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The sample mean is a simple but essential statistic that helps us understand the central tendency of a data set. It gives a rough idea of where the middle of the data lies. To find the sample mean, you add up all the observed values from your sample and then divide this sum by the total number of values.
For example, if we have heart rates measured for 34 participants in a Zumba class, we simply sum all these rates and divide by 34.
The formula to calculate the sample mean is:

• \( \overline{X} = \dfrac{1}{n}\sum_{i=1}^{n}X_i \)

Where:

• \( \overline{X} \) is the sample mean
• \( n \) is the sample size
• \( X_i \) are the individual data points

Calculating the sample mean is an initial step in many statistical analyses and serves as a point estimate of the population mean.

The sample standard deviation is a measure that tells us how much the values in a sample deviate or vary from the sample mean. It helps us understand the spread or dispersion within the data.
First, you calculate the variance, which involves finding the average of the squared differences between each data point and the sample mean.
Once you have the variance, you find the standard deviation by taking its square root.
The formula for the variance is:

• \( s^2 = \dfrac{1}{n - 1}\sum_{i=1}^{n}(X_i - \overline{X})^2 \)

Here,

• \( s^2 \) is the sample variance
• \( X_i \) are the data points
• \( \overline{X} \) is the sample mean
• \( n \) is the number of data points

Finally, the standard deviation \( s \) is obtained as \( s = \sqrt{s^2} \).

• s
• \(\overline{X}\)
• n

Knowing the standard deviation helps in calculating confidence intervals and determining data variability.

A Z-score is a statistical measure that describes a value's relation to the mean of a group of values. It tells you how many standard deviations an element is from the mean.
In the context of creating a confidence interval, the Z-score indicates the number of standard deviations by which our estimated parameter differs from the hypothesized parameter value. In other words, it provides a way to quantify the likelihood of a result occurring under a normal distribution.
When calculating a 98% confidence interval, we rely on the Z-table to find the appropriate Z-score, which is 2.33 for 98% confidence. This Z-score helps us expand from the mean both below and above to set the range of the confidence interval.

• When the Z-score is high, it suggests a point is far from the mean.
• For confidence intervals, higher Z-scores are used for higher confidence levels.

The population mean is the average of a set of characteristics from an entire population. It is a parameter that is usually unknown in practice, as obtaining data from an entire population can be challenging or impossible.
A sample mean is often used as an estimate or representation of the population mean. Therefore, the population mean can be seen as the true average that we aim to approximate using our sample statistics.
While constructing confidence intervals, the goal is to estimate the population mean by defining a range that, with a certain probability, is likely to contain the true population mean.

• In practice, the population mean is theoretical and is predicted through sampling.
• Confidence intervals provide insight into where this theoretical mean lies with the given level of confidence.

By understanding the relationship between the sample mean and the population mean, along with the variability within data, we can make educated guesses about the population characteristics.