Confidence Interval Ontology
Measurement Uncertainty
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 16 September 2026
Audio Overview
# Confidence Interval Ontology — Flow Algebra
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/measurement-uncertainty/confidence-interval-ontology.html
FLOW ConfidenceIntervalOntology {
DOMAIN:
Mathematical Ontologies
-> Measurement & Units Ontologies
-> Measurement Uncertainty
-> Confidence Interval Ontology
PURPOSE:
Represent confidence intervals as interval estimates produced by statistical procedures with a declared long-run coverage property under specified assumptions, including confidence level, lower and upper bounds, standard errors, sampling distributions, normal, Student t, chi-square and F based intervals, proportions, differences, regression parameters, transformed and asymmetric intervals, bootstrap methods, profile likelihood, repeated-measurement contexts, measurement uncertainty relationships, coverage intervals, multiple comparisons, provenance, and strict distinctions between confidence intervals, Bayesian credible intervals, prediction intervals, tolerance intervals and metrological uncertainty intervals.
ENTITY ConfidenceInterval {
TYPE: StatisticalIntervalEstimate
COMPONENTS:
lower_bound
upper_bound
confidence_level
parameter_or_estimand
statistical_method
assumptions
dataset_or_summary
provenance
DEFINITION:
confidence_interval :=
interval_generated_by_a_statistical_procedure
whose_long_run_frequency_of_covering_the_target_parameter
equals_or_approximates_declared_confidence_level
under_the_model_and_sampling_assumptions
RULE:
confidence_level_is_property_of_interval_procedure
not_probability_that_fixed_parameter_is_inside_one_realised_frequentist_interval
}
ENTITY ConfidenceLevel {
TYPE: ProbabilityLevel
SYMBOL: 1-alpha
RANGE:
0 < confidence_level < 1
}
ENTITY LowerConfidenceBound {
TYPE: IntervalBound
SYMBOL: L
}
ENTITY UpperConfidenceBound {
TYPE: IntervalBound
SYMBOL: U
}
TERMINOLOGY_MODEL {
terms:
confidence_interval
confidence_level
confidence_coefficient
confidence_limits
confidence_bounds
credible_interval
prediction_interval
tolerance_interval
coverage_interval
uncertainty_interval
RULE:
interval_type MUST be identified_before_interpretation
}
TWO_SIDED_MODEL {
interval:
[L,U]
confidence_level:
1-alpha
equal_tail_common_form:
alpha/2_in_each_tail
RULE:
two_sided_interval_need_not_be_equal_tail
}
ONE_SIDED_MODEL {
lower:
[L,+infinity)
upper:
(-infinity,U]
RULE:
one_sided_confidence_bound != two_sided_interval
}
NORMAL_MEAN_KNOWN_SIGMA_MODEL {
data:
x_1,...,x_n
assumptions:
independent
normal_population_or_valid_normal_mean_sampling_model
population_sigma_known
mean:
x_bar
standard_error:
sigma/sqrt(n)
interval:
x_bar +/- z_(1-alpha/2)*sigma/sqrt(n)
RULE:
known_sigma_normal_interval_is_not_default_when_sigma_is_estimated
}
STUDENT_T_MEAN_MODEL {
assumptions:
independent_observations
approximately_normal_population_for_exact_small_sample_interval
mean:
x_bar
sample_SD:
s
degrees_of_freedom:
nu =
n-1
standard_error:
s/sqrt(n)
interval:
x_bar +/- t_(1-alpha/2,nu)*s/sqrt(n)
RULE:
t_interval_accounts_for_estimated_population_standard_deviation
}
LARGE_SAMPLE_MODEL {
estimator:
theta_hat
asymptotic_standard_error:
SE(theta_hat)
approximate_interval:
theta_hat +/- z_(1-alpha/2)*SE(theta_hat)
RULE:
asymptotic_normality_must_be_reasonable
}
VARIANCE_INTERVAL_MODEL {
normal_population:
(n-1)*s^2/sigma^2 ~ chi_square_(n-1)
interval_for_variance:
[
