calc.engineer

confidence interval

confidence_interval

Calculate the confidence interval for a sample mean. Given a sample mean, sample size, standard deviation, and confidence level, computes the margin of error, lower and upper bounds, critical z-score, and standard error. Supports finite population correction (FPC) when a population size is provided, which narrows the interval for samples that are a large fraction of the population. Uses the Abramowitz & Stegun rational approximation for the inverse normal CDF to derive the critical z-value. Common in survey analysis, A/B testing, and quality control.

Input Schema

TableJSON Schema

Name	Required	Description
`sample_mean`	Yes	The observed sample mean (x-bar) around which the confidence interval is centered.
`sample_size`	Yes	The number of observations in the sample (n). Must be a positive integer.
`population_size`	No	Total population size for finite population correction (FPC). Omit for infinite population assumption.
`confidence_level`	No	Confidence level as a decimal between 0 and 1 (e.g. 0.95 for 95%). Default is 0.95.
`standard_deviation`	Yes	The standard deviation of the sample or population. Must be a positive number.

Output Schema

TableJSON Schema

Name	Required	Description
`z_score`	Yes	Critical z-value used for the given confidence level.
`lower_bound`	Yes	Lower bound of the confidence interval.
`upper_bound`	Yes	Upper bound of the confidence interval.
`standard_error`	Yes	Standard error of the mean, optionally adjusted with finite population correction.
`margin_of_error`	Yes	Half-width of the confidence interval (z_score * standard_error).
`confidence_level`	Yes	The confidence level used (echoed back).

TDQS

A4.5/5.0

Behavior4/5

Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

With no annotations, the description carries the burden of behavioral transparency. It discloses the algorithm ('Abramowitz & Stegun rational approximation for the inverse normal CDF') and the effect of finite population correction. It does not mention any side effects, but as a calculation tool, none are expected. The description is sufficiently transparent.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Conciseness5/5

Is the description appropriately sized, front-loaded, and free of redundancy?

The description is concise with two well-structured sentences. The first sentence states the core function and outputs, and the second adds important details about FPC and the algorithm. Every sentence earns its place with no wasted words.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Completeness5/5

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

Given the tool has 5 parameters and an output schema, the description provides enough context for correct invocation. It explains the purpose, inputs, special features (FPC), and expected outputs. It is complete for an AI agent to understand and use the tool correctly.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Parameters4/5

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Although schema description coverage is 100%, the description adds meaning beyond the schema by explaining the role of finite population correction, the FPC parameter, and the approximation method. It also lists the computed outputs. This adds value beyond the basic parameter descriptions.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Purpose5/5

Does the description clearly state what the tool does and how it differs from similar tools?

The description clearly states the tool's purpose: 'Calculate the confidence interval for a sample mean.' It specifies the inputs (sample mean, size, standard deviation, confidence level) and outputs (margin of error, bounds, critical z-score, standard error). This distinguishes it from sibling tools, which are various other calculators.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Usage Guidelines4/5

Does the description explain when to use this tool, when not to, or what alternatives exist?

The description provides usage context: 'Common in survey analysis, A/B testing, and quality control.' This implies appropriate scenarios but does not explicitly state when not to use the tool or contrast with alternatives. It lacks explicit exclusions or sibling differentiation.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.

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TDQS

A3.9/5.0

Disambiguation4/5

Despite 89 tools, each has a clearly distinct purpose with detailed descriptions that often reference related tools. Overlap exists (e.g., multiple LoRa/RF tools), but the descriptions are sufficient to distinguish them. Some confusion possible among similar-sounding tools like attenuator_pi and attenuator_tee, but the descriptions explicitly compare them.

Naming Consistency4/5

Consistent underscore-separated lowercase naming. Most tools follow a verb_noun pattern (e.g., capacitor_charge, wire_gauge) or noun_noun (power_cost). Minor inconsistencies such as 'bmi_calculator' vs 'solar_sizing' but overall predictable.

Tool Count2/5

89 tools is far too many for a single MCP server. This scope is more appropriate for multiple specialized servers. The sheer number will slow agent selection and increase cognitive load, reducing coherence.

Completeness3/5

Covers many domains (RF, solar, PCB, networking, math, etc.) but lacks depth in some areas (e.g., no three-phase power, no airflow calculations). Some domains have comprehensive coverage (LoRa/Meshtastic), but others feel incomplete for the tool count.

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