To calculate the standard deviation from the given data, we must first understand the distribution of scores based on the frequency represented in the bar graph. The data shows the following scores and their respective frequencies:
- Score 7: Frequency 8
- Score 8: Frequency 6
- Score 9: Frequency 8
Given this information, we can summarize the data as follows:
- Scores: 7, 8, 9
- Frequencies: 8, 6, 8
Step 1: Calculate the Mean
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Calculate the total number of scores (N): \( N = 8 + 6 + 8 = 22 \)
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Calculate the weighted total for the scores: \[ \text{Weighted Score} = (7 \times 8) + (8 \times 6) + (9 \times 8) \] \[ = 56 + 48 + 72 = 176 \]
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Calculate the Mean (\(\mu\)): \[ \mu = \frac{\text{Weighted Score}}{N} = \frac{176}{22} = 8 \]
Step 2: Calculate Each Score's Deviation from the Mean, Square It, and Multiply by Frequency
Next, we calculate the squared deviations and their weighted sum:
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For score \(7\): \[ (7 - 8)^2 \times 8 = (-1)^2 \times 8 = 1 \times 8 = 8 \]
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For score \(8\): \[ (8 - 8)^2 \times 6 = (0)^2 \times 6 = 0 \times 6 = 0 \]
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For score \(9\): \[ (9 - 8)^2 \times 8 = (1)^2 \times 8 = 1 \times 8 = 8 \]
Step 3: Sum the Weighted Squared Deviations
Now sum these values: \[ \text{Total Sum of Squared Deviations} = 8 + 0 + 8 = 16 \]
Step 4: Calculate the Variance and Standard Deviation
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Calculate Variance (\(\sigma^2\)): \[ \sigma^2 = \frac{\text{Total Sum of Squared Deviations}}{N} = \frac{16}{22} = \frac{8}{11} \approx 0.7272727273 \]
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Calculate Standard Deviation (\(\sigma\)): \[ \sigma = \sqrt{\sigma^2} = \sqrt{\frac{8}{11}} \approx 0.853 \]
Step 5: Round the Standard Deviation
Finally, rounding to the nearest hundredth gives us: \[ \sigma \approx 0.85 \]
Conclusion
The standard deviation is approximately 0.85.