Question

Find the standard deviation for the following data set: -3, -2, -2, -1, -1, -1, 0, 0, 1. Round to the nearest hundredth.

Answer 1

To calculate the standard deviation for the given data set \(-3, -2, -2, -1, -1, -1, 0, 0, 1\), follow these steps:

1. **Calculate the mean (average)** of the data set:

\[
\text{Mean} = \frac{\sum x}{n}
\]

where \(n\) is the number of values. Here, we have:

\[
n = 9 \quad (\text{There are 9 numbers})
\]
\[
\sum x = -3 + -2 + -2 + -1 + -1 + -1 + 0 + 0 + 1 = -9
\]

So,

\[
\text{Mean} = \frac{-9}{9} = -1
\]

2. **Calculate the squared differences from the mean** for each data point:

\[
(-3 - (-1))^2 = (-3 + 1)^2 = (-2)^2 = 4
\]
\[
(-2 - (-1))^2 = (-2 + 1)^2 = (-1)^2 = 1
\]
\[
(-2 - (-1))^2 = (-1)^2 = 1
\]
\[
(-1 - (-1))^2 = (0)^2 = 0
\]
\[
(-1 - (-1))^2 = (0)^2 = 0
\]
\[
(-1 - (-1))^2 = (0)^2 = 0
\]
\[
(0 - (-1))^2 = (0 + 1)^2 = (1)^2 = 1
\]
\[
(0 - (-1))^2 = (1)^2 = 1
\]
\[
(1 - (-1))^2 = (1 + 1)^2 = (2)^2 = 4
\]

3. **Sum the squared differences**:

\[
\text{Sum of squared differences} = 4 + 1 + 1 + 0 + 0 + 0 + 1 + 1 + 4 = 12
\]

4. **Calculate the variance**. For a sample, divide by \(n - 1\):

\[
\text{Variance} = \frac{\text{Sum of squared differences}}{n-1} = \frac{12}{9-1} = \frac{12}{8} = 1.5
\]

5. **Calculate the standard deviation** by taking the square root of the variance:

\[
\text{Standard deviation} = \sqrt{1.5} \approx 1.22474487139
\]

Rounding to the nearest hundredth gives:

\[
\text{Standard deviation} \approx 1.22
\]

Therefore, the standard deviation of the data set is:

\[
\boxed{1.22}
\]