Question

How to calculate the standard deviation of 12 15 18 20 19 21 17 16 14

Answer 1

To calculate the standard deviation of a data set, you can follow these steps:

1. **Calculate the Mean (Average):**
\[
\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
\]
For the given data set: \(12, 15, 18, 20, 19, 21, 17, 16, 14\)

Sum of values:
\[
12 + 15 + 18 + 20 + 19 + 21 + 17 + 16 + 14 = 152
\]
Number of values:
\[
9
\]
So, the Mean is:
\[
\text{Mean} = \frac{152}{9} \approx 16.89
\]

2. **Calculate Each Deviation from the Mean:**
Subtract the Mean from each value and square the result:
- \( (12 - 16.89)^2 \approx 24.49 \)
- \( (15 - 16.89)^2 \approx 3.57 \)
- \( (18 - 16.89)^2 \approx 1.24 \)
- \( (20 - 16.89)^2 \approx 9.79 \)
- \( (19 - 16.89)^2 \approx 4.49 \)
- \( (21 - 16.89)^2 \approx 16.44 \)
- \( (17 - 16.89)^2 \approx 0.01 \)
- \( (16 - 16.89)^2 \approx 0.79 \)
- \( (14 - 16.89)^2 \approx 8.23 \)

3. **Calculate the Variance:**
- Sum of the squared deviations:
\[
24.49 + 3.57 + 1.24 + 9.79 + 4.49 + 16.44 + 0.01 + 0.79 + 8.23 \approx 68.05
\]

- Divide by the number of values (if calculating the population standard deviation) or \( n - 1 \) (if calculating the sample standard deviation). Here, we'll use \( n - 1 = 8 \) since it's usually more common in statistics.

So the variance is:
\[
\text{Variance} = \frac{68.05}{8} \approx 8.50625
\]

4. **Calculate the Standard Deviation:**
Finally, take the square root of the variance:
\[
\text{Standard Deviation} = \sqrt{8.50625} \approx 2.91
\]

Therefore, the standard deviation of the data set \(12, 15, 18, 20, 19, 21, 17, 16, 14\) is approximately \(2.91\).