To compute the standard deviation of the given scores, we'll follow these steps:
1. Calculate the Mean:
First, find the mean (average) of the scores.
The scores given are: 10, 12, 15, 35, 40, 43, 47, 49, 50, 55.
The mean is calculated as:
[tex]\[
\text{Mean} = \frac{\sum \text{scores}}{n}
\][/tex]
Where [tex]\( n \)[/tex] is the number of scores. Here, [tex]\( n = 10 \)[/tex].
[tex]\[
\text{Sum of scores} = 10 + 12 + 15 + 35 + 40 + 43 + 47 + 49 + 50 + 55 = 356
\][/tex]
[tex]\[
\text{Mean} = \frac{356}{10} = 35.6
\][/tex]
2. Calculate the Squared Differences from the Mean:
Next, compute the squared difference between each score and the mean.
[tex]\[
\text{Squared Difference} = (\text{Score} - \text{Mean})^2
\][/tex]
We organize this in a table for clarity:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Score} & \text{Mean} & (\text{Score} - \text{Mean})^2 \\
\hline
10 & 35.6 & (10 - 35.6)^2 = 660.49 \\
12 & 35.6 & (12 - 35.6)^2 = 556.96 \\
15 & 35.6 & (15 - 35.6)^2 = 428.49 \\
35 & 35.6 & (35 - 35.6)^2 = 0.36 \\
40 & 35.6 & (40 - 35.6)^2 = 19.36 \\
43 & 35.6 & (43 - 35.6)^2 = 54.76 \\
47 & 35.6 & (47 - 35.6)^2 = 128.44 \\
49 & 35.6 & (49 - 35.6)^2 = 178.56 \\
50 & 35.6 & (50 - 35.6)^2 = 204.84 \\
55 & 35.6 & (55 - 35.6)^2 = 380.25 \\
\hline
\end{array}
\][/tex]
3. Calculate the Variance:
The variance is the average of these squared differences.
[tex]\[
\text{Variance} = \frac{\sum (\text{Score} - \text{Mean})^2}{n}
\][/tex]
Sum of squared differences:
[tex]\[
660.49 + 556.96 + 428.49 + 0.36 + 19.36 + 54.76 + 128.44 + 178.56 + 204.84 + 380.25 = 2604.40
\][/tex]
[tex]\[
\text{Variance} = \frac{2604.40}{10} = 260.44
\][/tex]
4. Calculate the Standard Deviation:
The standard deviation is the square root of the variance.
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{260.44} \approx 16.138
\][/tex]
So, the mean of the scores is 35.6, the variance is 260.44, and the standard deviation is approximately 16.138.
