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:
- Calculate the mean (average) of the data set.
- Calculate the variance (the average of the squared differences from the mean).
- 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.