Mean, median, mode.
Three classic measures of center. Each one is a different answer to "where does this data sit?" Plus a powerful visual intuition: the mean is the value that balances the bars.
A teacher hands back a quiz and tells the class: "The average was 75." Half the room feels relieved (they did better than 75); the other half feels deflated (they didn't). The teacher then adds: "the median was 80, and the most common score was an 85." Suddenly the same quiz has three different numbers describing the same data set.
The word average is colloquially used for one of three different things. Mean is the arithmetic average (sum ÷ count). Median is the middle value when the data is sorted. Mode is the most frequent value. Each one answers a slightly different question about "where does this data sit?" and a careful reader can tell the difference at a glance.
By the end of this lesson, you'll compute all three on demand and know which one to reach for in different situations. ALEKS will test each of them as a stand-alone question and as a combined three-answer question. Same three moves every time.
Three different answers to "where does this sit?"
Median sorts the data and picks the middle value. With an odd count, it's the single middle entry. With an even count, it's the average of the two middle entries. The median is robust: one extreme outlier can drag the mean wildly off, but it barely moves the median.
Mode is the value (or values) that appears most often. A data set can have one mode, two modes ("bimodal"), three modes, or no mode at all (every value appears exactly once).
Left: the same seven-number data set produces three different "centers" (mean ≈ 6.57, median = 6, mode = 4). Right: the ALEKS Q6 picture — two black bars at heights 8 and 4 balance at a mean height of 6, exactly halfway between them.
A measure of center is a single number that summarizes where a data set sits on the number line. The three classical measures — mean, median, and mode — each answer that question differently.
Words you'll see on ALEKS
- Mean (arithmetic average) mean = (x₁ + x₂ + … + xₙ) / n. The bar atop a variable, x̄, is the standard symbol for sample mean. Round when the answer isn't a whole number; ALEKS typically asks for one decimal.
- Median Sort the data first. With odd n, the median is the single middle value. With even n, average the two middle values. Don't forget to sort before reading off the middle.
- Mode Count how often each value appears. The most frequent value (or values) is the mode. A data set with two tied highest frequencies is bimodal; one with no repeats has no mode.
- "No mode" When every value in the data set appears exactly once, the mode is undefined. ALEKS gives you a "No mode" button for this case; do not type a number.
- Bimodal / multimodal Two values tied for most frequent → bimodal; three or more → multimodal. List all the modes, separated by commas, in the order they appear in the data set.
Compute mean, median, and modes of one data set.
"The 10 completion times in minutes are: 38, 43, 40, 27, 45, 39, 37, 26, 27, 40. Find the median, the mean, and the mode(s)."
Sort the data first.
Every measure benefits from sorted data, but the median requires it. Reorder from smallest to largest:
Median: average the two middle values.
With n = 10, there is no single middle entry. The two middle entries are the 5th (38) and the 6th (39). Average them:
Mean: sum, divide by count.
Add all ten values. Order doesn't matter for the mean, but a quick re-add of the sorted list reduces typos.
Mode: count each value's frequency.
Scan the sorted list and count repeats. Most values appear once; 27 appears twice, and 40 appears twice. The data set is bimodal.
Sanity check.
Mean (36.2) is below median (38.5). That tells us the smaller values (26, 27, 27) are pulling the mean down more than the larger values are pulling it up. The median, by contrast, doesn't care about the size of those outliers — only their position in the sort.
Three problems. Mean, median, mode.
Here are the weights in pounds of a sample of 10 male eleventh graders: 150, 164, 165, 157, 163, 148, 164, 151, 174, 168. Find the median weight.
Here are the numbers of children in 12 elementary school classes: 19, 18, 18, 20, 16, 17, 17, 17, 16, 18, 18, 16. What is the mode of this data set?
A veterinarian weighs 7 dogs treated this morning. The weights in pounds are: 26, 71, 56, 3, 5, 26, 24. Find the mean weight, rounded to one decimal place.
Three fast questions before you move on.
Q1. A data set has an even number of values. How do you find the median?
Q2. Every value in a data set appears exactly once. What is the mode?
Q3. Two black bars sit on a chart: one at height 10, one at height 2. What height should the "mean" bar be so it has the same average as the two black bars?
Now name the spread.
Center is half the story. A data set with mean 75 and another with mean 75 can look completely different if one is tightly bunched and the other is wildly spread out. The number that captures that difference is the standard deviation, and it's the move that powers everything inferential in the back half of this chapter.
Next: Lesson 3 introduces range (the quick read on spread) and standard deviation (the precise read). The empirical rule in Lesson 5 will pile directly on top of standard deviation, so getting comfortable with it now pays off twice.
Continue to Lesson 03Different angle? Need another rep? These are optional — tap any that look helpful.
Mean, Median, and Mode — Crash Course Statistics #3
Defines all three measures of center and shows how outliers skew the mean (Middle Ages life-expectancy example, then income skew). The central insight of this section in one video.
Mean, Median, Mode, and Range — How To Find It!
Procedural companion to the Crash Course conceptual video. Methodical worked examples for odd and even data sets — the plug-and-chug skill students need.