Find the standard deviation for the data set.
​$68​, ​$43​, ​$37​, ​$31​, ​$39

To find the standard deviation of a data set, we follow these steps:

1. **Find the mean (average):**
   \[
   \text{Mean} = \frac{68 + 43 + 37 + 31 + 39}{5} = \frac{218}{5} = 43.6
   \]

2. **Find the squared differences from the mean for each data point:**
   \[
   (68 - 43.6)^2 = (24.4)^2 = 595.36
   \]
   \[
   (43 - 43.6)^2 = (-0.6)^2 = 0.36
   \]
   \[
   (37 - 43.6)^2 = (-6.6)^2 = 43.56
   \]
   \[
   (31 - 43.6)^2 = (-12.6)^2 = 158.76
   \]
   \[
   (39 - 43.6)^2 = (-4.6)^2 = 21.16
   \]

3. **Find the average of these squared differences (variance):**
   \[
   \text{Variance} = \frac{595.36 + 0.36 + 43.56 + 158.76 + 21.16}{5} = \frac{819.2}{5} = 163.84
   \]

4. **Take the square root of the variance to get the standard deviation:**
   \[
   \text{Standard deviation} = \sqrt{163.84} \approx 12.8
   \]

Thus, the standard deviation is approximately **$12.8**.

Would you like a detailed explanation of any of the steps? Here are five related questions for further exploration:

1. What is the difference between population standard deviation and sample standard deviation?
2. How is variance related to standard deviation?
3. Can standard deviation be negative?
4. How do outliers affect standard deviation?
5. How would the standard deviation change if we added or removed a data point?

**Tip:** Standard deviation provides insight into the spread of data. The larger the value, the more spread out the data is from the mean.

Find the standard deviation for the data set: $68, $43, $37, $31, $39

Learn how to calculate the standard deviation for the data set $68, $43, $37, $31, and $39. Follow a step-by-step guide covering key statistical concepts like mean, variance, and standard deviation. Suitable for students in Grades 8-10.

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