To find the standard deviation based on the bar graph data provided, we first need to construct a frequency distribution and then calculate the standard deviation using the following steps:
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Data Collection: The data collected from the bar graph is:
- Score 7: Frequency 7
- Score 8: Frequency 5
- Score 9: Frequency 7
We also note that the scores 5, 6, 10, and 11 have a frequency of 0.
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Construct a Frequency Table: \[ \begin{array}{|c|c|} \hline \text{Score (x)} & \text{Frequency (f)} \ \hline 5 & 0 \ 6 & 0 \ 7 & 7 \ 8 & 5 \ 9 & 7 \ 10 & 0 \ 11 & 0 \ \hline \end{array} \]
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Calculate Mean (μ): The mean \( \mu \) is calculated as follows: \[ \mu = \frac{\sum (x \cdot f)}{\sum f} \] where \( x \) is the score and \( f \) is the frequency.
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\(\sum (x \cdot f) = (5 \cdot 0) + (6 \cdot 0) + (7 \cdot 7) + (8 \cdot 5) + (9 \cdot 7) + (10 \cdot 0) + (11 \cdot 0)\)
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\(= 0 + 0 + 49 + 40 + 63 + 0 + 0 = 152\)
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\(\sum f = 0 + 0 + 7 + 5 + 7 + 0 + 0 = 19\)
So, the mean \( \mu \) is: \[ \mu = \frac{152}{19} \approx 8 \]
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Calculate Variance (σ²): The variance is calculated as: \[ \sigma^2 = \frac{\sum f(x - \mu)^2}{\sum f} \]
We compute \( (x - \mu)^2 \) for each score:
- For \( x = 7 \): \[ (7 - 8)^2 = (-1)^2 = 1 \]
- For \( x = 8 \): \[ (8 - 8)^2 = (0)^2 = 0 \]
- For \( x = 9 \): \[ (9 - 8)^2 = (1)^2 = 1 \]
Now we compute \( f(x - \mu)^2 \): \[ \begin{align*} f(x - \mu)^2 &= 0 \cdot 0 + 0 \cdot 0 + 7 \cdot 1 + 5 \cdot 0 + 7 \cdot 1 + 0 \cdot 0 + 0 \cdot 0 \ &= 0 + 0 + 7 + 0 + 7 + 0 + 0 = 14 \end{align*} \]
Plugging it into the variance formula: \[ \sigma^2 = \frac{14}{19} \approx 0.736842 \]
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Calculate Standard Deviation (σ): \[ \sigma = \sqrt{\sigma^2} = \sqrt{0.736842} \approx 0.859 \]
Finally, rounding to two decimal places, the standard deviation is approximately: \[ \boxed{0.86} \]