Statistics · Descriptive

Mean, median, mode, and when each one lies

Anyone can define them. The interview question is "which would you report for salaries, and why?" Answering that is the whole lesson.

Your win todayPick the right measure of centre for any dataset, and explain skew with one picture.

The three definitions

Measure	What it is	Excel	SQL
Mean	Sum ÷ count	`AVERAGE`	`AVG(x)`
Median	Middle value after sorting. If the count is even, average the two middles.	`MEDIAN`	`PERCENTILE_CONT(0.5) WITHIN GROUP (ORDER BY x)`
Mode	Most frequent value. There can be none, or several.	`MODE.SNGL`	`GROUP BY x ORDER BY COUNT(*) DESC LIMIT 1`

Definitions follow OpenStax Statistics §2.5. Note that SQL has no universal MEDIAN(). PostgreSQL, SQL Server and Oracle use PERCENTILE_CONT.

One outlier, two answers

Five employees earn 30k, 35k, 40k, 45k and the founder earns 400k.

The mean uses every value, so it gets dragged toward extremes. The median only cares about position, so it doesn't move.

Skew: where the tail goes, the mean follows

As OpenStax §2.6 shows, in right-skewed data the mean is the largest of the three, and in left-skewed data the smallest. In symmetric data all three coincide.

Decision rule

Numbers, roughly symmetric

→ Mean
Heights, test scores, daily temperatures

Numbers, skewed or outliers

→ Median
Income, house prices, delivery times, session length

Bonus: say "and the spread"

Centre alone hides variation. Mention the standard deviation (with the mean) or the IQR (with the median).

Say it in the interview"Mean is the sum over the count, median is the middle value, and mode is the most frequent. The mean is sensitive to outliers. In right-skewed data like salaries, a few high earners pull it above the median. So for skewed data I'd report the median, for symmetric data the mean, and for categories the mode. Ideally I'd also report a measure of spread."

Retrieval check

What is the median of 7, 3, 9, 1, 5?

What is the median of 2, 4, 6, 8?

Which measure would you report for typical house prices in a city?

In right-skewed data, what is the usual ordering?

What is the best "average" for the most popular T-shirt size (S/M/L/XL)?

Type it

Drop the founder. What's the new median (in k) of 30, 35, 40, 45?

Excel: the median of salaries in A2:A101.

Your turn

In Excel, enter 30, 35, 40, 45, 400. Compute AVERAGE, MEDIAN and MODE.SNGL. (Mode → #N/A: there's no repeat.)
Change 400 to 4000. Which number moved and which didn't?
Use the Lesson 3 sales data: the mean vs median of Revenue. Is it skewed?
Sketch the right-skew curve from memory and label mode, median and mean.
Think of 3 business metrics at your target company and say which centre measure you'd use for each.

Go deeper

Primary source: OpenStax Introductory Business Statistics 2e, §2.6: Skewness and the Mean, Median, and Mode. It's a free, peer-reviewed textbook with worked histograms.

💬 Ask your teacher: "explain standard deviation visually" or "what's a bimodal distribution?"