How old are professional football players? The 11th edition...

How old are professional football players? The 11th edition of The Pro Football Encyclopedia gave the following information. Random sample of pro football player ages in years: $$\begin{array}{llllllllll}24 & 23 & 25 & 23 & 30 & 29 & 28 & 26 & 33 & 29 \\24 & 37 & 25 & 23 & 22 & 27 & 28 & 25 & 31 & 29 \\25 & 22 & 31 & 29 & 22 & 28 & 27 & 26 & 23 & 21 \\25 & 21 & 25 & 24 & 22 & 26 & 25 & 32 & 26 & 29\end{array}$$ (a) Compute the mean, median, and mode of the ages. (b) Interpretation Compare the averages. Does one seem to represent the age of the pro football players most accurately? Explain.

How old are professional football players? The 11th edition of The Pro Football Encyclopedia gave the following information. Random sample of pro football player ages in years:
$$\begin{array}{llllllllll}24 & 23 & 25 & 23 & 30 & 29 & 28 & 26 & 33 & 29 \\24 & 37 & 25 & 23 & 22 & 27 & 28 & 25 & 31 & 29 \\25 & 22 & 31 & 29 & 22 & 28 & 27 & 26 & 23 & 21 \\25 & 21 & 25 & 24 & 22 & 26 & 25 & 32 & 26 & 29\end{array}$$
(a) Compute the mean, median, and mode of the ages.
(b) Interpretation Compare the averages. Does one seem to represent the age of the pro football players most accurately? Explain.

Key Concepts

Mean

The mean is the arithmetic average of a dataset, calculated by summing all values and then dividing by the number of values. It provides an overall summary of the data but can be sensitive to extreme values, which may skew the measurement away from the central tendency of the majority.

Median

The median is the middle value in a dataset that has been ordered from smallest to largest. It is a robust measure of central tendency, meaning that it is less affected by outliers or skewed data, and often provides a better central location of the data in non-normal distributions.

Mode

The mode is the most frequently occurring value in a dataset. It is particularly useful for categorical data and can indicate the most common or popular response in a distribution, though it may not always provide a complete picture when a dataset has multiple modes or no repeated values.

Comparison and Interpretation of Measures

Comparing the mean, median, and mode can provide insights into the shape and spread of the data distribution. In datasets with skewed distributions or outliers, the median may serve as a more accurate measure of central tendency than the mean. Evaluating these measures together allows for a deeper interpretation of how well any single average represents the data.

Transcript

00:01 For this problem, the first thing i'm going to want to do is to put my data in order.

00:07 So i'm going to use a stem and leaf plot to help arrange my numbers.

00:18 And since i have numbers in both the 20s and the 30s, i'm going to do stems of 2 and 3.

00:26 And i'm doing it as a double row stem and leaf plot because i do have a lot of numbers in the 20s.

00:36 So when i look through, i'm going to find that there are 221s, there are 4 222s, there are 423s, there are 324s, there are 725s, there are 4 23s, there are 424s, there are 424s, four 26es.

01:18 There are two 27s.

01:23 There are three 28s.

01:29 There are five 29s.

01:35 There's one 30.

01:39 There are two 31s.

01:42 There's a 32, a 33, and a 37.

01:48 And for the sake of someone being able to read this, we should provide a key that three with a vertical bar between the seven represents 37 years old.

02:06 All right, so this data has to do with professional football players ages, and a random sample of 40 was taken.

02:16 And your part a is asking you to find the mean, and to find the mean, we have to sum up all of our data, and divide by how many pieces of data there are.

02:32 And if you add up all of these numbers in this chart, you will end up with a total of 1 ,050, and there happen to be 40 numbers in the chart, so you'll divide by 40, and you will get the mean of 26 .25.

02:53 The next part of part a is to find the median.

02:57 And the median refers to the middle number or numbers as long as our data is arranged in order low to high.

03:16 So i'm going to put a little white dot next to numbers as i count them off.

03:21 Now remember, there's 40 numbers.

03:23 So i'm going to count 10 from the beginning.

03:27 One, two, three, four, five, six, seven, eight, nine, ten.

03:33 And then i'm going to count 10 from the end.

03:36 1, 2, 3, 4, 5, 6, 7, 8, 9, 10...