套餐

Z 分数计算器（含百分位数）

这个 z 分数计算器求一个值离平均数有多少个标准差，z = (x − μ)/σ。它还给出正态分布中低于和高于该值的百分比，便于把 z 分数换算成百分位数。

结果

输入数值并点击“计算”查看结果。

公式

z=x−μσz = \frac{x - \mu}{\sigma}

Subtract the mean, then divide by the standard deviation.

x=μ+zσx = \mu + z\sigma

Rearranged, to find the value for a given z-score.

如何使用此计算器

  1. Enter the value, the mean and the standard deviation (which must be positive).
  2. Press Calculate to see the z-score and the percentages below and above it.

例题

A test score of 85 when the mean is 70 and σ = 15

85 − 70 = 15, and 15 ÷ 15 = 1, so z = 1.

The score is 1 standard deviation above the mean: about 84.13% of a normal distribution is below it.

A value below the mean: x = 62, μ = 70, σ = 4

62 − 70 = −8, and −8 ÷ 4 = −2.

z = −2: only about 2.28% of values are lower.

z = 1.5: x = 130, μ = 100, σ = 20

(130 − 100) ÷ 20 = 1.5.

About 93.32% of values are below 130.

常见错误

  • Dividing by the variance instead of the standard deviation.
  • Subtracting the wrong way round, which flips the sign of z.
  • Using the percentages for data that is far from normally distributed.

常见问题

What does a z-score tell you?

How many standard deviations a value is above (positive) or below (negative) the mean. z = 0 is exactly the mean; z = 2 is unusually high.

How do you turn a z-score into a percentile?

Use the normal distribution (a z-table): the percentage below z is the percentile. z = 1 is about the 84th percentile.

What is a good z-score?

It depends on the context. About 68% of normal data has z between −1 and 1, and about 95% between −2 and 2.