Question

The San Luis ObispoTelegram-Tribune(October1,1994) reported the following monthly salaries for supervisors from six different counties: \(\$ 5,354\) (Kern), \(\$ 5,166\) (Monterey), \(\$ 4,443\) (Santa Cruz), \(\$ 4,129\) (Santa Barbara), \(\$ 2,500\) (Placer), and \$2,220 (Merced). San Luis Obispo County supervisors are supposed to be paid the average of the two counties in the middle of this salary range. Which measure of center determines this salary, and what is its value? Find the value of the other measure of center featured in this chapter. Why is it not as favorable to the San Luis Obispo County supervisors (although it might appeal to taxpayers)?

Short answer

The measure of center that determines the salary of San Luis Obispo County supervisors is the 'Median', which is \$ 4,286. The other measure of center, 'Mean' is \$ 4,135 and this value is lower than the median, hence, not as favorable for the supervisors would result in a lower salary but would be more appealing to taxpayers.

Step-by-step solution

Arrange the Salaries in Ascending Order

First of all, let's arrange the given salaries in ascending order. This is necessary to determine the middle values of the range. The salaries in ascending order are: \(\$ 2,220\), \(\$ 2,500\), \(\$ 4,129\), \(\$ 4,443\), \(\$ 5,166\) and \(\$ 5,354\).

Determine the Median Salary

Now we need to find the Median (middle value) of the salary range. Since we have an even number of salaries, the Median is the average of the two middle values. The two middle values in our range are \(\$ 4,129\) and \(\$ 4,443\). Using the formula for median for even number of observations which is (n/2th observation + (n/2)+1 th observation)/2, we get: \((\$ 4,129 + \$ 4,443) / 2= \$ 4,286\). So, the median salary is \$ 4,286.

Determine the Mean Salary

The mean salary is calculated as the sum of all salaries divided by the number of salaries. Adding all the salaries and dividing by 6, we get: \((\$ 2,220 + \$ 2,500 + \$ 4,129 + \$ 4,443 + \$ 5,166 + \$ 5,354) / 6 = \$ 4,135\). So, the mean salary is \$ 4,135.

Comparison of the Measures

The median salary (\$ 4,286) is more than the mean salary (\$ 4,135). The median is not as favorable to San Luis Obispo County supervisors as they would be paid less if the mean was used to calculate their salary. On the other hand, the mean might appeal to taxpayers as the supervisor salaries would be less.

Key concepts

Median

To understand the median, imagine lining up all the values in your data set from the smallest to the largest. The median is the number that lands in the middle of this ordered list. If there is an odd number of values, the median is simply the middle number. However, when there is an even number of values, like in our example with supervisors' salaries, the median is found by taking the average of the two middle numbers.

For the supervisors' salaries exercise, after lining up all the salaries in ascending order, the middle two numbers were \(\$ 4,129\) and \(\$ 4,443\). Averaging these gives us the median salary of \(\$ 4,286\). This calculation offers a resistant measure of central tendency, meaning it is not greatly influenced by extremely low or high values, hence providing a better reflection of a typical salary in this context.

Mean

The mean, often referred to as the average, is probably the most familiar measure of central tendency. To calculate it, you add together all of the values in your data set, and then divide that sum by the number of values.

In the case of the supervisors' salaries, adding up all six salaries and dividing by six yields a mean of \(\$ 4,135\). The mean salary is often used because it takes into account all of the salaries in the data set, but it is sensitive to extreme values. This is why, in some cases, it may not be the best representation of central tendency, especially when outliers are present in the data.

Statistical Analysis

Statistical analysis is a broad term for various methods used to process, interpret, and draw conclusions from data. Measures like the median and mean we've discussed are fundamental tools in statistical analysis, enabling us to summarize and understand large data sets quickly and easily.

Doing a thorough statistical analysis involves choosing appropriate measures and correctly interpreting them. Selecting between the mean and median depends on the nature of the data and the specific circumstances of the analysis, just like the supervisors' salaries scenario showcases.

Data Interpretation

Data interpretation involves making sense of numerical information. It's not only about applying formulas but also about understanding what the numbers mean in a given context. For the supervisors' salary exercise, while the calculation of both median and mean is straightforward, interpreting why the median might be a more favorable measure for the supervisors requires deeper analysis.

This is because the median is less affected by the low salary from Placer and the high salary from Kern, which might skew the mean. Interpreting data correctly helps us make informed decisions—like ensuring salaries are fair and represent what is typical for the role.