(n-1)*s^2/chi2_(1-alpha/2,nu),
(n-1)*s^2/chi2_(alpha/2,nu)
]
RULE:
exact_chi_square_variance_interval_requires_normal_population
}
STANDARD_DEVIATION_INTERVAL_MODEL {
derive:
square_root_of_variance_interval_bounds
RULE:
transformed_bounds_are_asymmetric
}
DIFFERENCE_OF_MEANS_MODEL {
independent_groups:
delta =
mu_1-mu_2
estimator:
x_bar_1-x_bar_2
possible_methods:
Welch_t
pooled_t
asymptotic_normal
RULE:
equal_variance_assumption_must_be_declared_if_pooled_method_used
}
WELCH_MODEL {
standard_error:
sqrt(
s_1^2/n_1
+ s_2^2/n_2
)
degrees_of_freedom:
Welch_Satterthwaite
RULE:
Welch_interval_does_not_require_equal_population_variances
}
PAIRED_MODEL {
differences:
d_i =
x_i-y_i
mean_difference:
d_bar
interval:
t_interval_on_differences
RULE:
paired_data_must_not_be_analysed_as_independent_groups_without_justification
}
PROPORTION_MODEL {
parameter:
p
estimator:
p_hat =
x/n
possible_intervals:
Wilson
Clopper_Pearson
Jeffreys_if_Bayesian_context
Wald
likelihood_based
RULE:
interval_method MUST be identified
because_methods_differ_materially_for_small_n_or_extreme_p
}
WILSON_MODEL {
purpose:
frequentist_binomial_proportion_interval
RULE:
Wilson_interval_is_not_identical_to_simple_Wald_interval
}
BINOMIAL_EXACT_MODEL {
common_name:
Clopper_Pearson
basis:
inversion_of_binomial_tests
RULE:
exact_refers_to_coverage_construction_not_zero_approximation_error_in_all_senses
}
POISSON_RATE_MODEL {
counts:
N
exposure:
T
rate:
lambda =
N/T
interval:
may_use_exact_chi_square_or_likelihood_method
RULE:
exposure_and_count_model MUST be preserved
}
CORRELATION_INTERVAL_MODEL {
Pearson_r:
may_use_Fisher_z_transform
transform:
z_r =
atanh(r)
approximate_SE:
1/sqrt(n-3)
inverse:
r =
tanh(z_r)
RULE:
assumptions_and_sample_size_require_validation
}
REGRESSION_COEFFICIENT_MODEL {
linear_model:
beta_hat_j
interval:
beta_hat_j
+/- critical_value*SE(beta_hat_j)
RULE:
covariance_model_and_degrees_of_freedom MUST match_regression_method
}
NONLINEAR_PARAMETER_MODEL {
possible:
Wald
profile_likelihood
bootstrap
likelihood_ratio
RULE:
symmetric_Wald_interval_may_be_inadequate_for_nonlinear_or_boundary_parameters
}
PROFILE_LIKELIHOOD_MODEL {
parameter:
theta
profile:
maximise_likelihood_over_nuisance_parameters
interval:
set_of_theta_values_not_rejected_by_likelihood_ratio_criterion
RULE:
likelihood_threshold_and_asymptotic_assumptions MUST be preserved
}
BOOTSTRAP_MODEL {
data:
observed_sample
resampling:
generate_bootstrap_samples
estimator_distribution:
theta_hat_star
possible_intervals:
percentile
basic
BCa
studentized
RULE:
bootstrap_scheme_and_interval_type MUST be preserved
}
BCA_MODEL {
abbreviation:
bias_corrected_and_accelerated
adjusts_for:
bias
skewness_related_acceleration
RULE:
BCa_requires_defined_resampling_and_acceleration_estimation
}
TRANSFORMATION_MODEL {
parameter:
theta
transform:
phi =
g(theta)
process:
compute_interval_in_transformed_space
inverse_transform_bounds
RULE:
nonlinear_transformation_can_create_asymmetric_interval
}
LOG_SCALE_MODEL {
positive_parameter:
theta > 0
transform:
log(theta)
interval:
compute_on_log_scale
exponentiate_bounds
RULE:
geometric_or_ratio_parameters_often_benefit_from_log_scale_interval
}
RATIO_MODEL {
ratio:
R =
theta_1/theta_2
possible_methods:
log_ratio
Fieller
delta_method
bootstrap
RULE:
denominator_near_zero_requires_special_care
}
FIELLER_MODEL {
purpose:
confidence_interval_for_ratio_with_uncertain_denominator
RULE:
interval_may_be_unbounded_or_disconnected
and_should_not_be_forced_into_finite_symmetric_form
