When determining the variance of a data set, particularly for a sample rather than the entire population, we need to consider an important concept: the bias in the estimation of the population variance. Let's go through the steps to understand why [tex]\( n-1 \)[/tex] is used in the formula for calculating sample variance.
### Step-by-Step Explanation:
1. Population vs. Sample:
- Population: The entire set of individuals or items that we're interested in.
- Sample: A smaller subset taken from the population.
2. Variance Formula:
- For a population with [tex]\( n \)[/tex] items, the variance is calculated as:
[tex]\[
\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2
\][/tex]
Here, [tex]\( \mu \)[/tex] is the population mean.
3. Bias in Sampling:
- When we draw a sample from the population, the sample mean [tex]\( \bar{x} \)[/tex] is typically used to estimate the population mean [tex]\( \mu \)[/tex]. The variance estimation based on the sample mean using [tex]\( n \)[/tex] can underestimate the true population variance.
4. Correcting the Bias - Bessel's Correction:
- To correct for this bias, we use [tex]\( n-1 \)[/tex] instead of [tex]\( n \)[/tex] in the denominator. This adjustment is known as Bessel's Correction.
- The formula for sample variance, [tex]\( s^2 \)[/tex], becomes:
[tex]\[
s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2
\][/tex]
Here, [tex]\( \bar{x} \)[/tex] is the sample mean.
5. Why [tex]\( n-1 \)[/tex]?:
- Using [tex]\( n-1 \)[/tex] rather than [tex]\( n \)[/tex] accounts for the fact that we're using a sample to estimate the population variance, leading to a slightly higher variance value, which corrects the underestimation.
- This makes the sample variance an unbiased estimator of the population variance.
6. Conclusion:
- By using [tex]\( n-1 \)[/tex] in the denominator, we expect the data in the sample to be more dispersed from the mean than if we had used [tex]\( n \)[/tex].
### Final Answer:
- The formula uses " [tex]\( n-1 \)[/tex] " in the denominator because the data is a sample and it is expected to be more dispersed from the mean. This adjustment corrects the bias and provides an unbiased estimate of the population variance.
Thus, the correct statement is:
- The data is a sample and is expected to be more dispersed from the mean.
Hence, the final choice is:
- The data is a sample and is expected to be more dispersed from the mean.
