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Why Is \(n-1\) in the Sample Standard Deviation? Why do we calculate \(s\) by dividing by \(n-1\), rather than just \(n\) ? $$ s^{2}=\frac{\sum(x-\bar{x})^{2}}{n-1} $$ The reason is that if we divide by \(n-1\), then \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\), the population variance. We want to show that \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\), sigma squared. The mathematical proof that this is true is beyond the scope of an introductory statistics course, but we can use an example to demonstrate that it is. First we will use a very small population that consists only of these three numbers: 1,2, and 5 . You can determine that the population standard deviation, \(\sigma\), for this population is \(1.699673\) (or about \(1.70\) ), as shown in the \(\mathrm{TI}-84\) output. So the population variance, sigma squared, \(\sigma^{2}\), is \(2.888889\) (or about 2.89). Now take all possible samples, with replacement, of size 2 from the population, and find the sample variance, \(s^{2}\), for each sample. This process is started for you in the table. Average these sample variances \(\left(s^{2}\right)\), and you should get approximately \(2.88889 .\) If you do. then you have demonstrated that \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\), sigma squared. Show your work by filling in the accompanying table and show the average of \(s^{2}\).

Step by step solution

01

Finding Population Variance

Make a note of the population data which are the numbers 1, 2, and 5. The population standard deviation (\(\sigma\)) is given as roughly 1.70. The population variance (\(\sigma^{2}\)) is the square of the standard deviation which gives us approximately 2.89.

02

Determining Sample Variance

We need to take all samples of size 2 from our population, with replacement. That means we are looking at all combinations of two numbers which could be the same number twice. For each of these samples calculate the sample variance using the formula for \(s^{2}\).

03

Averaging Sample Variance

Next, find the average of all these sample variances. This entails adding up all the sample variances calculated in step 2 and dividing by the total number of samples.

04

Compare Population and Sample Variance

After calculating the average of the sample variances, compare this value with the population variance calculated in step 1. If these values are approximately equal, it shows that the sample variance is an unbiased estimator of the population variance.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Variance

Population variance is a measure used in statistics to describe how data points in a population are spread out from their mean. It provides information about the consistency of a data set. The more spread out the data points are, the higher the population variance will be.

To calculate population variance, first determine the population mean, which is the average of all data points in the population. Then, for each data point, subtract the population mean and square the result, which diminishes the impact of negative differences. Add these squared differences together, and divide by the number of data points in the population to get the variance.

• Formula: \(σ^2 = \frac{1}{N} \sum (x_i - \mu)^2\)

Population variance gives us a snapshot of the variability within a full set of data, making it a foundational concept in statistics.

Unbiased Estimator

An unbiased estimator is a statistical term for an estimate that accurately reflects the true value of the parameter being estimated. Essentially, an unbiased estimator does not systematically overestimate or underestimate the population parameter.

One of the key characteristics of unbiased estimators is that the expected value of the estimator equals the parameter it estimates. For instance, when estimating the population variance using sample data, the sample variance is considered an unbiased estimator if it, on average, equals the population variance when derived from multiple samples.

• Why it matters: Unbiased estimators are desirable because they provide a "fair" estimate without consistent error, improving the reliability of statistical analysis.

This concept is vital in ensuring statistical conclusions are as accurate as possible.

Sample Variance

Sample variance is a representative measure of the variability or spread of a set of sample data points. It works similarly to population variance but is used when data is obtained from a sample rather than an entire population.

Calculating sample variance involves taking the same steps as calculating population variance, but with a slight difference. Instead of dividing by the number of data points in the sample, you divide by one less than that, which addresses bias in the estimation process.

• The formula is: \(s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2\)

The adjustment of dividing by \(n-1\) rather than \(n\) corrects the bias and makes sample variance a better estimate of population variance.

n-1 in Statistics

The role of \(n-1\) in statistics is crucial, especially when calculating sample variance and sample standard deviation. This is often referred to as "Bessel's correction."

When calculating sample variance, using \(n\) where \(n\) is the number of data points in your sample, can lead to a biased estimate of the population variance. Dividing by \(n-1\) instead compensates for this bias.

• The rationale: By sampling, we lose one degree of freedom—there's always one less independent piece of information, hence the subtraction of one.
• Mathematically, it ensures that, on average, the sample variance equals the true population variance.

Understanding this correction is key to ensuring accurate and unbiased statistical analysis, making it an essential concept in inferential statistics.