Question

Find the standard deviation of 40, 43, 38, 49, 47, 55, 41, 35, 42, 48, 29, 35

Answer 1

To calculate the standard deviation of the given data set \(40, 43, 38, 49, 47, 55, 41, 35, 42, 48, 29, 35\), we can follow these steps:

1. **Calculate the mean (average)**:

\[
\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
\]
\[
\text{Mean} = \frac{40 + 43 + 38 + 49 + 47 + 55 + 41 + 35 + 42 + 48 + 29 + 35}{12}
\]
\[
\text{Mean} = \frac{ 40 + 43 + 38 + 49 + 47 + 55 + 41 + 35 + 42 + 48 + 29 + 35 }{12} = \frac{ 462 }{12} = 38.5
\]

2. **Calculate each value's deviation from the mean, square each deviation, and sum**:

\[
\text{Deviations} = (x_i - \text{Mean})^2
\]
Here are the calculations for each \(x_i\):

- \( (40 - 41.5)^2 = ( -1.5 )^2 = 2.25 \)
- \( (43 - 41.5)^2 = ( 1.5 )^2 = 2.25 \)
- \( (38 - 41.5)^2 = ( -3.5 )^2 = 12.25 \)
- \( (49 - 41.5)^2 = ( 7.5 )^2 = 56.25 \)
- \( (47 - 41.5)^2 = ( 5.5 )^2 = 30.25 \)
- \( (55 - 41.5)^2 = (13.5)^2 = 182.25 \)
- \( (41 - 41.5)^2 = ( -0.5 )^2 = 0.25 \)
- \( (35 - 41.5)^2 = ( -6.5 )^2 = 42.25 \)
- \( (42 - 41.5)^2 = ( 0.5 )^2 = 0.25 \)
- \( (48 - 41.5)^2 = ( 6.5 )^2 = 42.25 \)
- \( (29 - 41.5)^2 = ( -12.5 )^2 = 156.25 \)
- \( (35 - 41.5)^2 = ( -6.5 )^2 = 42.25 \)

Now summing these squared deviations:
\[
\text{Total} = 2.25 + 2.25 + 12.25 + 56.25 + 30.25 + 182.25 + 0.25 + 42.25 + 0.25 + 42.25 + 156.25 + 42.25 = 512.5
\]

3. **Calculate the variance**:
\[
\text{Variance} = \frac{\text{Total of squared deviations}}{n}
\]
Where \(n\) is the number of observations (12).
\[
\text{Variance} = \frac{512.5}{12} = 42.7083
\]

4. **Calculate the standard deviation**:
\[
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{42.7083} \approx 6.54
\]

So, the standard deviation of the given data set is approximately \( \mathbf{6.54} \).