Question

Use the display of data to find the standard deviation.
A bar graph titled Score with a horizontal axis labeled from 5 to 11 in increments of 1 and a vertical axis labeled Frequency from 0 to 12 in increments of 2 contains three vertical bars with labels and heights as follows, where the horizontal axis label is listed first and the height is listed second: 7, 7; 8, 5; 9, 7.
Score
5
6
7
8
9
10
11
0
2
4
6
8
10
12
Frequency
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Part 1
The standard deviation is approximately
  
enter your response here.
​(Do not round until the final answer. Then round to the nearest hundredth as​ needed.)

Answer 1

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:

1. **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.

2. **Construct a Frequency Table:**
\[
\begin{array}{|c|c|}
\hline
\text{Score (x)} & \text{Frequency (f)} \\
\hline
5 & 0 \\
6 & 0 \\
7 & 7 \\
8 & 5 \\
9 & 7 \\
10 & 0 \\
11 & 0 \\
\hline
\end{array}
\]

3. **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.

- \(\sum (x \cdot f) = (5 \cdot 0) + (6 \cdot 0) + (7 \cdot 7) + (8 \cdot 5) + (9 \cdot 7) + (10 \cdot 0) + (11 \cdot 0)\)
- \(= 0 + 0 + 49 + 40 + 63 + 0 + 0 = 152\)

- \(\sum f = 0 + 0 + 7 + 5 + 7 + 0 + 0 = 19\)

So, the mean \( \mu \) is:
\[
\mu = \frac{152}{19} \approx 8
\]

4. **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:
\[
\sigma^2 = \frac{14}{19} \approx 0.736842
\]

5. **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:
\[
\boxed{0.86}
\]