Answer :
- Calculate the sample mean: $\bar{x} = 49.57$.
- Find the sum of squared differences: $\sum_{i=1}^{n}(x_i - \bar{x})^2 \approx 2235.300$.
- Divide by $n-1 = 9$: $\frac{2235.300}{9} \approx 248.367$.
- Take the square root and round to 3 decimal places: $s = \sqrt{248.367} \approx \boxed{15.760}$.
### Explanation
1. Understand the problem and provided data
We are given a sample of size $n=10$ with the following data points:
33.4, 41.9, 37.9, 49, 59.5, 67.7, 47.6, 23.3, 67.7, 67.7.
The range of the data set is given as 44.4. We need to calculate the sample standard deviation and round it to 3 decimal places.
2. State the formula for sample standard deviation
The formula for the sample standard deviation is given by:
$$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}$$
where $\bar{x}$ is the sample mean and $n$ is the sample size.
3. Calculate the sample mean
First, we calculate the sample mean $\bar{x}$.
$$\bar{x} = \frac{33.4 + 41.9 + 37.9 + 49 + 59.5 + 67.7 + 47.6 + 23.3 + 67.7 + 67.7}{10} = \frac{495.7}{10} = 49.57$$
So, the sample mean is 49.57.
4. Calculate the sum of squared differences
Next, we calculate the sum of squared differences from the mean:
$$\sum_{i=1}^{n}(x_i - \bar{x})^2 = (33.4-49.57)^2 + (41.9-49.57)^2 + (37.9-49.57)^2 + (49-49.57)^2 + (59.5-49.57)^2 + (67.7-49.57)^2 + (47.6-49.57)^2 + (23.3-49.57)^2 + (67.7-49.57)^2 + (67.7-49.57)^2$$
$$= (-16.17)^2 + (-7.67)^2 + (-11.67)^2 + (-0.57)^2 + (9.93)^2 + (18.13)^2 + (-1.97)^2 + (-26.27)^2 + (18.13)^2 + (18.13)^2$$
$$= 261.4689 + 58.8289 + 136.1889 + 0.3249 + 98.6049 + 328.6969 + 3.8809 + 689.9129 + 328.6969 + 328.6969$$
$$= 2235.300$$ (approximately).
5. Divide by n-1
Now, we divide the sum of squared differences by $n-1 = 10-1 = 9$:
$$\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1} = \frac{2235.300}{9} = 248.366666...$$
6. Take the square root and round
Finally, we take the square root of the result to find the sample standard deviation:
$$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}} = \sqrt{248.366666...} = 15.76$$ (rounded to 3 decimal places).
7. State the final answer
Therefore, the sample standard deviation of the given data set, rounded to 3 decimal places, is $15.760$.
### Examples
Understanding standard deviation is crucial in finance. For instance, when evaluating investment risks, a higher standard deviation indicates greater volatility, meaning the investment's returns can vary significantly. Investors use this measure to assess whether an investment aligns with their risk tolerance, helping them make informed decisions about their portfolios.
- Find the sum of squared differences: $\sum_{i=1}^{n}(x_i - \bar{x})^2 \approx 2235.300$.
- Divide by $n-1 = 9$: $\frac{2235.300}{9} \approx 248.367$.
- Take the square root and round to 3 decimal places: $s = \sqrt{248.367} \approx \boxed{15.760}$.
### Explanation
1. Understand the problem and provided data
We are given a sample of size $n=10$ with the following data points:
33.4, 41.9, 37.9, 49, 59.5, 67.7, 47.6, 23.3, 67.7, 67.7.
The range of the data set is given as 44.4. We need to calculate the sample standard deviation and round it to 3 decimal places.
2. State the formula for sample standard deviation
The formula for the sample standard deviation is given by:
$$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}$$
where $\bar{x}$ is the sample mean and $n$ is the sample size.
3. Calculate the sample mean
First, we calculate the sample mean $\bar{x}$.
$$\bar{x} = \frac{33.4 + 41.9 + 37.9 + 49 + 59.5 + 67.7 + 47.6 + 23.3 + 67.7 + 67.7}{10} = \frac{495.7}{10} = 49.57$$
So, the sample mean is 49.57.
4. Calculate the sum of squared differences
Next, we calculate the sum of squared differences from the mean:
$$\sum_{i=1}^{n}(x_i - \bar{x})^2 = (33.4-49.57)^2 + (41.9-49.57)^2 + (37.9-49.57)^2 + (49-49.57)^2 + (59.5-49.57)^2 + (67.7-49.57)^2 + (47.6-49.57)^2 + (23.3-49.57)^2 + (67.7-49.57)^2 + (67.7-49.57)^2$$
$$= (-16.17)^2 + (-7.67)^2 + (-11.67)^2 + (-0.57)^2 + (9.93)^2 + (18.13)^2 + (-1.97)^2 + (-26.27)^2 + (18.13)^2 + (18.13)^2$$
$$= 261.4689 + 58.8289 + 136.1889 + 0.3249 + 98.6049 + 328.6969 + 3.8809 + 689.9129 + 328.6969 + 328.6969$$
$$= 2235.300$$ (approximately).
5. Divide by n-1
Now, we divide the sum of squared differences by $n-1 = 10-1 = 9$:
$$\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1} = \frac{2235.300}{9} = 248.366666...$$
6. Take the square root and round
Finally, we take the square root of the result to find the sample standard deviation:
$$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}} = \sqrt{248.366666...} = 15.76$$ (rounded to 3 decimal places).
7. State the final answer
Therefore, the sample standard deviation of the given data set, rounded to 3 decimal places, is $15.760$.
### Examples
Understanding standard deviation is crucial in finance. For instance, when evaluating investment risks, a higher standard deviation indicates greater volatility, meaning the investment's returns can vary significantly. Investors use this measure to assess whether an investment aligns with their risk tolerance, helping them make informed decisions about their portfolios.