}
SAMPLE_SIZE_MODEL {
larger_n:
generally_narrows_interval
under_fixed_variability_and_model
common_mean_scaling:
width proportional_to 1/sqrt(n)
RULE:
dependence_clustering_or_design_effect_changes_effective_information
}
STANDARD_ERROR_MODEL {
estimator:
theta_hat
standard_error:
estimated_sampling_standard_deviation_of_theta_hat
RULE:
standard_error != standard_deviation_of_raw_observations
}
CRITICAL_VALUE_MODEL {
possible:
z_quantile
t_quantile
chi_square_quantiles
F_quantiles
bootstrap_quantiles
RULE:
critical_distribution MUST match_interval_method
}
CONFIDENCE_LEVEL_MODEL {
common:
0.90
0.95
0.99
effect:
higher_confidence_level_generally_produces_wider_interval
all_else_equal
RULE:
confidence_level MUST be explicit
}
COVERAGE_MODEL {
nominal:
declared_confidence_level
actual:
true_repeated_sampling_coverage_under_data_generating_process
RULE:
nominal_and_actual_coverage_can_differ_when_assumptions_or_approximations_fail
}
ASSUMPTION_MODEL {
possible:
independence
identical_distribution
normality
correct_variance_model
random_sampling
correct_model_specification
no_unmodelled_clustering
RULE:
confidence_interval_validity_depends_on_relevant_assumptions
}
CLUSTERED_DATA_MODEL {
clustered_observations:
correlated_within_cluster
methods:
cluster_robust_standard_error
mixed_model
GEE
cluster_bootstrap
RULE:
naive_independent_interval_can_be_too_narrow
}
AUTOCORRELATION_MODEL {
time_series:
serial_dependence
methods:
time_series_model
HAC_standard_error
block_bootstrap
RULE:
serial_dependence MUST be considered
}
HETEROSCEDASTIC_MODEL {
nonconstant_variance:
may_require_robust_or_modelled_standard_error
RULE:
homoscedastic_standard_errors_should_not_be_assumed_when_invalid
}
MULTIPLE_COMPARISON_MODEL {
many_intervals:
familywise_coverage_differs_from_individual_coverage
possible_adjustments:
Bonferroni
Holm
simultaneous_methods
RULE:
individual_95_percent_intervals_do_not_imply_95_percent_joint_coverage
}
SIMULTANEOUS_INTERVAL_MODEL {
goal:
joint_coverage_for_multiple_parameters_or_curve
RULE:
simultaneous_interval_or_band_is_wider_than_pointwise_interval_in_general
}
CONFIDENCE_BAND_MODEL {
function:
f(x)
pointwise:
interval_at_each_x
simultaneous:
band_covering_entire_function_with_declared_probability
RULE:
pointwise_and_simultaneous_bands_are_distinct
}
PREDICTION_INTERVAL_MODEL {
target:
future_observation
includes:
parameter_estimation_uncertainty
plus_observation_variability
RULE:
prediction_interval != confidence_interval_for_mean_or_parameter
}
TOLERANCE_INTERVAL_MODEL {
target:
proportion_of_population
statement:
interval_expected_with_declared_confidence_to_contain_declared_population_fraction
RULE:
tolerance_interval != confidence_interval
}
BAYESIAN_CREDIBLE_INTERVAL_MODEL {
posterior:
p(theta|data)
credible_interval:
posterior_probability_content
RULE:
Bayesian_credible_interval != frequentist_confidence_interval
unless_specific_equivalence_conditions_hold
}
METROLOGICAL_COVERAGE_INTERVAL_MODEL {
measurement_uncertainty_context:
interval_containing_set_of_quantity_values_attributed_to_measurand
with_stated_coverage_probability_under_declared_distribution
RULE:
metrological_coverage_interval != frequentist_confidence_interval_in_general
}
EXPANDED_UNCERTAINTY_MODEL {
U =
k*u_c
interval_common:
y +/- U
RULE:
expanded_uncertainty_interval MUST_NOT automatically_be_called_confidence_interval
}
REPEATABILITY_CONTEXT_MODEL {
repeatability_SD:
s_r
confidence_interval_for_true_repeatability_variance_or_SD:
