To find the standard deviation of the given dataset [tex]\([14, 23, 31, 29, 33]\)[/tex], we will follow several steps. Let's go through the process step by step.
### Step 1: Calculate the Mean (μ)
The first step is to calculate the mean (average) of the dataset.
[tex]\[
\mu = \frac{\sum_{i=1}^{N} x_i}{N}
\][/tex]
where [tex]\( N \)[/tex] is the number of elements in the dataset, and [tex]\( x_i \)[/tex] represents each individual data point.
The dataset consists of 5 values, so [tex]\( N = 5 \)[/tex].
[tex]\[
\mu = \frac{14 + 23 + 31 + 29 + 33}{5} = \frac{130}{5} = 26
\][/tex]
### Step 2: Calculate the Variance for Each Data Point
Next, we calculate the variance for each data point, using the formula:
[tex]\[
(x_i - \mu)^2
\][/tex]
Let’s compute this for each data point:
For [tex]\( x_1 = 14 \)[/tex]:
[tex]\[
(14 - 26)^2 = (-12)^2 = 144
\][/tex]
For [tex]\( x_2 = 23 \)[/tex]:
[tex]\[
(23 - 26)^2 = (-3)^2 = 9
\][/tex]
For [tex]\( x_3 = 31 \)[/tex]:
[tex]\[
(31 - 26)^2 = 5^2 = 25
\][/tex]
For [tex]\( x_4 = 29 \)[/tex]:
[tex]\[
(29 - 26)^2 = 3^2 = 9
\][/tex]
For [tex]\( x_5 = 33 \)[/tex]:
[tex]\[
(33 - 26)^2 = 7^2 = 49
\][/tex]
So, the variances for each data point are [tex]\([144, 9, 25, 9, 49]\)[/tex].
### Step 3: Calculate the Mean of These Variances
To find the standard deviation, we need to calculate the mean of the variances:
[tex]\[
\text{Mean of variances} = \frac{144 + 9 + 25 + 9 + 49}{5} = \frac{236}{5} = 47.2
\][/tex]
### Step 4: Calculate the Standard Deviation (σ)
Finally, the standard deviation (σ) is the square root of the mean of the variances:
[tex]\[
\sigma = \sqrt{47.2} \approx 6.870225614927067
\][/tex]
Therefore, the standard deviation for this set of population data is closest to:
[tex]\[
\boxed{6.9}
\][/tex]
