Question

Find the standard deviation of the distribution 12,6,7,3,15,10,18,5

a. 4.87
b. 4.97
c. 2.21
d. 5.81

Answer 1

Find the standard deviation of the distribution 12,6,7,3,15,10,18,5

a. 4.87 b. 4.97 c. 2.21 d. 5.81

Solution

The standard deviation of the discrete distribution is

$$\sigma = \sqrt {\frac {1}{N} \sum_ {i = 1} ^ {N} (x _ {i} - \bar {x}) ^ {2}},$$

where $\bar{x} = \frac{\sum_{i=1}^{N} x_i}{N}$ - the mean of distribution.

For our distribution:

$$\bar {x} = \frac {1 2 + 6 + 7 + 3 + 1 5 + 1 0 + 1 8 + 5}{8} = 9. 5,$$

Variance

$$\begin{array}{l} V a r = \frac {1}{N} \sum_ {i = 1} ^ {N} (x _ {i} - \bar {x}) ^ {2} = \frac {1}{8} ((1 2 - 9. 5) ^ {2} + (6 - 9. 5) ^ {2} + (7 - 9. 5) ^ {2} + (3 - 9. 5) ^ {2} \\ + (1 5 - 9. 5) ^ {2} + (1 0 - 9. 5) ^ {2} + (1 8 - 9. 5) ^ {2} + (5 - 9. 5) ^ {2}) \\ \end{array}$$ $$V a r = \frac {1}{8} (2. 5 ^ {2} + 3. 5 ^ {2} + 2. 5 ^ {2} + 6. 5 ^ {2} + 5. 5 ^ {2} + 0. 5 ^ {2} + 8. 5 ^ {2} + 4. 5 ^ {2}) = 2 3. 7 5.$$

The standard deviation of the distribution

$$\sigma = \sqrt {V a r} = \sqrt {2 3 . 7 5} = 4. 8 7.$$

Answer: a. 4.87.