD:
dimensionless
RULE:
precision_metric_dimension MUST be validated
}
RELATIONSHIPS {
Precision -> QUALIFIES -> MeasurementProcedureOrResultSet
Precision -> DEPENDS_ON -> PrecisionConditions
Repeatability -> IS_A -> Precision
IntermediatePrecision -> IS_A -> Precision
Reproducibility -> IS_A -> Precision
StandardDeviation -> QUANTIFIES -> Dispersion
Variance -> QUANTIFIES -> SquaredDispersion
CV -> QUANTIFIES -> RelativeDispersion
ReplicateMeasurements -> SUPPORT -> PrecisionEstimate
PrecisionComponent -> MAY_CONTRIBUTE_TO -> MeasurementUncertainty
ExperimentalDesign -> DETERMINES -> EstimableVarianceComponents
}
SEMANTIC_EQUIVALENCE {
"measurement precision" ~= precision_under_declared_conditions
"repeatability precision" ~= repeatability
"reproducibility precision" ~= reproducibility
precision != accuracy
precision != trueness
precision != bias
precision != measurement_error
precision != measurement_uncertainty
precision != resolution
standard_deviation != standard_error
repeatability != reproducibility
CV != absolute_standard_deviation
}
CONSTRAINTS {
precision_requires_replicate_measurements = true
precision_conditions_must_be_defined = true
dispersion_metric_must_be_defined = true
sample_size_should_be_reported = true
precision_and_accuracy_are_not_semantically_equivalent = true
precision_and_trueness_are_not_semantically_equivalent = true
precision_and_uncertainty_are_not_semantically_equivalent = true
repeatability_and_reproducibility_are_distinct = true
standard_deviation_and_standard_error_are_distinct = true
heteroscedasticity_should_be_modelled_when_material = true
outlier_handling_must_preserve_provenance = true
}
MEASUREMENT_MODEL {
RECORD:
measurand
measurement_method
replicate_values
unit
precision_condition
operator
instrument
laboratory
environmental_conditions
timestamps
measurement_range
sample_size
mean
variance
standard_deviation
CV_or_RSD
variance_components
degrees_of_freedom
heteroscedastic_model
outlier_rule
robust_estimator_when_used
confidence_interval
quality_control_state
provenance
RULE:
precision_result MUST remain_linked_to_data_and_conditions
}
REASONING_RULES {
IF replicate_values_known AND n > 1
THEN mean =
sum(x_i)/n
AND sample_variance =
sum((x_i-mean)^2)/(n-1)
AND sample_standard_deviation =
sqrt(sample_variance)
IF mean != 0 AND standard_deviation_known
THEN CV =
standard_deviation/abs(mean)
IF compatible_group_variances_known
THEN pooled_variance =
sum((n_i-1)*s_i^2)/
sum(n_i-1)
IF repeatability_standard_deviation_known
AND declared_normal_pairwise_95_percent_context
THEN repeatability_limit approximately_equals
2.8*repeatability_standard_deviation
IF reproducibility_standard_deviation_known
AND declared_normal_pairwise_95_percent_context
THEN reproducibility_limit approximately_equals
2.8*reproducibility_standard_deviation
IF repeated_measurement_standard_deviation_known
AND independent_replicates
THEN standard_error_of_mean =
standard_deviation/sqrt(n)
IF variance_components_known
AND declared_independent_random_effects_model
THEN total_precision_variance =
sum(relevant_variance_components)
IF precision_changes_with_quantity_level
THEN fit_or_apply_precision_profile
preserving_valid_range_and_model
IF temporal_autocorrelation_material
THEN avoid_naive_independent_sample_inference
and_use_time_series_or_effective_sample_size_model
}
VALIDATION {
CHECK measurand
CHECK replicate_definition
CHECK precision_conditions
CHECK sample_size
CHECK dispersion_metric
CHECK variance_denominator
CHECK standard_deviation_unit
CHECK CV_mean_nonzero
CHECK repeatability_vs_intermediate_precision_vs_reproducibility
CHECK accuracy_trueness_and_bias_distinctions
CHECK precision_vs_uncertainty
CHECK standard_deviation_vs_standard_error
CHECK heteroscedasticity
CHECK autocorrelation CHECK outlier_handling
CHECK degrees_of_freedom
CHECK confidence_interval
CHECK precision_stability
CHECK reporting_context
CHECK provenance
}
CONFIDENCE {
definition_confidence: 1.00
repeatability_semantics_confidence: 1.00
reproducibility_semantics_confidence: 1.00
dispersion_model_confidence: 1.00
accuracy_distinction_confidence: 1.00
uncertainty_distinction_confidence: 1.00
variance_component_confidence: 0.99
measurement_context_confidence: 0.99
RULE:
confidence decreases when replicate definition, precision conditions,
