Explanation:

The sample standard deviation is a biased estimator for the population standard deviation. Let's understand why this is the case.

1. Definition of Sample Standard Deviation:
The sample standard deviation measures the variability or dispersion of a set of data points from their mean. It is calculated by taking the square root of the sample variance.

2. Bias in Estimators:
An estimator is said to be biased if, on average, it consistently underestimates or overestimates the true parameter it is trying to estimate. In the case of the sample standard deviation, it consistently underestimates the true population standard deviation.

3. Bias in Sample Standard Deviation:
The bias in the sample standard deviation arises from the fact that it uses the sample mean as an estimate of the population mean. Since the sample mean is itself an estimator, it is subject to sampling variability and can deviate from the true population mean. This deviation affects the calculation of the sample standard deviation.

4. Bessel's Correction:
The sample standard deviation formula uses Bessel's correction, which adjusts the denominator by subtracting 1 from the sample size (n-1) instead of just using the sample size (n). This correction is applied to make the sample variance an unbiased estimator of the population variance.

5. Biased Estimator for Population Standard Deviation:
Despite Bessel's correction, the sample standard deviation remains a biased estimator for the population standard deviation. This bias arises from the fact that the sample standard deviation is based on a single sample and does not take into account the variability of multiple samples from the same population. In other words, the sample standard deviation does not account for the uncertainty associated with estimating the population standard deviation from a single sample.

Conclusion:
In summary, the sample standard deviation is a biased estimator for the population standard deviation. This bias arises from the use of the sample mean as an estimate of the population mean and the fact that it is based on a single sample. It is important to be aware of this bias when interpreting and using the sample standard deviation in statistical analysis.