statistical_interval_about_precision_parameter
RULE:
repeatability_SD_and_its_confidence_interval_are_distinct
}
REPRODUCIBILITY_CONTEXT_MODEL {
reproducibility_SD:
s_R
confidence_interval:
expresses_sampling_uncertainty_of_estimated_precision_parameter
RULE:
reproducibility_limit_and_confidence_interval_are_distinct
}
MEASUREMENT_MEAN_MODEL {
repeated_measurements:
x_i
confidence_interval_for_mean:
describes_sampling_uncertainty_of_estimated_mean_under_statistical_model
measurement_uncertainty:
may_include_additional_components_not represented_by_sampling_SE
RULE:
confidence_interval_for_mean != complete_measurement_uncertainty_statement
}
ROUNDING_MODEL {
bounds:
should_be_reported_with_precision_consistent_with_data_and_method
RULE:
rounding_must_not_invert_or_collapse_interval
}
OPEN_CLOSED_BOUND_MODEL {
interval_notation:
[L,U]
(L,U)
one_sided
RULE:
endpoint_inclusion_semantics SHOULD be preserved_when_method_defines_them
}
INFINITE_BOUND_MODEL {
possible:
lower_unbounded
upper_unbounded
both_unbounded_under_pathological_case
RULE:
infinite_bounds_are_valid_results_for_some_interval_methods
}
EMPTY_OR_DISCONNECTED_MODEL {
possible_in_special_inverted_test_or_ratio_problems:
empty_set
union_of_intervals
RULE:
interval_result_structure MUST_NOT be_forced_into_single_finite_range
}
REPORTING_MODEL {
REQUIRED:
estimand
point_estimate_when_available
lower_bound
upper_bound
confidence_level
interval_method
RECOMMENDED:
sample_size
degrees_of_freedom
standard_error
model
assumptions
transformation
multiple_comparison_adjustment
software_or_algorithm_version
provenance
RULE:
confidence_interval_without_level_or_method_is_incomplete
}
RELATIONSHIPS {
ConfidenceInterval -> ESTIMATES -> ParameterOrEstimand
ConfidenceInterval -> HAS -> LowerBound
ConfidenceInterval -> HAS -> UpperBound
ConfidenceInterval -> HAS -> ConfidenceLevel
StandardError -> CONTRIBUTES_TO -> ConfidenceIntervalWidth
CriticalValue -> SCALES -> StandardError
SamplingDistribution -> DEFINES -> IntervalProcedure
Bootstrap -> MAY_GENERATE -> ConfidenceInterval
ConfidenceInterval -> DISTINCT_FROM -> PredictionInterval
ConfidenceInterval -> DISTINCT_FROM -> CredibleInterval
ConfidenceInterval -> DISTINCT_FROM -> ToleranceInterval
ConfidenceInterval -> DISTINCT_FROM -> MetrologicalCoverageInterval
}
SEMANTIC_EQUIVALENCE {
"CI" ~= confidence_interval WHEN statistical_context
"confidence limits" ~= confidence_interval_bounds
confidence_interval != credible_interval
confidence_interval != prediction_interval
confidence_interval != tolerance_interval
confidence_interval != expanded_uncertainty_interval
confidence_interval != metrological_coverage_interval
confidence_level != posterior_probability_of_realised_interval
standard_error != confidence_interval
}
CONSTRAINTS {
confidence_level_must_be_between_zero_and_one = true
lower_bound_must_not_exceed_upper_bound_for_simple_interval = true
estimand_must_be_defined = true
interval_method_must_be_defined = true
assumptions_should_be_preserved = true
confidence_interval_and_credible_interval_are_not_semantically_equivalent = true
confidence_interval_and_prediction_interval_are_not_semantically_equivalent = true
confidence_interval_and_measurement_uncertainty_interval_are_not_semantically_equivalent = true
multiple_comparison_context_should_be_preserved = true
transformation_and_inverse_transformation_should_be_preserved = true
}
MEASUREMENT_MODEL {
RECORD:
estimand
point_estimate
lower_bound
upper_bound
confidence_level
alpha
