🎯 Central Tendency

1. The big idea — one number for the whole pile

Central tendency is a single value that summarizes where a batch of scores piles up — the data's center of gravity. It answers the most natural question you can ask of any dataset: "So… what's typical?" There are three classic ways to answer, and the most important thing to know up front is that they don't always agree. When they disagree, the gap between them is itself information — it tells you the shape of your data.

We write the mean of a sample as x̄ ("x-bar") and the mean of a whole population as μ ("mu"). Same recipe, different label — a distinction that matters more in later units. For now, treat them as the same idea: the balance point of your numbers.

2. The mean (the "average")

Add all the scores, then divide by how many there are. That's the whole thing. Five students sleep 5, 6, 7, 7, 10 hours:

Mean = (5 + 6 + 7 + 7 + 10) ÷ 5 = 35 ÷ 5 = 7 hours.

In stats shorthand, the recipe is written x̄ = Σx ÷ n, where the funny Σ ("sigma") just means "add up all the x's" and n is the count. So Σx = 35, n = 5, and x̄ = 7. Don't let the symbols spook you — you've been doing this since grade school.

That "uses every value" property is exactly why the mean is the workhorse of statistics — almost every test you'll meet later (z-scores, t-tests, ANOVA) is built on it. But it also means the mean is only honest when no single score is wildly out of line with the rest.

3. The median (the "middle")

Put the scores in order, then find the one sitting dead center. In order, 5, 6, 7, 7, 10 → the middle (3rd of 5) is 7. The median splits the data into two equal halves: half the scores fall at or below it, half at or above.

Two small wrinkles handle every case:

• Odd count — there's a single middle value; read it straight off. (5 scores → the 3rd.)
• Even count — there's no single middle, so average the middle two. For 4, 6, 8, 10 the middle two are 6 and 8, so the median is (6 + 8) ÷ 2 = 7.

4. The mode (the "most common")

The mode is simply the score that appears most often. In 5, 6, 7, 7, 10, the mode is 7 — it shows up twice while everything else appears once. To find it, you just count repeats; no arithmetic at all.

The mode has one property the other two lack: it's the only measure that works for categories. You can't add up Freshman and Senior and divide by two, and you can't line up Omnivore, Vegetarian, and Vegan to find a "middle" — but you can ask which one appears most. In our class dataset the modal class_year is Freshman and the modal diet is Omnivore.

5. Outliers & skew — when the average lies

Here's the single most important lesson of the unit, and it's easiest to see. Start with a tidy little set, compute all three centers, then drop in one monster value and watch what happens. Predict each step before you reveal it.

That stretched-out shape has a name from Unit 2: skew. When a long tail pulls the data to one side, the three centers spread out in a predictable order. The mean, chasing the tail, ends up farthest out; the mode (the tallest bump) stays put; the median sits between them. Here's the classic right-skewed picture:

6. So which one should you use?

Two questions decide it: What's the shape? and What's the scale of measurement? (the NOIR ladder from Unit 1). Here's the whole decision in two tables.

And here's how the scale of measurement sets a ceiling on which centers are even allowed — a measure needs the math its scale supports:

✱ Means of ordinal data (e.g., averaging a 1–5 Likert scale) are extremely common in psychology but technically assume the steps are equal — a debate you'll hear about for the rest of your research life. Mode and median are always safe on ordinal data; the mean is the contested one.

Try it now — the Descriptive Statistics calculator

Paste a list of numbers (or hit Load class data to drop in real commute minutes), then read the mean, median, and mode straight off — the same three numbers you'd get from the Sheets formulas below. Load the class commute data and watch the mean sit above the median: that's the right-skew you just learned to read.

🐮 The typical animal life

Central tendency reframes hard-to-feel numbers into a single graspable one. Each year roughly 83 billion land animals are slaughtered for food worldwide — a number too big for the mind to hold. But express it as a center: the typical broiler chicken (about 90% of those animals) lives only about 42 days before slaughter, against a natural lifespan of several years. One median-style number — "about six weeks" — does more moral work than the billions ever could.

Our class dataset carries the quieter version. Run the three measures on animal_compassion by diet and the centers separate cleanly: omnivores cluster lower, vegans higher, vegetarians between — a preview of the group comparisons we'll formalize in later blocks. Zoom out and the framing sharpens further: animal agriculture uses 83% of the world's farmland but provides only 18% of our calories (Poore & Nemecek, 2018). A single well-chosen "average" can carry real weight — and choosing it honestly is the skill this unit teaches.

💻 In Google Sheets

You won't compute these by hand on exam day — Sheets does it in one function each. Open the SJNY Student Life dataset and try these on the stress column (column J):

=AVERAGE(J2:J201)     → the mean stress score   (≈ 20.1)
=MEDIAN(J2:J201)      → the middle stress score (≈ 19)
=MODE(J2:J201)        → the most common score   (= 18)

Because the mean (≈20.1) and median (≈19) sit close together, stress is roughly symmetric — the mean is a fine summary here. Now try the same three on commute_minutes (column W) and you'll see the gap open up:

=AVERAGE(W2:W201)     → mean commute   (≈ 22.8 min)
=MEDIAN(W2:W201)      → median commute (= 20.5 min)

The mean sits above the median because a handful of brutal 70-minute-plus commutes pull it up — right-skewed, so the median (20.5) is the typical commuter. Two notes: newer Sheets uses =MODE.SNGL for one mode and =MODE.MULT when several tie, and =AVERAGE politely ignores blank cells (good) but counts a stray 0 (be careful). When you want a quick read on shape, drop both =AVERAGE and =MEDIAN side by side — far apart means skew.

Key terms