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
0.25
0.25
0.50
0.50
5.00
5.00
0.71
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To calculate the standard deviation of the given data set, we first need to identify the values along with their frequencies.
The data set is as follows:
- Value 4 occurs 1 time.
- Value 5 occurs 3 times.
- Value 6 occurs 1 time.
We can summarize this data set as:
- \(4, 5, 5, 5, 6\) (expanding the values based on their frequency).
### Step 1: Calculate the mean (average).
\[
\text{Mean} = \frac{\text{Total Sum of Values}}{\text{Total Count of Values}}
\]
\[
\text{Total Sum} = (4 \times 1) + (5 \times 3) + (6 \times 1) = 4 + 15 + 6 = 25
\]
\[
\text{Total Count} = 1 + 3 + 1 = 5
\]
\[
\text{Mean} = \frac{25}{5} = 5
\]
### Step 2: Calculate the variance.
The variance is calculated using the formula:
\[
\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}
\]
where \(x_i\) is each value, \(\mu\) is the mean, and \(N\) is the number of observations.
Calculating the squared differences from the mean:
- For 4: \((4 - 5)^2 = (-1)^2 = 1\)
- For 5: \((5 - 5)^2 = (0)^2 = 0\) (3 times, so add 0 three times)
- For 6: \((6 - 5)^2 = (1)^2 = 1\)
Thus, we have:
\[
\sum (x_i - \mu)^2 = 1 + 0 + 0 + 0 + 1 = 2
\]
Now, we can compute the variance:
\[
\text{Variance} = \frac{2}{5} = 0.4
\]
### Step 3: Calculate the standard deviation.
Standard deviation is the square root of variance:
\[
\text{Standard Deviation} = \sqrt{0.4} \approx 0.6325
\]
Rounding this to the nearest hundredth gives:
\[
\text{Standard Deviation} \approx 0.63
\]
Among the provided options, **0.71** is the closest representation, but a value of **0.63** does not appear in your responses. Based on the computation, the nearest hundredth is indeed **0.63** which may imply a slight error in available options.
The answer is not explicitly listed among your provided options, and the calculated result is approximately **0.63**.
The data set is as follows:
- Value 4 occurs 1 time.
- Value 5 occurs 3 times.
- Value 6 occurs 1 time.
We can summarize this data set as:
- \(4, 5, 5, 5, 6\) (expanding the values based on their frequency).
### Step 1: Calculate the mean (average).
\[
\text{Mean} = \frac{\text{Total Sum of Values}}{\text{Total Count of Values}}
\]
\[
\text{Total Sum} = (4 \times 1) + (5 \times 3) + (6 \times 1) = 4 + 15 + 6 = 25
\]
\[
\text{Total Count} = 1 + 3 + 1 = 5
\]
\[
\text{Mean} = \frac{25}{5} = 5
\]
### Step 2: Calculate the variance.
The variance is calculated using the formula:
\[
\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}
\]
where \(x_i\) is each value, \(\mu\) is the mean, and \(N\) is the number of observations.
Calculating the squared differences from the mean:
- For 4: \((4 - 5)^2 = (-1)^2 = 1\)
- For 5: \((5 - 5)^2 = (0)^2 = 0\) (3 times, so add 0 three times)
- For 6: \((6 - 5)^2 = (1)^2 = 1\)
Thus, we have:
\[
\sum (x_i - \mu)^2 = 1 + 0 + 0 + 0 + 1 = 2
\]
Now, we can compute the variance:
\[
\text{Variance} = \frac{2}{5} = 0.4
\]
### Step 3: Calculate the standard deviation.
Standard deviation is the square root of variance:
\[
\text{Standard Deviation} = \sqrt{0.4} \approx 0.6325
\]
Rounding this to the nearest hundredth gives:
\[
\text{Standard Deviation} \approx 0.63
\]
Among the provided options, **0.71** is the closest representation, but a value of **0.63** does not appear in your responses. Based on the computation, the nearest hundredth is indeed **0.63** which may imply a slight error in available options.
The answer is not explicitly listed among your provided options, and the calculated result is approximately **0.63**.
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