Calculate the standard deviation of the sample quantitative data shown, to two decimal places.

x
1.4
23.8
27.5
22.6
12
23.6
29.2
5.3

Standard deviation:

To calculate the standard deviation of a sample, follow these steps:

1. **Find the mean (average) of the sample data**:
   \[
   \text{Mean} (\bar{x}) = \frac{\sum x_i}{n}
   \]
   where \( x_i \) are the data points and \( n \) is the number of data points.

2. **Calculate the variance**:
   \[
   \text{Variance} (s^2) = \frac{\sum (x_i - \bar{x})^2}{n-1}
   \]
   where \( \bar{x} \) is the mean calculated in step 1.

3. **Take the square root of the variance to get the standard deviation**:
   \[
   \text{Standard Deviation} (s) = \sqrt{s^2}
   \]

Let's compute the standard deviation step by step:

1. **Calculate the mean**:
   \[
   \bar{x} = \frac{1.4 + 23.8 + 27.5 + 22.6 + 12 + 23.6 + 29.2 + 5.3}{8} = \frac{145.4}{8} = 18.175
   \]

2. **Calculate each squared deviation from the mean and sum them up**:
   \[
   \sum (x_i - \bar{x})^2 = (1.4 - 18.175)^2 + (23.8 - 18.175)^2 + \ldots + (5.3 - 18.175)^2
   \]
   
   Let's compute each squared deviation:

   \[
   (1.4 - 18.175)^2 = (-16.775)^2 = 281.414
   \]
   \[
   (23.8 - 18.175)^2 = (5.625)^2 = 31.641
   \]
   \[
   (27.5 - 18.175)^2 = (9.325)^2 = 86.963
   \]
   \[
   (22.6 - 18.175)^2 = (4.425)^2 = 19.583
   \]
   \[
   (12 - 18.175)^2 = (-6.175)^2 = 38.129
   \]
   \[
   (23.6 - 18.175)^2 = (5.425)^2 = 29.428
   \]
   \[
   (29.2 - 18.175)^2 = (11.025)^2 = 121.550
   \]
   \[
   (5.3 - 18.175)^2 = (-12.875)^2 = 165.734
   \]

   \[
   \sum (x_i - \bar{x})^2 = 281.414 + 31.641 + 86.963 + 19.583 + 38.129 + 29.428 + 121.550 + 165.734 = 774.442
   \]

3. **Calculate the variance**:
   \[
   s^2 = \frac{774.442}{8 - 1} = \frac{774.442}{7} \approx 110.635
   \]

4. **Calculate the standard deviation**:
   \[
   s = \sqrt{110.635} \approx 10.52
   \]

So, the standard deviation of the sample data is approximately **10.52**.

Would you like any more details or explanations on how this was calculated?

---

Here are 5 related questions to consider:

1. How does standard deviation differ between a sample and a population?
2. What is the impact of an outlier on the standard deviation?
3. How would you calculate the variance for this sample data?
4. How do you interpret the standard deviation in the context of data spread?
5. What are the assumptions underlying the calculation of standard deviation?

**Tip:** Always check whether you're working with a sample or population data to use the correct formula for variance and standard deviation.

Learn how to calculate the standard deviation of sample quantitative data step-by-step with a detailed solution. This problem covers key statistical concepts including mean, variance, and standard deviation formula applications. Suitable for college-level students.

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