When you get a big set of data (a class of 45 students, an event with 60 invitees, a building with a 300 residents, a theater that seats 100 patrons), there are all sorts of ways to “describe” the data using math. Mean, median, and mode are all types of averages.
MEAN = AVERAGE OF ALL NUMBERS IN DATA
When we use the word “average” in daily life, we are usually talking about the mean. You can figure out the mean by adding up all the numbers in the data and then dividing by the how many numbers you have.
For example, if you have 10 numbers, you add up all the numbers and divide the sum by 10. This would give you the mean of the data.
You can calculate the mean by adding up all of the numbers in the set and then dividing by how many numbers there are in the set.
Here are some examples:
EXAMPLE 1 – AVERAGE HEIGHT OF BASKETBALL PLAYER IN COACH SLATER’S TEAM
Coach Slater’s basketball team has 10 kids. Some of the kids are really shooting up in height this year. What is the average height of the ballers in Coach Slater’s team?
Heights ordered in 40 in, 44 in, 46 in, 46 in, 47 in, 47 in, 51 in, 51 in, 52 in, 58 in,
Step 1: Add up all of the heights.
40 + 44 + 46 + 46 + 47 + 47 + 51 + 51 + 52 + 58 = 482 in.
Step 2: Divide the sum by the number of heights (10 kids, remember?).So we will divide by 10.
482 ÷ 10 = 48.2 in.
Answer: So, the average height of Coach Slater’s team is 48.2 in.
EXAMPLE 2 – WILLIAM’S AVERAGE GRADE IN MATH CLASS
William’s grade in Math class is an average of all his homework, quiz and test grades during the 6 week period.
You have a total of 20 grades, all from a total of 100 points. The zero’s are homework you missed when you were out sick:
100, 100, 100, 80, 82, 85, 91, 99, 87, 100, 100, 0, 0, 0, 45, 90, 94, 100, 100, 100
Step 1: Add up all of the grades.
100 + 100 + 100 + 80 + 82 + 85 + 91 + 99 + 87 + 100 + 100 + 0 + 0 + 0 + 45 + 90 + 94 + 100 + 100 + 100 = 1553
Step 2: Divide the sum by the total number of assignments (20, remember?).So we will divide by 20.
1553 ÷ 20 = 77.65.
Answer: Average grade = 77.65. A solid C+. What happened here? The zero’s really brought this down but the 100’s helped even them out.
EXAMPLE 3 – WILLIAM’S AVERAGE GRADE IN MATH CLASS IF THE TEACHER DROPS THE ZEROS
William brings a doctor’s note about the time he had the flu, shows his teacher how well he had been doing in class, and asks her if she can drop the zeros or let him make up the work. Since he had a valid reason, the teacher decides to drop the zeros.
Let’s calculate this again, shall we?
100, 100, 100, 80, 82, 85, 91, 99, 87, 100, 100, 0, 0, 0, 45, 90, 94, 100, 100, 100
Step 1: Add up all of the grades.
100 + 100 + 100 + 80 + 82 + 85 + 91 + 99 + 87 + 100 + 100 + 0 + 0 + 0 + 45 + 90 + 94 + 100 + 100 + 100 = 1553
Yes, calm down, we noticed too that the sum is the same.
Step 2: Divide the sum by the total number of assignments – we only have 17 now. So we will divide by 17.
1553 ÷ 17 = 91. 35
Answer: Average grade = 91.3! A solid A-. Great job, William – both with the grades and with convincing your teacher.
WHEN DOES MEAN NOT WORK?
We saw something interesting in the above data. High and low numbers (100’s and 0’s) can really change the mean. That’s why we have median and mode.
Here’s an example. David is interviewing for a small startup that has 10 employees. The owner and CEO tells him that most of the staff are like him, beginners at their very first job. Some have been with the company for an year and so have a little more experience.
William doesn’t want to ask him what he’ll be paid, and so he asks what the average pay is. The CEO tells him that the average pay is $37,200 per year.
William is excited and accepts the position. He’s then offered an annual salary of $24,000.
Uh-oh. What went wrong here? Let’s take a closer look at the data then, shall we? Here are all the salaries of the staff members.
| Title | Salary |
|---|---|
| Junior Software Developer | $24,000 |
| Software Developer | $26,000 |
| Senior Software Developer | $27,000 |
| Web Designer | $26,000 |
| Graphic Designer | $24,000 |
| Web Developer | $24,000 |
| Senior Software Developer | $28,000 |
| Marketing Manager | $29,000 |
| Financial Analyst | $32,000 |
| CEO | $132,000 |
While the CEO was right about the “mean salary” for these staff, there is an outlier here – the CEO’s own salary of $132,000.
This outlier “skews” the numbers and makes the mean much higher than the average would be without it.This is where you need the <drumroll> MEDIAN </drumroll>.
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