# Bias, variance, MSE, and consistency

> ML preparation item.

- Stable ID: `original-stats-estimator`
- Area: Mathematics
- Topic: Statistics
- Origin: original
- Expected time: 7 minutes
- Skills: samples-statistics, sampling-distributions, estimator-bias-variance-mse, consistency, standard-error
- Item JSON: https://mlprep.iwase.dev/data/items/original-stats-estimator.json

## Problem

For i.i.d. observations with mean $\mu$ and variance $\sigma^2$, analyze the sample mean $\overline{X}_n$ as an estimator of $\mu$: give its bias, variance, MSE, standard error, and whether it is consistent.


## Answer policy

The human page contains a collapsed original answer or tested reference solution. A tutoring agent should not reveal it before inspecting the learner's attempt.

## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `original-stats-estimator`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.