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The mean is a critical concept in statistics, often referred to as the "average." It is a measure of central tendency that summarises the central point of a data set. To calculate the mean, you add up all the values and then divide by the number of values. In this exercise, you're given that the mean daily wage for a sample of employees is \(18.

• Formula: \ \mu = \frac{\sum x}{n} \ where \ \mu is the mean, \ \sum x is the sum of all values, \ n is the number of values.

The mean tells us that, on average, employees earn \)18 per day. It is a single number representing the overall "balance point" of the data. While the mean provides a valuable snapshot, it does not inform us about the spread or variability of the data, which is where our next concept, standard deviation, comes in.

The standard deviation is a measure of how spread out the data points are around the mean. It indicates how much individual data points deviate from the average. In our case, the standard deviation is \(2.50, which illustrates the variability in daily wages.

• Formula: \ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}} \ where \ \sigma is the standard deviation, \(x_i\) represents each data point, \ \mu is the mean, and \ n is the number of data points.

A low standard deviation signifies that most data points are close to the mean, while a high standard deviation indicates greater variation. In our exercise, a standard deviation of \)2.50 points to some fluctuation in daily wages around the mean of $18. Understanding standard deviation is crucial as it tells us if the data points are clustered closely around the mean or if there is significant variability. It helps in assessing the reliability and predictability of the data.

The Empirical Rule is an essential guideline that helps describe how data is distributed within a normal distribution. For data that follows a bell-shaped curve like our wages example, the Empirical Rule helps us understand where most of the data falls. According to this rule:

• About 68% of data lies within one standard deviation from the mean
• 95% of data falls within two standard deviations
• 99.7% of data is within three standard deviations

In our example, we want to find where 95% of the data falls. Given the mean (8) and standard deviation (49), you calculate this by moving two standard deviations away from the mean.
Thus, approximately 95% of employees earn between $13 and $23 per day. The Empirical Rule provides a quick and intuitive way to understand the distribution and spread of data within a normal distribution, helping to make informed predictions and insights. Understanding this allows statisticians to estimate the degree of certainty with which they can describe data behavior.