Question
VALUE 4 5 6
TOTAL OBSERVATIONS 1 3 1
Calculate the standard deviation of the data set. Round the answer to the nearest hundredth. (1 point) Responses 0.50 0.71 0.25 5.00
TOTAL OBSERVATIONS 1 3 1
Calculate the standard deviation of the data set. Round the answer to the nearest hundredth. (1 point) Responses 0.50 0.71 0.25 5.00
Answers
GPT-4o mini
To calculate the standard deviation, we first need to determine the mean (average) of the data set, and then use it to find the variance and standard deviation.
### Step 1: Calculate the Mean
We have the values and their corresponding total observations (frequencies):
- Value 4: 1 observation
- Value 5: 3 observations
- Value 6: 1 observation
To find the mean:
\[
\text{Mean} = \frac{\sum (x \cdot f)}{\sum f}
\]
Where \( x \) is the value, and \( f \) is the frequency.
Calculating \( \sum (x \cdot f) \):
\[
4 \cdot 1 + 5 \cdot 3 + 6 \cdot 1 = 4 + 15 + 6 = 25
\]
Calculating \( \sum f \):
\[
1 + 3 + 1 = 5
\]
Now, calculate the mean:
\[
\text{Mean} = \frac{25}{5} = 5
\]
### Step 2: Calculate the Variance
Using the formula for variance:
\[
\sigma^2 = \frac{\sum f(x - \text{mean})^2}{\sum f}
\]
Calculating \( (x - \text{mean})^2 \) for each value:
- For value 4: \( (4 - 5)^2 = (-1)^2 = 1 \)
- For value 5: \( (5 - 5)^2 = (0)^2 = 0 \)
- For value 6: \( (6 - 5)^2 = (1)^2 = 1 \)
Now, calculate \( \sum f(x - \text{mean})^2 \):
\[
1 \cdot 1 + 3 \cdot 0 + 1 \cdot 1 = 1 + 0 + 1 = 2
\]
Put it all into the variance formula:
\[
\sigma^2 = \frac{2}{5} = 0.4
\]
### Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[
\sigma = \sqrt{0.4} \approx 0.632455532
\]
Rounding to the nearest hundredth:
\[
\sigma \approx 0.63
\]
### Conclusion
The value provided does not match the options given. To find the closest answer, we recognize that the calculated standard deviation is approximately 0.63, which is not exactly in the provided options. Assuming rounding may influence it, the closest simple value is:
- 0.71
Thus, if you have to select from the options, pick **0.71** as the best fitting response.
### Step 1: Calculate the Mean
We have the values and their corresponding total observations (frequencies):
- Value 4: 1 observation
- Value 5: 3 observations
- Value 6: 1 observation
To find the mean:
\[
\text{Mean} = \frac{\sum (x \cdot f)}{\sum f}
\]
Where \( x \) is the value, and \( f \) is the frequency.
Calculating \( \sum (x \cdot f) \):
\[
4 \cdot 1 + 5 \cdot 3 + 6 \cdot 1 = 4 + 15 + 6 = 25
\]
Calculating \( \sum f \):
\[
1 + 3 + 1 = 5
\]
Now, calculate the mean:
\[
\text{Mean} = \frac{25}{5} = 5
\]
### Step 2: Calculate the Variance
Using the formula for variance:
\[
\sigma^2 = \frac{\sum f(x - \text{mean})^2}{\sum f}
\]
Calculating \( (x - \text{mean})^2 \) for each value:
- For value 4: \( (4 - 5)^2 = (-1)^2 = 1 \)
- For value 5: \( (5 - 5)^2 = (0)^2 = 0 \)
- For value 6: \( (6 - 5)^2 = (1)^2 = 1 \)
Now, calculate \( \sum f(x - \text{mean})^2 \):
\[
1 \cdot 1 + 3 \cdot 0 + 1 \cdot 1 = 1 + 0 + 1 = 2
\]
Put it all into the variance formula:
\[
\sigma^2 = \frac{2}{5} = 0.4
\]
### Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[
\sigma = \sqrt{0.4} \approx 0.632455532
\]
Rounding to the nearest hundredth:
\[
\sigma \approx 0.63
\]
### Conclusion
The value provided does not match the options given. To find the closest answer, we recognize that the calculated standard deviation is approximately 0.63, which is not exactly in the provided options. Assuming rounding may influence it, the closest simple value is:
- 0.71
Thus, if you have to select from the options, pick **0.71** as the best fitting response.