Mean, Median & Mode

These three numbers are all “averages” — single values that summarize a whole data set — but they answer different questions and behave differently when data is skewed.

Mean

The mean is the familiar arithmetic average: add everything, divide by the count.

Data: 4, 8, 6, 5, 12 → sum = 35, count = 5, mean = 7

The mean uses every value, which makes it sensitive to outliers. Change that 12 to 40 and the mean jumps to 12.6 — even though most of the data did not move.

Median

The median is the middle value when the data is sorted.

Data sorted: 4, 5, 6, 8, 12 → median = 6

With an even count, average the two middle values: for 4, 5, 6, 8 the median is (5 + 6) ÷ 2 = 5.5.

The median resists outliers. In the data 4, 5, 6, 8, 40 the median is still 6, while the mean climbs to 12.6. This is why house prices and incomes are usually reported as medians.

Mode

The mode is the most frequent value.

Data: 2, 3, 3, 5, 7 → mode = 3

A data set can have no mode (all values appear once), one mode, or several. The mode is the only average that works for non-numeric data — the “mode” of red, blue, blue, green is blue.

The mode calculator reports every tied mode instead of silently picking one, and says “no mode” outright when nothing repeats.

A full worked example

Here is a five-value data set processed end to end, so you can see every measure on the same numbers:

Data: 3, 7, 12, 7, 21

• Sort it: 3, 7, 7, 12, 21. (Always sort before finding the median.)
• Mean: 3 + 7 + 12 + 7 + 21 = 50, then 50 ÷ 5 = 10
• Median: the third value = 7
• Mode: 7 appears twice, everything else once → 7
• Range: 21 − 3 = 18 (range is a spread measure, not an average)

Notice the mean (10) sits well above the median (7). That gap is a signal: 21 is pulling the mean upward, so the data is skewed right. When mean and median disagree like this, the median usually describes a “typical” value better.

Outliers change the mean, not the story

Salaries at a five-person firm: 35,000, 40,000, 42,000, 39,000, and one founder at 250,000.

• Sorted: 35,000, 39,000, 40,000, 42,000, 250,000 → median = 40,000
• Sum = 406,000 → mean = 81,200

The mean is double the median because a single extreme value drags it. That is exactly why “average salary” headlines feel wrong to people who live them: the average being quoted is usually the mean, while the experience readers have looks like the median.

On data-entry outliers: if the 250,000 were a typo for 25,000, fixing it changes everything — so investigate implausible values rather than deleting them on sight. An outlier that is a real measurement belongs in the data; an outlier that is an error does not.

Weighted mean

Not every contribution is equal. Course grades are the classic case: a midterm worth 40% of the grade and a final worth 60% should not count equally.

Scores: midterm 78, final 90. The weighted mean is:

weighted mean = (weight × value) summed, divided by the total weight = (0.40 × 78 + 0.60 × 90) ÷ (0.40 + 0.60) = (31.2 + 54) ÷ 1 = 85.2

Compare the plain mean: (78 + 90) ÷ 2 = 84. The true grade (85.2) is higher because the bigger weight sat on the better score. Whenever values carry unequal importance, use the weighted form — averaging the raw numbers first is wrong.

Spread: standard deviation, with the divisor question

Center is only half the story; two data sets can share a mean and differ wildly in scatter. Take the earlier data 4, 8, 6, 5, 12 (mean 7). Squared distances from the mean are:

(4−7)² + (8−7)² + (6−7)² + (5−7)² + (12−7)² = 9 + 1 + 1 + 4 + 25 = 40

Now the divide: if your data is the whole population (all 5 employees), divide by 5 — variance = 8, standard deviation = √8 ≈ 2.83. If it is a sample meant to represent a larger group, divide by 4 (one less than the count) — variance = 10, standard deviation = √10 ≈ 3.16.

The two conventions differ, and tools do not always agree on which they use. Divide by n for a complete population and by n − 1 for a sample; if a tool’s result differs slightly from your hand computation, check which convention it implements before assuming either side is wrong.

Choosing the right average

| Situation | Best measure | Why | | | | | | Typical income in a country | Median | A few very high earners drag the mean up | | Average test score | Mean | Every point matters and errors are symmetric | | Most popular shoe size | Mode | You cannot average sizes meaningfully | | Deciding staffing for a hospital | Mean (or distribution) | Total workload matters | | Home listing prices in a city | Median | Buyers think in “typical home”, not total market value | | Days late per delivery route | Mean | Totals across routes matter for planning |

Common mistakes

1. Taking a median without sorting. The middle of 7, 12, 3 is 12 in appearance and 7 in truth. Sort first, every time.
2. Even counts: forgetting to average the two middle values. For 1, 2, 8, 9 the median is (2 + 8) ÷ 2 = 5, not 2.
3. Using the mean when the data is skewed. With an outlier in play, the mean reports the outlier’s influence, not the typical case. Report both, or switch to the median.
4. Mode on continuous data. Measured values (heights like 171.2 cm) rarely repeat exactly, so “no mode” tells you nothing. Group into ranges before looking for a mode.
5. Comparing means across different-sized groups without weights. A class of 5 students scoring 90 does not outweigh a class of 200 scoring 70 — weight by count first.

Practice set

Work these by hand, then check with a tool.

1. Data: 2, 9, 4, 4, 11 → find mean, median, mode. (Mean = 30 ÷ 5 = 6; sorted 2, 4, 4, 9, 11 so median = 4; mode = 4.)
2. Data: 1, 3, 5, 7 → median? (The two middle values average: (3 + 5) ÷ 2 = 4.)
3. Weekly wages: 400, 450, 500, 4,000 → which average defends a “we’re typical” claim? (Median = 475; the mean is 1,337.5, obviously not typical.)

Check your work with a calculator

Hand arithmetic on five values is fine; on fifty values it is error-prone. The mean, median and mode calculator computes all three measures at once from a pasted list, so skew becomes visible immediately: mean far from median means look at your outliers before quoting either number. If you only need the middle, the plain median calculator sorts and splits even-length lists correctly — the exact “forget to sort” trap that trips hand computation. And once the center is settled, the standard deviation calculator measures the spread that paired with it.