This range, standard deviation, and variance calculator finds the measures of variability for a sample or population. First, the calculator will give you a quick answer. Then it will guide you through a step-by-step solution to easily learn how to do the problem yourself.

Before calculating the measures of variability, you may want to check out the Variance and Standard Deviation Definition and Standard Deviation and Variance Formulas.

### Range, Variance, and Standard Deviation Calculator

### Solution:

Range:

The range is found by subtracting the smallest data value from the largest data value. Here, the smallest data value is 106 and the largest is 233. Therefore, the range is:
$$ 233 - 106 = 127 $$

Variance:

1. The steps that follow are also needed for finding the standard deviation. Start by writing the computational formula for the variance of a sample:
$$ {s^2}= \frac{{\sum}{x^2} - \frac{({\sum}{x})^2}{n}}{n-1}$$

2. Create a table of 2 columns and 7 rows. There will be a header row and a row for each data value. The header row should be labeled with ${x}$ and $ x^2$. Enter the data values in the ${x}$ column, with each data value in its own row. In the second column, put the square of each of the data values, ${x^2}$.

$x$ | $x^2$ |

220 | 48400 |

106 | 11236 |

233 | 54289 |

142 | 20164 |

121 | 14641 |

199 | 39601 |

3. Find the sum of all the values in the first column, ${\sum}{x}$.

$$
\sum{x} = 1021 $$

4. Square the answer from step 3, then divide that number by the size of the sample.

$$ \frac{({\sum}{x})^2}{n} = \frac{1042441}{6} = 173740.16666667 $$

5. Find the sum of all the values in the second column, ${\sum}{x^2}$.

$$
{\sum}{x^2} = 188331 $$

6. Subtract the answer in step 4 from the answer in step 5.

$$ {\sum}{x^2} - \frac{({\sum}{x})^2}{n} = 188331 - 173740.16666667 = 14590.833333333 $$

7. Divide the answer in step 6 by
n - 1, one less than the size of the sample. This answer is the **variance** of the sample.
$$ {s^2}= \frac{{\sum}{x^2} - \frac{({\sum}{x})^2}{n}}{n-1}
= \frac{ 14590.833333333 }{5} = 2918.1666666667$$

Standard Deviation

To find the standard deviation, first write the computational formula for the standard deviation of the sample.
$$ {s}= \sqrt{\frac{{\sum}{x^2} - \frac{({\sum}{x})^2}{n}}{n - 1}} $$

Take the square root of the answer found in step 7 above. This number is the *standard deviation* of the sample. It is symbolized by
${s}$
. Here, we round the standard deviation to at most 4 decimal places.

$$ {s} = \sqrt{2918.1666666667} = 54.0201$$