Top-Paid CEOs The data shown are the total compensation (in...

Top-Paid CEOs The data shown are the total compensation (in millions of dollars) for the 50 top-paid CEOs for a recent year. Compare the averages, and state which one you think is the best measure. $\begin{array}{rrrrr}17.5 & 18.0 & 36.8 & 31.7 & 31.7 \\ 17.3 & 24.3 & 47.7 & 38.5 & 17.0 \\ 23.7 & 16.5 & 25.1 & 17.4 & 18.0 \\ 37.6 & 19.7 & 21.4 & 28.6 & 21.6 \\ 19.3 & 20.0 & 16.9 & 25.2 & 19.8 \\ 25.0 & 17.2 & 20.4 & 20.1 & 29.1 \\ 19.1 & 25.2 & 23.2 & 25.9 & 24.0 \\ 41.7 & 24.0 & 16.8 & 26.8 & 31.4 \\ 16.9 & 17.2 & 24.1 & 35.2 & 19.1 \\ 22.9 & 18.2 & 25.4 & 35.4 & 25.5\end{array}$

Top-Paid CEOs The data shown are the total compensation (in millions of dollars) for the 50 top-paid CEOs for a recent year. Compare the averages, and state which one you think is the best measure.
$\begin{array}{rrrrr}17.5 & 18.0 & 36.8 & 31.7 & 31.7 \\ 17.3 & 24.3 & 47.7 & 38.5 & 17.0 \\ 23.7 & 16.5 & 25.1 & 17.4 & 18.0 \\ 37.6 & 19.7 & 21.4 & 28.6 & 21.6 \\ 19.3 & 20.0 & 16.9 & 25.2 & 19.8 \\ 25.0 & 17.2 & 20.4 & 20.1 & 29.1 \\ 19.1 & 25.2 & 23.2 & 25.9 & 24.0 \\ 41.7 & 24.0 & 16.8 & 26.8 & 31.4 \\ 16.9 & 17.2 & 24.1 & 35.2 & 19.1 \\ 22.9 & 18.2 & 25.4 & 35.4 & 25.5\end{array}$

Key Concepts

Arithmetic Mean

The arithmetic mean is calculated by summing all the values in a dataset and dividing by the number of values. It provides a measure of central tendency, but it is sensitive to extreme values or outliers, which can skew the result.

Median

The median is the middle number in a dataset when the values are arranged in order. It is a robust measure of central tendency because it is less affected by outliers and skewed data, making it a reliable indicator of the center for data with extreme values.

Outliers

Outliers refer to values that are significantly higher or lower than most of the other data points. Their presence can distort measures like the mean, which is why alternative statistics such as the median are often considered for datasets with such anomalies.

Robustness of Statistics

The robustness of a statistical measure refers to its ability to provide a reliable measure of central tendency even when the data contains outliers or is highly skewed. Measures like the median are robust, meaning they are less influenced by extreme values compared to measures like the arithmetic mean.

Transcript

00:01 Okay, so we are given a huge list of 50 numbers from ceo's salaries and compensation, and we're asked to find and compare the averages and then find out which way of measuring the average is the best way.

00:15 So to start off, we want to list the ways we can possibly use.

00:20 So the four ways to find an average is through the mean, the median mode, and something called mid -range.

00:40 So what we're going to do is we're going to find all four of these to then find out which one is the best.

00:46 So starting with mean, mean is another way to say average, and how you find it is you add up the sum of all the xes.

00:56 This symbol right here means some, and this represents all of the x's, the x values.

01:02 And you put it over n, which can represent the number of the ques.

01:08 Quantity and amount of numbers you have.

01:11 So if you add all of the x's together, all the numbers we have, you get 1 ,221 .1.

01:23 And then you divide that by how many numbers there are.

01:26 And we're told in the problem that we have 50 numbers.

01:31 So you do this, you do 1 ,221 .1 divided by 50, and you get the mean which is 2442.

01:40 And of course this is, this is 1 ,22.

01:42 And of course, this in dollars, right? so that right there is your mean.

01:47 Second is median.

01:48 So median is finding the middle number.

01:51 If you were to arrange all the 50 numbers from lowest to highest and then find the middle numbers.

01:57 So when you do that, it takes a while to do, but it's worth it.

02:02 When you put them all from lowest to highest and you start scratching them off one by one to find the middle number, you actually get two middle numbers.

02:10 And those numbers are 23 .2 and 23 .7.

02:17 So in this case, how you find which one's the middle being that you have two, you have to, you have to add them together and divide by two.

02:26 And what that kind of stimulates in axas is you're basically finding the average between these two numbers.

02:33 So when you do this, you get 2345.

02:40 That is your median.

02:42 Then on to mode...

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