Short answer
The arithmetic mean summarizes the total per observation, the median locates the ordered middle, and the mode identifies the most frequent value. Extreme values can pull the mean away from the median, while repeated values determine the sample mode. These are different answers to what counts as typical, especially in skewed data. 1 2
On this page
At a glance
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| Question | Arithmetic mean | Median | Mode |
|---|---|---|---|
| What determines it? | Sum divided by count | Middle sorted position(s) | Greatest frequency |
| What information matters? | Every numerical magnitude | Order and middle values | Counts of repeated values |
| Main caution | Tail extremes affect the result | Not a frequency summary | Continuous values may not repeat |
These operations describe sample summaries; histogram-based modes require a separate interpretation. 1
What each thing is
For numerical observations, the mean is an arithmetic calculation, the median is an ordering calculation, and the mode is a counting calculation. With an even number of observations, the numerical median averages the two middle values. Neither that median nor the mean must be an observed value; a sample mode is drawn from the observed values. 1
Key differences
Magnitude, position, and repetition are the central boundaries. Changing a large observation can change the mean even if it stays last in sorted order; the median can remain unchanged. Repeating a value affects its claim to be the mode. Under skew, the balance point, halfway location, and peak can describe noticeably different places. 1 2
How to tell them apart
Identify the operation, not just the label average: adding and dividing indicates a mean; sorting and finding the middle indicates a median; counting occurrences indicates a sample mode. The limit is continuous data: a reported mode may instead be the midpoint of the tallest histogram interval, rather than a repeated observation. 1
Where they overlap
For a normal population distribution, mean, median, and mode coincide. Finite samples from that distribution can produce nearby but unequal estimates, as NIST’s example demonstrates. Agreement therefore does not make the definitions interchangeable, and numerical equality alone does not establish that data follow a normal distribution. 1
Edge cases
Continuous measurements may contain no repeated values, leaving frequency counting without a distinctive winner. A histogram peak offers a less exact alternative. Also, symmetry alone is too broad a shortcut for claiming all three measures agree: NIST distinguishes the normal distribution’s well-behaved tails from the symmetric Cauchy distribution’s heavy tails. 1
Why the distinction exists
A skewed distribution has no single obvious center in the usual visual sense. The mean represents balance, the median represents the halfway location, and the mode represents the peak. Multiple summaries exist because those features answer different descriptive questions, not because one calculation always supplies the uniquely correct average. 2
Common misconceptions
Most frequent does not mean middle, and middle does not mean total divided by count. Nor is the mode necessarily a good stand-in for the center of severely skewed data. Calling all three averages without specifying the operation obscures their different meanings; NIST explicitly reserves average for mean in its handbook. 1 2
Examples
Hypothetical case 1: for 1, 2, 2, 3, 12, the mean is 4, while median and mode are both 2. Replacing 12 with 102 raises the mean to 22 without changing either other summary. Case 2: for 1, 2, 3, 4, the mean and median are 2.5, but no value occurs more often than the others. These calculations apply NIST’s definitions. 1