Statistics 简明教程

Statistics - Chebyshev’s Theorem

任何一组数字的一部分位于数字标准偏差 k 的平均值的 k 以内，至少为

The fraction of any set of numbers lying within k standard deviations of those numbers of the mean of those numbers is at least

其中——

Where −

  1. ${k = \frac{the\ within\ number}{the\ standard\ deviation}}$

且 ${k}$ 必须大于 1

and ${k}$ must be greater than 1

Example

Problem Statement −

Problem Statement −

使用切比雪夫定理，找出均值为 151、标准差为 14 的数据集中有多少百分比的值介于 123 和 179 之间。

Use Chebyshev’s theorem to find what percent of the values will fall between 123 and 179 for a data set with mean of 151 and standard deviation of 14.

Solution −

Solution −

  1. We subtract 151-123 and get 28, which tells us that 123 is 28 units below the mean.

  2. We subtract 179-151 and also get 28, which tells us that 151 is 28 units above the mean.

  3. Those two together tell us that the values between 123 and 179 are all within 28 units of the mean. Therefore the "within number" is 28.

  4. So we find the number of standard deviations, k, which the "within number", 28, amounts to by dividing it by the standard deviation −

现在我们知道 123 和 179 之间的值都在平均值的 28 个单位以内，这与平均值的 k=2 个标准偏差内相同。现在，由于 k > 1，我们可以使用切比雪夫公式找出数据中位于平均值的 k=2 个标准偏差内的部分。代入 k=2，我们有 −

So now we know that the values between 123 and 179 are all within 28 units of the mean, which is the same as within k=2 standard deviations of the mean. Now, since k > 1 we can use Chebyshev’s formula to find the fraction of the data that are within k=2 standard deviations of the mean. Substituting k=2 we have −

因此，${3/4}$ 的数据介于 123 和 179 之间。并且由于 ${3/4 = 75}$%，这意味着 75% 的数据值介于 123 和 179 之间。

So ${\frac{3}{4}}$ of the data lie between 123 and 179. And since ${\frac{3}{4} = 75}$% that implies that 75% of the data values are between 123 and 179.

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