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Understanding the concept of population mean is crucial in statistics as it represents the average value in a population. A population is the entire collection of elements or outcomes that you're interested in studying. In many situations, it is impractical or impossible to measure every individual in the population, so we use a sample - a smaller, manageable number of observations - to make inferences about the population mean, denoted by \(\mu\).

The sample mean, denoted by \(\bar{x}\), is used as an estimate of the true population mean. However, because it's derived from a sample, it's subject to sampling error, and it may not exactly match the true mean. That's where confidence intervals come in, as they provide a range of values within which the true population mean is likely to fall with a certain level of confidence.

The t-distribution, also known as Student's t-distribution, is a type of probability distribution that is symmetric and bell-shaped, like the normal distribution, but has heavier tails. These heavier tails mean there is a greater probability for extreme values, which compensates for the additional variability and uncertainty when estimating the population mean with a small sample size.

The t-distribution is used instead of the normal distribution when the sample size is small (commonly n<30) and/or when the population standard deviation is unknown. As the sample size increases, the t-distribution approaches the normal distribution. This distribution is pivotal when it comes to calculating confidence intervals for the population mean when using small samples. The t-value, which varies based on the sample size through degrees of freedom (df), allows us to determine how far our sample statistic lies from the population parameter, adjusting for sample size.

The sample standard deviation (denoted as \(s\)) is a measure of the amount of variation or dispersion of a set of values. A low \(s\) indicates that the data points tend to be close to the sample mean (\bar{x}) and hence to the population mean (\(\mu\)). Conversely, a high sample standard deviation signifies that the data points are spread out over a wider range of values.

It is important to note that when calculating the confidence interval for the population mean, we use the sample standard deviation as an estimate of the true population standard deviation. This is because, in many cases, the actual population standard deviation is unknown. The accuracy and reliability of the confidence interval heavily depend on the accuracy of the sample standard deviation. When applying this measure in the steps to find a confidence interval, it is squared to find the variance (\(s^2\)) before taking the square root to return to standard deviation, ensuring the units match those of the mean.