Question

the number of cars sold at a dealership over several weeks is given below.
14, 23, 31, 29, 33
what is the standard deviation for this set of population data?
standard deviation: $sigma = sqrt{\frac{(x_1 - mu)^2 + (x_2 - mu)^2 + dots + (x_n - mu)^2}{n}}$
$circ$ 6.9
$circ$ 12.4
$circ$ 15.4
$circ$ 47.2

Explanation:

Step1: Calculate the mean (μ)

The data set is \( 14, 23, 31, 29, 33 \). The number of data points \( N = 5 \).
The mean \( \mu=\frac{14 + 23+31 + 29+33}{5}=\frac{130}{5} = 26 \).

Step2: Calculate the squared differences from the mean

For \( x_1 = 14 \): \( (14 - 26)^2=(- 12)^2 = 144 \)
For \( x_2 = 23 \): \( (23 - 26)^2=(-3)^2 = 9 \)
For \( x_3 = 31 \): \( (31 - 26)^2=(5)^2 = 25 \)
For \( x_4 = 29 \): \( (29 - 26)^2=(3)^2 = 9 \)
For \( x_5 = 33 \): \( (33 - 26)^2=(7)^2 = 49 \)

Step3: Sum the squared differences

Sum \(=144 + 9+25 + 9+49=236\)

Step4: Calculate the variance

Variance \(=\frac{236}{5}=47.2\)

Step5: Calculate the standard deviation

Standard deviation \( \sigma=\sqrt{47.2}\approx6.9\)

Answer:

6.9