Imagine you and your friends are comparing how many minutes you spend on your favorite video game each day. Some of you might play a lot, some just a little. How can we describe the group’s playing habits in a fair, clear way—not just by looking at one person? That is where the ideas of center and variability of data come in.
In this lesson, you learn how to:
These ideas help you make sense of real life: from sports scores 🏀 to test results, from sleep hours to steps on a fitness tracker.
The dot plot in [Figure 1] shows how data values line up on a number line and helps us see center, spread, and unusual points.
Data are pieces of information, usually numbers, that tell us about something. For example:
When we collect many data values and display them (for example, in a dot plot, line plot, or histogram), we get a distribution. A distribution shows how often each value appears.
The distribution helps us answer questions like:
To describe a distribution in a useful way, we usually talk about:
Center gives us a number that describes what is “in the middle” or “typical” for the data.
The mean is what most people call the “average.” You find it by:
If the data values are numbers like 3, 4, and 7, the mean is: \(3 + 4 + 7 = 14\) and there are 3 numbers, so \(14 / 3\).
You can think of the mean as the fair share. If you took all the “points” or “amounts” and shared them equally among all the data points, each one would have the mean.
Example (concept): If three friends have 2, 4, and 6 pieces of candy, that is 12 pieces total. If they share fairly, each friend gets 4 pieces. So the mean number of candies is 4.
The median is the middle number when the data are lined up in order from smallest to largest.
The median is good for describing what is “typical” when there are a few really large or really small values that might pull the mean up or down.
The box plot in [Figure 2] shows where the median sits in the middle of the data, along with quartiles and the IQR.
Two data sets can have the same center but be very different. One can be tightly packed; another can be very spread out. Variability tells us how spread out the data are.
The interquartile range (IQR) measures the spread of the middle half of the data.
To find the IQR:
The IQR tells you how wide the “middle 50%” of the data is. A small IQR means the middle data are close together. A large IQR means the middle data are more spread out.
The mean absolute deviation (MAD) tells us, on average, how far the data values are from the mean.
To find the MAD:
We can write this idea as:
First find the mean of the data, call it \(m\). Then for each value \(x\), find its distance \(|x - m|\). The MAD is the mean of all those distances.
A small MAD means most values are close to the mean. A large MAD means values are often far from the mean.
When we look at a graph of data, like a line plot, histogram, or box plot, we want to describe the overall pattern. We also want to notice any striking deviations—values that do not fit the pattern.
Some things to describe in the overall pattern:
Striking deviations are unusual values, sometimes called outliers. These are points that are much higher or lower than most of the data.
When we spot a striking deviation, we always ask: What might this mean in the real-world context?
The line plot in [Figure 3] highlights one point far away from the rest, which is a striking deviation from the overall pattern.
Suppose five students track how many minutes they read after school one day:
Data (in minutes): 10, 20, 20, 30, 60
Add the numbers: \(10 + 20 + 20 + 30 + 60 = 140\).
There are 5 data values.
Mean: \[\textrm{mean} = \frac{140}{5} = 28\]
So the mean reading time is 28 minutes.
The data are already in order: 10, 20, 20, 30, 60.
There are 5 values, so the median is the 3rd value.
Median = 20 minutes.
Notice: the mean (28) is higher than the median (20) because of the 60-minute reading time, which is much larger than the others.
The middle half of the reading times covers 30 minutes (from 15 to 45).
Now find the mean of these distances:
Add them: \(18 + 8 + 8 + 2 + 32 = 68\)
There are 5 distances.
MAD: \[\textrm{MAD} = \frac{68}{5} = 13.6\]
This tells us that, on average, each student’s reading time is about 13.6 minutes away from the mean of 28 minutes.
Center: A typical reading time is around 20–28 minutes. The median is 20 minutes, and the mean is 28 minutes.
Variability: The IQR is 30 minutes, and the MAD is about 13.6 minutes, so students’ reading times are pretty spread out.
Striking deviation: The 60-minute reading time is much larger than the others (10, 20, 20, 30). This one value pulls the mean up and makes the spread larger. In context, this could mean one student really loves reading that day 📚, or had extra time.
Six students count their steps using fitness trackers:
Data (steps): 4,000; 5,000; 5,000; 6,000; 7,000; 20,000
Write them as: 4000, 5000, 5000, 6000, 7000, 20000.
Add the numbers:
4000 + 5000 + 5000 + 6000 + 7000 + 20000 = 46000
There are 6 values.
Mean: \[\textrm{mean} = \frac{46000}{6} \approx 7666.7\]
So the mean is about 7,667 steps.
The data in order: 4000, 5000, 5000, 6000, 7000, 20000.
With 6 values, the median is the average of the 3rd and 4th values.
3rd value = 5000; 4th value = 6000.
Median: \[\textrm{median} = \frac{5000 + 6000}{2} = 5500\]
A typical number of steps is around 5,500.
Most data values (4000–7000) are between 4,000 and 7,000 steps.
One value, 20,000, is much larger than all the others. This is a striking deviation or possible outlier.
Because of this one large value, the mean (about 7,667) is higher than what most students actually walked. The median (5,500) may better describe a “typical” student’s steps here.
Seven students take a quiz. Their scores (out of 10) are:
Data: 5, 6, 7, 7, 8, 9, 9
Add the scores: \(5 + 6 + 7 + 7 + 8 + 9 + 9 = 51\)
There are 7 scores.
Mean: \[\textrm{mean} = \frac{51}{7} \approx 7.29\]
The mean score is about 7.3 out of 10.
Scores in order are already: 5, 6, 7, 7, 8, 9, 9.
With 7 values, the median is the 4th value, which is 7.
Median score = 7.
Lower quartile Q1: median of 5, 6, 7 → Q1 = 6.
Upper quartile Q3: median of 8, 9, 9 → Q3 = 9.
IQR: \[\textrm{IQR} = Q3 - Q1 = 9 - 6 = 3\]
The middle half of the scores is from 6 to 9.
This tells us the class performed fairly consistently on this quiz 🎯.
These ideas of mean, median, IQR, MAD, overall pattern, and striking deviations appear everywhere in real life:
Knowing how to describe center and variability helps you understand when a number is “normal” for your situation or when something is surprising and worth a closer look 🔍.
Data and distributions: Data are numbers that describe something. A distribution shows how those numbers are spread out.
Center:
Variability:
Overall pattern: When you look at a graph, describe the shape, center, and spread of the data.
Striking deviations: Values that are far from most of the data are important to notice. In context, they might be mistakes, special events, or important discoveries.
By combining measures of center (mean and median) with measures of variability (IQR and MAD), and by paying attention to patterns and unusual values, you can turn raw numbers into clear stories about the world around you 🌍.
