Question

standard deviation
the number of cars sold at a dealership over several weeks is given.
14, 23, 31, 29, 33
what is the standard deviation for this set of population data?
standard deviation: $sigma = sqrt{\frac{(x_1-mu)^2+(x_2-mu)^2+dots+(x_n-mu)^2}{n}}$
12.4 \t\t47.2
15.4 \t\t6.9

Explanation:

Step1: Calculate the mean (μ)

First, find the sum of the data points: \(14 + 23 + 31 + 29 + 33 = 130\).
There are \(N = 5\) data points, so the mean \(\mu=\frac{130}{5}=26\).

Step2: Calculate each \((x_i - \mu)^2\)

• 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 is \(\frac{\sum (x_i - \mu)^2}{N}=\frac{236}{5}=47.2\)

Step5: Calculate the standard deviation (σ)

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

Answer:

6.9