*50*

**Chebyshev’s Theorem** states that for any number k greater than 1, at least **1 – 1/k ^{2} **of the data values in any shaped distribution lie within k standard deviations of the mean.

For example, for any shaped distribution at least 1 – 1/3^{2} = 88.89% of the values in the distribution will lie within 3 standard deviations of the mean.

This tutorial illustrates several examples of how to apply Chebyshev’s Theorem in Excel.

**Example 1: Use Chebyshev’s Theorem to find what percentage of values will fall between 30 and 70 for a dataset with a mean of 50 and standard deviation of 10.**

First, determine the value for k. We can do this by finding out how many standard deviations away 30 and 70 are from the mean:

(30 – mean) / standard deviation = (30 – 50) / 10 = -20 / 10 =** -2**

(70 – mean) / standard deviation = (70 – 50) / 10 = 20 / 10 = **2**

The values 30 and 70 are 2 standard deviations below and above the mean, respectively. Thus, **k = 2**.

We can then use the following formula in Excel to find the minimum percentage of values that fall within 2 standard deviations of the mean for this dataset:

The percentage of values that fall within 30 and 70 for this dataset will be **at least 75%**.

**Example 2: Use Chebyshev’s Theorem to find what percentage of values will fall between 20 and 50 for a dataset with a mean of 35 and standard deviation of 5.**

First, determine the value for k. We can do this by finding out how many standard deviations away 20 and 50 are from the mean:

(20 – mean) / standard deviation = (20 – 35) / 5 = -15 / 5 =** -3**

(50 – mean) / standard deviation = (50 – 35) / 5 = 15 / 5 = **3**

The values 20 and 50 are 3 standard deviations below and above the mean, respectively. Thus, **k = 3**.

We can then use the following formula in Excel to find the minimum percentage of values that fall within 3 standard deviations of the mean for this dataset:

The percentage of values that fall within 20 and 50 for this dataset will be **at least 88.89%**.

**Example 3: Use Chebyshev’s Theorem to find what percentage of values will fall between 80 and 120 for a dataset with a mean of 100 and standard deviation of 5.**

First, determine the value for k. We can do this by finding out how many standard deviations away 80 and 120 are from the mean:

(80 – mean) / standard deviation = (80 – 100) / 5 = -20 / 5 =** -4**

(120 – mean) / standard deviation = (120 – 100) / 5 = 20 / 5 = **4**

The values 80 and 120 are 4 standard deviations below and above the mean, respectively. Thus, **k = 4**.

We can then use the following formula in Excel to find the minimum percentage of values that fall within 4 standard deviations of the mean for this dataset:

The percentage of values that fall within 80 and 120 for this dataset will be **at least 93.75%**.