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 −
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${k = \frac{the\ within\ number}{the\ standard\ deviation}}$
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 −
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We subtract 151-123 and get 28, which tells us that 123 is 28 units below the mean.
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We subtract 179-151 and also get 28, which tells us that 151 is 28 units above the mean.
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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.
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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.