Variance Standard Deviation 6C8Fb3

1. **Stating the problem:** We want to understand what variance and standard deviation are and how to calculate them. 2. **Formulas:** - Variance ($\sigma^2$ for population, $s^2$ for sample) measures the average squared deviation from the mean. - Standard deviation ($\sigma$ for population, $s$ for sample) is the square root of variance, showing spread in original units. 3. **Important rules:** - For a population of $N$ values $x_1, x_2, ..., x_N$ with mean $\mu$, variance is $$\sigma^2 = \frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2$$ - For a sample of $n$ values with mean $\bar{x}$, sample variance is $$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$ 4. **Example calculation:** Suppose we have data points $2, 4, 6, 8$. - Calculate mean: $$\bar{x} = \frac{2+4+6+8}{4} = \frac{20}{4} = 5$$ - Calculate squared deviations: $(2-5)^2=9$, $(4-5)^2=1$, $(6-5)^2=1$, $(8-5)^2=9$ - Sum squared deviations: $9+1+1+9=20$ - Sample variance: $$s^2 = \frac{20}{4-1} = \frac{20}{3} \approx 6.67$$ - Sample standard deviation: $$s = \sqrt{6.67} \approx 2.58$$ 5. **Explanation:** Variance tells us how spread out the data is by averaging squared differences from the mean. Standard deviation is easier to interpret because it is in the same units as the data. **Final answer:** For the data $2,4,6,8$, sample variance is approximately $6.67$ and sample standard deviation is approximately $2.58$.