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The number of cars sold at a dealership over several weeks is given below:
[tex]\[14, 23, 31, 29, 33\][/tex]

What is the standard deviation for this set of population data?

Standard deviation: [tex]\[\sigma=\sqrt{\frac{(x_1-\mu)^2+(x_2-\mu)^2+\ldots+(x_N-\mu)^2}{N}}\][/tex]

A. 6.9
B. 12.4
C. 15.4
D. 47.2


Sagot :

To find the standard deviation of the number of cars sold at a dealership over several weeks, follow these steps:

1. List the Data Points:
The number of cars sold each week is: [tex]\(14, 23, 31, 29, 33\)[/tex].

2. Calculate the Mean (μ):
Mean (μ) is found by summing all the data points and dividing by the number of data points.
[tex]\[ \mu = \frac{14 + 23 + 31 + 29 + 33}{5} = \frac{130}{5} = 26 \][/tex]

3. Calculate the Squared Differences from the Mean:
For each data point, subtract the mean and square the result.
[tex]\[ (14 - 26)^2 = (-12)^2 = 144 \][/tex]
[tex]\[ (23 - 26)^2 = (-3)^2 = 9 \][/tex]
[tex]\[ (31 - 26)^2 = 5^2 = 25 \][/tex]
[tex]\[ (29 - 26)^2 = 3^2 = 9 \][/tex]
[tex]\[ (33 - 26)^2 = 7^2 = 49 \][/tex]
These squared differences are: [tex]\(144, 9, 25, 9, 49\)[/tex].

4. Calculate the Variance (σ²):
Variance is the average of these squared differences.
[tex]\[ \sigma^2 = \frac{144 + 9 + 25 + 9 + 49}{5} = \frac{236}{5} = 47.2 \][/tex]

5. Calculate the Standard Deviation (σ):
The standard deviation is the square root of the variance.
[tex]\[ \sigma = \sqrt{47.2} \approx 6.8702 \][/tex]

Thus, the standard deviation of the number of cars sold is approximately [tex]\(6.87\)[/tex].

Reviewing the options given:

- 6.9
- 12.4
- 15.4
- 47.2

The closest option to our calculated standard deviation is 6.9.

Therefore, the correct answer is 6.9.
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