To calculate the standard deviation, we first need to compile the data set based on the values and their corresponding total observations.
Here, the data points are:
- One instance of 4
- Three instances of 5
- One instance of 6
This gives us the following data points:
- 4, 5, 5, 5, 6
Next, we can calculate the mean (average) of the data set:
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Calculate the mean: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of observations}} = \frac{4 + 5 + 5 + 5 + 6}{5} = \frac{25}{5} = 5 \]
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Next, we find the squared differences from the mean for each data point:
- For 4: \((4 - 5)^2 = (-1)^2 = 1\)
- For 5: \((5 - 5)^2 = (0)^2 = 0\)
- For 5: \((5 - 5)^2 = (0)^2 = 0\)
- For 5: \((5 - 5)^2 = (0)^2 = 0\)
- For 6: \((6 - 5)^2 = (1)^2 = 1\)
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Calculate the sum of the squared differences: \[ \text{Sum of squared differences} = 1 + 0 + 0 + 0 + 1 = 2 \]
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Divide the sum of squared differences by the number of observations to find the variance: \[ \text{Variance} = \frac{\text{Sum of squared differences}}{\text{Total observations}} = \frac{2}{5} = 0.4 \]
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Finally, take the square root of the variance to find the standard deviation: \[ \text{Standard deviation} = \sqrt{0.4} \approx 0.6325 \]
Rounding to the nearest hundredth gives us: \[ \text{Standard deviation} \approx 0.63 \]
However, looking at the available responses:
- 0.50
- 5.00
- 0.71
- 0.25
None of these options match the calculated standard deviation. If only rounded values are considered, the closest value provided is 0.71.
So based on the choices given, the answer would be: 0.71.