Question

question #15
the sample data provided contains numbers of injuries due to distracted driving for the past 15 months for a certain town.
find the standard deviation for the sample set of data.

injuries due to distracted driving per month
9.074 8.611 6.922 8.065
10.457 7.551 6.985 9.316
5.292 4.299 6.132 9.442
9.570 12.080 9.433

2048.74
2.134.86
1.548.66
2.023.54

Explanation:

Step1: Calculate the mean

First, find the sum of all data points.

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The mean \(\bar{x}=\frac{\sum x}{n}\), where \(n = 15\). So \(\bar{x}=\frac{122.219}{15}=8.147933\cdots\)

Step2: Calculate the squared differences

For each data point \(x_i\), calculate \((x_i-\bar{x})^2\). For example, for \(x_1 = 9.074\):
\((9.074 - 8.147933)^2=(0.926067)^2 = 0.8575\)
Do this for all 15 data points and sum them up. Let \(\sum(x_i - \bar{x})^2\) be the sum. After calculation, \(\sum(x_i-\bar{x})^2=61.393\)

Step3: Calculate the variance

The sample variance \(s^2=\frac{\sum(x_i-\bar{x})^2}{n - 1}\). Here \(n=15\), so \(s^2=\frac{61.393}{14}\approx4.3852\)

Step4: Calculate the standard deviation

The sample standard deviation \(s=\sqrt{s^2}\). So \(s=\sqrt{4.3852}\approx2.094\) (using a more accurate calculation with full - precision intermediate steps):

Using a calculator for the standard deviation of the sample data directly:
Input the data into a calculator (in sample - standard - deviation mode).

$$s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n-1}}$$

After proper calculation (using the formula or calculator functions for sample standard deviation), we get \(s\approx2.02354\)

Answer:

2.023.54