Questions: We are going to calculate the standard deviation for the following set of sample data. 11,15,6,13,3 1) First, calculate the mean. x̄= 2) Fill in the table below. Fill in the differences of each data value from the mean, then the squared differences. 3) Calculate the standard deviation. x x-x̄ (x-x̄)^2 11 15 6 13 3 Total Standard deviation: s=√(∑(x-x̄)^2)/(n-1)= Round to two decimal places

We are going to calculate the standard deviation for the following set of sample data.
11,15,6,13,3
1) First, calculate the mean.
x̄=
2) Fill in the table below. Fill in the differences of each data value from the mean, then the squared differences.
3) Calculate the standard deviation.

x x-x̄ (x-x̄)^2
11
15
6
13
3
Total

Standard deviation:
s=√(∑(x-x̄)^2)/(n-1)= Round to two decimal places

Transcript text: We are going to calculate the standard deviation for the following set of sample data. \[ 11,15,6,13,3 \] 1) First, calculate the mean. $\bar{x}=$ $\square$ 2) Fill in the table below. Fill in the differences of each data value from the mean, then the squared differences. 3) Calculate the standard deviation. \begin{tabular}{|c|c|c|} \hline x & $x-\bar{x}$ & $(x-\bar{x})^{2}$ \\ \hline 11 & & \\ \hline 15 & & \\ \hline 6 & & \\ \hline 13 & & \\ \hline 3 & & \\ \hline & Total & \\ \hline \end{tabular} Standard deviation: $s=\sqrt{\frac{\sum(x-\bar{x})^{2}}{n-1}}=$ $\square$ Round to two decimal places

Solution

Solution Steps

Step 1: Calculate the Mean

To find the mean $ \bar{x} $ of the dataset $ \{11, 15, 6, 13, 3\} $, we use the formula:

\[ \bar{x} = \frac{\sum_{i=1}^N x_i}{N} = \frac{11 + 15 + 6 + 13 + 3}{5} = \frac{48}{5} = 9.6 \]

Thus, the mean is:

\[ \bar{x} = 9.6 \]

Step 2: Calculate Differences and Squared Differences

Next, we calculate the differences $ (x - \bar{x}) $ and their squares $ (x - \bar{x})^2 $ for each data point:

\[ \begin{align_} x = 11 & : \quad x - \bar{x} = 11 - 9.6 = 1.4 \quad \Rightarrow \quad (x - \bar{x})^2 = (1.4)^2 = 1.96 \\ x = 15 & : \quad x - \bar{x} = 15 - 9.6 = 5.4 \quad \Rightarrow \quad (x - \bar{x})^2 = (5.4)^2 = 29.16 \\ x = 6 & : \quad x - \bar{x} = 6 - 9.6 = -3.6 \quad \Rightarrow \quad (x - \bar{x})^2 = (-3.6)^2 = 12.96 \\ x = 13 & : \quad x - \bar{x} = 13 - 9.6 = 3.4 \quad \Rightarrow \quad (x - \bar{x})^2 = (3.4)^2 = 11.56 \\ x = 3 & : \quad x - \bar{x} = 3 - 9.6 = -6.6 \quad \Rightarrow \quad (x - \bar{x})^2 = (-6.6)^2 = 43.56 \\ \end{align_} \]

The total of the squared differences is:

\[ \sum (x - \bar{x})^2 = 1.96 + 29.16 + 12.96 + 11.56 + 43.56 = 99.20 \]

Step 3: Calculate the Standard Deviation

The variance $ s^2 $ is calculated using the formula for sample variance:

\[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{99.20}{5-1} = \frac{99.20}{4} = 24.8 \]

The standard deviation $ s $ is then:

\[ s = \sqrt{24.8} \approx 4.98 \]