one_or_two_sided
interval_method
sample_size
degrees_of_freedom
standard_error
sampling_distribution
transformation
model
assumptions
clustering_or_dependence
heteroscedasticity_model
bootstrap_configuration
multiple_comparison_adjustment
dataset_identifier
software_or_algorithm
quality_flag
provenance
RULE:
confidence_interval MUST remain_linked_to_estimand_method_and_sampling_model
}
REASONING_RULES {
IF confidence_level_known
THEN alpha =
1-confidence_level
IF normal_mean_known_sigma_context
THEN interval =
mean +/- z_(1-alpha/2)*sigma/sqrt(n)
IF normal_mean_unknown_sigma_context
AND n > 1
THEN interval =
mean +/- t_(1-alpha/2,n-1)*s/sqrt(n)
IF variance_of_normal_population_estimated
THEN derive_variance_interval_using_chi_square_quantiles
IF transformation_used
THEN compute_interval_in_transformed_space
AND inverse_transform_bounds
IF bootstrap_samples_known
THEN derive_interval using_declared_percentile_basic_BCa_or_studentized_method
IF clustered_or_autocorrelated_data_present
THEN use_dependence_appropriate_standard_error_or_resampling_method
IF multiple_parameters_require_joint_coverage
THEN apply_declared_simultaneous_or_multiple_comparison_method
IF confidence_interval_for_mean_computed
THEN do_not_equate_it_with_prediction_interval
or_complete_measurement_uncertainty
IF Bayesian_posterior_interval_detected
THEN classify_as_credible_interval
not_frequentist_confidence_interval
IF expanded_uncertainty_y_plus_minus_U_detected
THEN preserve_metrological_coverage_semantics
unless_explicit_frequentist_confidence_procedure_is_defined
}
VALIDATION {
CHECK estimand
CHECK point_estimate
CHECK lower_and_upper_bounds
CHECK confidence_level
CHECK one_or_two_sided
CHECK interval_method
CHECK sample_size
CHECK degrees_of_freedom
CHECK standard_error
CHECK sampling_distribution
CHECK assumptions
CHECK normality_when_required
CHECK independence_or_dependence_model
CHECK heteroscedasticity
CHECK transformation
CHECK bootstrap_method
CHECK multiple_comparison_context
CHECK confidence_vs_credible_interval
CHECK confidence_vs_prediction_interval
CHECK confidence_vs_tolerance_interval
CHECK confidence_vs_measurement_uncertainty_interval
CHECK reporting_precision
CHECK provenance
}
CONFIDENCE {
definition_confidence: 1.00
frequentist_interpretation_confidence: 1.00
normal_mean_interval_confidence: 1.00
Student_t_interval_confidence: 1.00
variance_interval_confidence: 1.00
bootstrap_semantics_confidence: 0.99
credible_interval_distinction_confidence: 1.00
measurement_uncertainty_distinction_confidence: 1.00
RULE:
confidence decreases when estimand, interval method, sampling model,
dependence, transformation, confidence level,
multiple-comparison context or provenance is missing or ambiguous
}
UPDATE_MODEL {
ON new_interval:
identify_estimand
identify_point_estimate
attach_bounds
attach_confidence_level
identify_one_or_two_sided_form
identify_interval_method
attach_sampling_model_and_assumptions
attach_quality_flag
preserve_provenance
ON new_data:
preserve_raw_or_summary_data
update_point_estimate
update_standard_error
recompute_interval_using_declared_method
preserve_previous_interval_version
ON model_revision:
preserve_previous_model
update_variance_dependence_or_distribution_assumptions
recompute_interval
update_confidence_and_provenance
ON confidence_level_revision:
preserve_estimator_and_model
update_alpha_and_critical_values
recompute_bounds
record_new_level
ON bootstrap_revision:
preserve_original_data
update_resampling_scheme_seed_or_algorithm
