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Understanding how to calculate the mean salary is fundamental in statistics, as it gives us a picture of the average earnings an individual or group receives. To define the mean, add together all the individual salaries and then divide by the number of salaries added. In the context of our exercise, the employer has separate mean salaries for weekdays and weekends. The mean weekly salary is simply the sum of these two mean values. That's to say, if an employer pays a mean salary of \(\$1250\) for the 5-day workweek and \(\$450\) on the weekends, the mean of total weekly salaries would be \(\$1250 + \$450 = \$1700\).

This figure represents the average total amount the employer pays out in salaries over the complete week. When considering mean salary calculations, it's important to note that if there are outliers or a wide range of salaries, the mean might not accurately reflect the typical earning amount. To fully understand salary distribution, it is often helpful to look at the median and mode salaries in conjunction with the mean.

The standard deviation is a measure of how much variation or 'spread' exists from the mean or average value. A low standard deviation means that most of the numbers are close to the mean, while a high standard deviation means that the numbers are more spread out. In salary terms, a low standard deviation indicates that most employees' salaries are close to the average salary, whereas a high standard deviation would imply a larger disparity among what employees are earning.

To calculate the standard deviation, one would typically square the difference between each salary and the mean salary, add these squared differences together, divide by the number of salaries (less one if using a sample to estimate for a larger population, known as 'n-1'), and then take the square root of that result. However, when combining two separate groups with their own standard deviations, as in our exercise, you cannot simply add the standard deviations. Instead, you calculate the square root of the sum of the squared standard deviations. This method, called the Pythagorean theorem of statistics, provides a combined standard deviation that accounts for the variability within both groups of salaries.

When dealing with the mean and standard deviation of total weekly salaries, the situation becomes a composite of the two separate financial distributions: weekday salaries and weekend salaries. The mean of the total weekly salaries has been determined through a simple addition, resulting in \(\$1700\).

Calculating the combined standard deviation requires a different approach. Instead of adding, we utilize the formula for the combination of standard deviations. This involves taking the sum of the squares of the individual standard deviations and then finding the square root of this sum. Based on the standard deviations given for weekdays (\(\$129\)) and weekends (\(\$57\)), the combined standard deviation for total weekly salaries is calculated as \(\sqrt{129^2 + 57^2} = \sqrt{16641 + 3249} = \sqrt{19890} = \$140.43\).

This combined standard deviation helps the employer understand the variability in total salary expenses from week to week. It's crucial for budgeting purposes and for setting salary ranges. A good understanding of the mean and standard deviation can assist in making informed financial decisions within a company.