Lesson 53: Solving Problems Using Measures of Central Tendency

New Concept

To summarize data, we often report an average of some kind, like the mean, median, or mode.

What’s next

Next, we'll dive into worked examples showing how to calculate each measure and choose the best one to describe a set of data.

Property

The mean is the sum of the data divided by the number of data points. The median of an ordered list of numbers is the middle number or the mean of the two central numbers. The mode is the most frequently occurring number in a set.

Examples

For the data set {4, 6, 6, 8, 11}, the mean is $\frac{4+6+6+8+11}{5} = 7$, the median is 6, and the mode is 6.
For an even data set {10, 20, 40, 50}, the median is the mean of the two middle values: $\frac{20+40}{2} = 30$.
In the set {Red, Blue, Red, Green}, the mode is Red, as it is the most frequent category.

Explanation

Think of these as different detectives trying to find the 'typical' number. The mean is the math nerd, adding everything up and dividing. The median is the diplomat, finding the exact middle person in the line. The mode is the trend-spotter, pointing out the most popular choice in the crowd. They all find an 'average' but in their own unique way.

Property

The range of data is the difference between the highest and lowest values in the set.

Examples

For the TV ratings data with a high of 16.1 and a low of 14.2, the range is $16.1 - 14.2 = 1.9$.
If your test scores are {75, 81, 82, 98}, the range is $98 - 75 = 23$.
For customer ages {27, 28, 31, 33, 35, 38, 41, 42, 73, 75}, the range is a wide $75 - 27 = 48$.

Explanation

Range tells you the spread of your data, like the distance from the tallest to the shortest person in a room. It's a quick way to see how varied your numbers are, but it's sensitive to extreme values and doesn't tell you anything about what's happening in the middle. It gives a full picture of the data's scope from end to end.

What happens when there’s a huge age gap in your club’s members?
Example Problem:
A music teacher lists the ages of her students:

Find the mean, median, mode, and range.

Step-by-Step:

1. Add all the ages to find the sum: $22 + 27 + 26 + 40 + 43 + 21 + 68 + 66 + 32 + 29 = 374$
2. To find the mean, divide total by the number of students ($10$):

3. Put the ages in order: $21, 22, 26, 27, 29, 32, 40, 43, 66, 68$
4. Since there are $10$ ages, the median is the mean of the $5$th and $6$th numbers: $29$ and $32$

5. There’s no repeated age, so there is no mode.
6. The range is the difference, highest minus lowest:

Takeaway:
Big outliers (like two students much older than the rest) pull the mean up, but the median stays closer to most students’ ages. Sometimes, the median is a better measure of what’s typical!

Property

The mean can be misleadingly high or low when a dataset includes a few values that are much greater or smaller than the others. These extreme values are often called outliers.

Examples

For salaries {32, 34, 32, 30, 95} thousands, the mean is 44.6 thousand dollars, but the median is 32 thousand dollars, which is more typical.
Five friends have allowances of {5, 10, 10, 15, 100} dollars. The mean is 28 dollars, but the median is 10 dollars, a more representative amount.

Explanation

Watch out for the mean when you have outliers—super high or low numbers! They can pull the mean in their direction, making it a fibbing tour guide for what's 'typical.' The median often tells a more honest story by ignoring the drama from extreme values and sticking to the true middle ground. It's the most reliable player on the team!