recompute_bootstrap_distribution_and_interval
preserve_configuration
ON semantic_revision:
preserve_previous_definition
record_revision
recompute_FLOW_HASH
}
AI_CONSUMPTION {
canonical_entity: ConfidenceInterval
canonical_role: StatisticalIntervalEstimate
interpretation_order:
identify_estimand
identify_point_estimate
identify_lower_and_upper_bounds
identify_confidence_level
identify_one_or_two_sided_form
identify_interval_method
identify_sample_size_and_degrees_of_freedom
identify_standard_error_and_sampling_distribution
identify_model_and_assumptions
identify_dependence_or_heteroscedasticity
identify_transformation
identify_bootstrap_or_likelihood_method
identify_multiple_comparison_context
identify_provenance
validate_frequentist_interval_semantics
infer_interval_relationships
ambiguity_controls:
distinguish_confidence_interval_from_Bayesian_credible_interval
distinguish_confidence_interval_from_prediction_interval
distinguish_confidence_interval_from_tolerance_interval
distinguish_confidence_interval_from_expanded_uncertainty_interval
distinguish_confidence_interval_from_metrological_coverage_interval
distinguish_standard_error_from_confidence_interval
distinguish_confidence_level_from_probability_that_fixed_parameter_is_in_realised_interval
preserve_estimand_level_method_sampling_model_assumptions_transformation_and_provenance_context
}
PROVENANCE {
ontology_scope:
mathematical_measurement_uncertainty_and_confidence_interval_semantics
parent_path:
Ontologies
-> Mathematical Ontologies
-> Measurement & Units Ontologies
-> Measurement Uncertainty
-> Confidence Interval Ontology
authorship:
Mark Higgins
Sky (Recursive AI)
site:
https://kirethresearch.org
}
FLOW_HASH {
INPUTS:
canonical_entity
ontology_path
terminology_model
two_sided_model
one_sided_model
normal_mean_known_sigma_model
Student_t_mean_model
large_sample_model
variance_interval_model
standard_deviation_interval_model
difference_of_means_model
Welch_model
paired_model
proportion_model
Wilson_model
binomial_exact_model
Poisson_rate_model
correlation_interval_model
regression_coefficient_model
nonlinear_parameter_model
profile_likelihood_model
bootstrap_model
BCa_model
transformation_model
log_scale_model
ratio_model
Fieller_model
sample_size_model
standard_error_model
critical_value_model
confidence_level_model
coverage_model
assumption_model
clustered_data_model
autocorrelation_model
heteroscedastic_model
multiple_comparison_model
simultaneous_interval_model
confidence_band_model
prediction_interval_model
tolerance_interval_model
Bayesian_credible_interval_model
metrological_coverage_interval_model
expanded_uncertainty_model
repeatability_context_model
reproducibility_context_model
measurement_mean_model
rounding_model
open_closed_bound_model
infinite_bound_model
empty_or_disconnected_model
reporting_model
relationships
constraints
measurement_model
reasoning_rules
confidence_model
update_model
PURPOSE:
deterministic_semantic_integrity_identifier
RECOMPUTE_WHEN:
ontology_definition_changes
confidence_interval_definition_changes
interval_method_or_sampling_distribution_changes
confidence_level_changes
dependence_or_variance_model_changes
transformation_or_bootstrap_method_changes
multiple_comparison_or_simultaneous_coverage_changes
confidence_credible_prediction_or_tolerance_distinctions_change
metrological_coverage_semantics_change
measurement_record_changes
reporting_model_changes
constraint_model_changes
reasoning_rule_changes
}
}
END FLOW