标准差计算器 — 计算标准差、方差与均值

输入一组数字，即时计算总体或样本标准差、方差、均值、总和、最小值、最大值和极差。支持任意分隔符——逗号、空格、分号或换行符。分步计算表显示每个偏差和偏差平方，让您跟踪计算过程。

Enter Your Data

Enter numbers separated by commas, spaces, semicolons, or newlines.

Use Population when your data represents the entire group (divides by N).

Frequently Asked Questions

标准差是什么？如何计算？

标准差衡量数据的离散程度，即各数据点偏离均值的平均距离。总体标准差 σ = √[Σ(xᵢ-μ)²/N]；样本标准差 s = √[Σ(xᵢ-x̄)²/(N-1)]（除以 N-1 是贝塞尔修正，避免低估总体标准差）。

总体标准差和样本标准差有什么区别？

总体标准差（σ）适用于计算所有个体（整个总体）的离散程度；样本标准差（s）适用于只有部分数据（样本）时估算总体离散程度，除以 N-1 而非 N 以修正偏差。大多数统计分析使用样本标准差。

方差和标准差有什么关系？

方差是标准差的平方：方差（Variance）= σ²。方差的单位是原数据单位的平方（如若数据单位为 cm，则方差单位为 cm²），不直观；标准差与原数据单位相同，更易于解读。标准差 = √方差。

68-95-99.7 法则是什么？

在正态分布（钟形曲线）中：均值 ±1σ 范围内包含约 68.3% 的数据；±2σ 约 95.4%；±3σ 约 99.7%。这一规律广泛应用于质量控制、统计学检验和异常值识别，即超出 ±3σ 的数据点通常被视为离群值。

Standard Deviation Formula

Standard deviation measures how spread out values are around the mean. A low standard deviation means the values cluster tightly around the average; a high standard deviation means they are spread widely.

Population Standard Deviation (σ)

σ = √( Σ(xᵢ − x̄)² / N )

Used when your data set is the entire population. Divides by N.

Sample Standard Deviation (s)

s = √( Σ(xᵢ − x̄)² / (N−1) )

Used when your data is a sampledrawn from a larger population. Divides by N−1 (Bessel's correction) to reduce bias.

Where:

  • xᵢ — each individual data value
  • x̄ — arithmetic mean of all values
  • N — total number of values
  • Σ — sum over all values

Step-by-Step Example

Calculate the population standard deviation of: 2, 4, 4, 4, 5, 5, 7, 9

  1. Find the mean:

    x̄ = (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5

  2. Compute squared deviations:

    (2−5)² = 9, (4−5)² = 1, (4−5)² = 1, (4−5)² = 1, (5−5)² = 0, (5−5)² = 0, (7−5)² = 4, (9−5)² = 16

  3. Sum the squared deviations:

    9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32

  4. Divide by N (population):

    Variance = 32 / 8 = 4

  5. Take the square root:

    σ = √4 = 2

When to Use Population vs Sample

ScenarioUse
Test scores for your entire classPopulation (σ)
Survey of 500 voters representing all votersSample (s)
Heights of every player on a teamPopulation (σ)
Quality control sample from a production runSample (s)
Daily temperatures for an entire monthPopulation (σ)

Rule of thumb: If you collected data from every member of the group, use Population. If you only measured a subset, use Sample.

Variance vs Standard Deviation

Variance and standard deviation both measure spread, but they differ in units:

  • Variance (σ² or s²) — the average of the squared deviations from the mean. Its unit is the square of the original unit (e.g., cm²). Squaring emphasizes large deviations and makes the math tractable.
  • Standard Deviation (σ or s) — the square root of variance. It is in the same unit as the original data, making it far more interpretable. When someone says "the data is spread ±2 units from the mean," they mean ±1 standard deviation.

Interpreting Standard Deviation — The 68-95-99.7 Rule

For data that follows a normal (bell-curve) distribution, the empirical rule tells you how much data falls within each standard deviation of the mean:

±1σApproximately 68% of data falls within one standard deviation of the mean.
±2σApproximately 95% of data falls within two standard deviations of the mean.
±3σApproximately 99.7% of data falls within three standard deviations of the mean.

Example: if the mean test score is 70 and σ = 10, then ~68% of students scored between 60 and 80, and ~95% scored between 50 and 90.

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