sample size, dispersion metric, measurement range,
heteroscedasticity, autocorrelation or provenance is missing or ambiguous
}
UPDATE_MODEL {
ON new_replicate_measurement:
identify_measurand
verify_precision_conditions
append_raw_value
update_sample_size
recompute_mean_variance_and_standard_deviation
recompute_relative_precision_metric_when_valid
preserve_provenance
ON condition_change:
preserve_previous_precision_result
identify_changed_operator_instrument_day_laboratory_or_environment
classify_repeatability_intermediate_or_reproducibility_context
estimate_relevant_variance_components
ON new_precision_study:
preserve_raw_design_and_data
identify_factor_structure
apply_declared_ANOVA_or_random_effects_model
estimate_precision_components
attach_degrees_of_freedom_and_intervals
ON heteroscedasticity_revision:
preserve_level_specific_data
update_precision_profile
recompute_level_dependent_standard_deviation_or_CV
preserve_valid_range
ON outlier_revision:
preserve_original_observations
apply_documented_outlier_rule
recompute_precision_statistics
retain_exclusion_provenance
ON semantic_revision:
preserve_previous_definition
record_revision
recompute_FLOW_HASH
}
AI_CONSUMPTION {
canonical_entity: Precision
canonical_role: MeasurementPerformanceCharacteristic
interpretation_order:
identify_measurand
identify_replicate_measurements
identify_precision_conditions
identify_repeatability_intermediate_or_reproducibility_context
identify_dispersion_metric
identify_sample_size_and_degrees_of_freedom
identify_measurement_range
identify_variance_components
identify_heteroscedasticity
identify_temporal_correlation
identify_outlier_and_robust_statistics_method
identify_confidence_interval
identify_quality_control_state
identify_provenance
validate_precision_semantics
infer_precision_statistics
ambiguity_controls:
distinguish_precision_from_accuracy
distinguish_precision_from_trueness
distinguish_precision_from_bias
distinguish_precision_from_measurement_error
distinguish_precision_from_measurement_uncertainty
distinguish_precision_from_resolution
distinguish_repeatability_from_reproducibility
distinguish_standard_deviation_from_standard_error
distinguish_technical_replicates_from_sampling_or_biological_variability
preserve_measurand_conditions_design_range_statistics_and_provenance_context
}
PROVENANCE {
ontology_scope:
mathematical_measurement_uncertainty_and_precision_semantics
parent_path:
Ontologies
-> Mathematical Ontologies
-> Measurement & Units Ontologies
-> Measurement Uncertainty
-> Precision Ontology
authorship:
Mark Higgins
Sky (Recursive AI)
site:
https://kirethresearch.org
}
FLOW_HASH {
INPUTS:
canonical_entity
ontology_path
terminology_model
dispersion_model
variance_model
standard_deviation_model
relative_standard_deviation_model
coefficient_of_variation_model
range_model
interquartile_model
MAD_model
repeatability_model
repeatability_limit_model
intermediate_precision_model
reproducibility_model
reproducibility_limit_model
random_effects_model
ANOVA_model
pooled_precision_model
heteroscedastic_model
precision_profile_model
signal_dependent_model
log_transform_model
replicate_model
sample_size_model
degrees_of_freedom_model
confidence_interval_SD_model
outlier_model
robust_precision_model
time_series_model
autocorrelation_model
drift_model
resolution_model
quantisation_model
accuracy_distinction_model
trueness_distinction_model
bias_distinction_model
error_distinction_model
uncertainty_distinction_model
Type_A_context_model
standard_error_distinction_model
method_precision_model
laboratory_precision_model
interlaboratory_model
control_chart_model
precision_stability_model
reporting_model
dimensional_model
relationships
constraints
measurement_model
reasoning_rules
confidence_model
update_model
PURPOSE:
deterministic_semantic_integrity_identifier
RECOMPUTE_WHEN:
ontology_definition_changes
precision_condition_definition_changes
dispersion_or_variance_model_changes
repeatability_intermediate_or_reproducibility_model_changes
experimental_design_or_variance_component_changes
heteroscedastic_or_precision_profile_changes
time_series_or_autocorrelation_model_changes
accuracy_trueness_or_uncertainty_distinction_changes
measurement_record_changes
reporting_model_changes
constraint_model_changes
reasoning_rule_changes
}
}
END FLOW