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

Annotations already declare readOnlyHint=true, idempotentHint=true, and openWorldHint=false, so the safety profile is covered and the bar is lower. The description adds a worked example but omits meaningful behavioral detail: whether 'tail probability' is one- or two-sided, precision conventions, or what happens when inputs conflict (e.g., both x/mean/sd and pct supplied).

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

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

Three compact sentences with no filler, and the trailing web-page link is only mild overhead for an API-facing tool. It leads with an example rather than the purpose statement, so the core capability is slightly back-loaded, but nothing is redundant.

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

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

With no output schema, the description must convey what comes back, and it does roughly (z-scores, tail probabilities, percentile value) plus a sample result. For a 6-parameter, all-optional, idempotent read tool with annotations covering safety, that is close to sufficient; the missing piece is the precise definition of the probability outputs.

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

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

Schema coverage is 100%, so the baseline is 3, but the description actually earns extra credit by decoding the cryptic mode enum ('Calculate' with values z/x/p) into z-scores, tail probabilities, and percentile values. The worked example also ties x, mean, and sd together, which the schema's plain labels do not.

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

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

The description names a specific operation set: computing a z-score, tail probabilities, or the raw value for a percentile, backed by a concrete worked example (85, mean 70, SD 10 → z=1.5, 93.32nd percentile). It is clearly distinct from generic siblings like scientific_calculator. It stops short of explicitly differentiating from near-neighbors such as standard_deviation_calculator, p_value_calculator, or confidence_interval_calculator.

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

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

The three output modes are listed, which implies when each is relevant, but there is no explicit 'use this when...' guidance and no named alternatives (e.g., p_value_calculator for hypothesis testing). An agent must infer the choice from the mode enum rather than from stated conditions.

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