To calculate the standard deviation of the dataset provided, we first need to organize the data based on the values and their respective frequencies (observations):
- Value 4: Frequency 1
- Value 5: Frequency 3
- Value 6: Frequency 1
Next, we can summarize the dataset:
- Data points:
- One 4
- Three 5s
- One 6
This gives us the following data points:
\[ 4, 5, 5, 5, 6 \]
Now, we can follow these steps to calculate the standard deviation:
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Calculate the mean (average): \[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Total number of data points}} = \frac{4 + 5 + 5 + 5 + 6}{5} = \frac{25}{5} = 5 \]
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Calculate the variance:
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Find the squared differences from the mean for each data point:
- For 4: \((4 - 5)^2 = 1\)
- For 5: \((5 - 5)^2 = 0\) (three occurrences)
- For 6: \((6 - 5)^2 = 1\)
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Now we gather these:
- Squared differences: \(1, 0, 0, 0, 1\)
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Calculate the average of these squared differences (variance): \[ \text{Variance} = \frac{1 + 0 + 0 + 0 + 1}{5} = \frac{2}{5} = 0.4 \]
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Calculate the standard deviation: \[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0.4} \approx 0.6325 \]
Rounding this value to the nearest hundredth, the final answer is: \[ \text{Standard Deviation} \approx 0.63 \]
Among the given options, none listed correspond to this value. Therefore, the calculated standard deviation is approximately 0.63.
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