Question

Leah asked five of her friends how many sports they play. She received the following responses: 1, 1, 2, 3, and 3. Calculate the standard deviation of the data. What is the stand deviation?

Answer 1

To calculate the standard deviation of the data set consisting of the number of sports Leah's friends play (1, 1, 2, 3, 3), we will follow these steps:

1. **Calculate the mean (average)** of the data set.
2. **Calculate the variance** (the average of the squared differences from the mean).
3. **Calculate the standard deviation** (the square root of the variance).

### Step 1: Calculate the Mean
The mean \( \mu \) is calculated as follows:

\[
\mu = \frac{\text{sum of all data points}}{\text{number of data points}}
\]

Calculating the sum:

\[
1 + 1 + 2 + 3 + 3 = 10
\]

Number of data points \( n = 5 \).

So, the mean is:

\[
\mu = \frac{10}{5} = 2
\]

### Step 2: Calculate the Variance
Next, we find the squared differences from the mean for each data point, then calculate the average of those squared differences.

- For the first data point (1):

\[
(1 - 2)^2 = (-1)^2 = 1
\]

- For the second data point (1):

\[
(1 - 2)^2 = (-1)^2 = 1
\]

- For the third data point (2):

\[
(2 - 2)^2 = 0^2 = 0
\]

- For the fourth data point (3):

\[
(3 - 2)^2 = (1)^2 = 1
\]

- For the fifth data point (3):

\[
(3 - 2)^2 = (1)^2 = 1
\]

Now we can sum these squared differences:

\[
1 + 1 + 0 + 1 + 1 = 4
\]

Now, we calculate the variance \( \sigma^2 \):

\[
\sigma^2 = \frac{\text{sum of squared differences}}{\text{number of data points}} = \frac{4}{5} = 0.8
\]

### Step 3: Calculate the Standard Deviation
Finally, we calculate the standard deviation \( \sigma \):

\[
\sigma = \sqrt{\sigma^2} = \sqrt{0.8} \approx 0.8944
\]

### Conclusion
The standard deviation of the data set is approximately **0.8944**.