MCAT content library / Research & data / Mean, median, and spread: describing a dataset
Describe the center, spread, and tails of a dataset without losing the differences that matter.
About 14 minutes, including the worked examples and checks.
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Concept: the center does not describe the whole distribution
A report says that two groups have the same average reaction time. That tells you something about their centers, but little about how predictable an individual response is. One group could cluster tightly around that average while the other contains both much faster and much slower observations. To compare the groups, ask both where the observations lie and how widely they are spread.
The mean uses every observation’s numerical value: add the values and divide by their number. The median uses their order: sort the observations and take the middle value, or the mean of the two middle values when the count is even. The mode describes the most frequent value; a dataset may have several equally prominent values or no single most frequent value. These summaries answer related but different questions.1
AAMC includes mean, median, mode, range, interquartile range, and standard deviation within data-based statistical reasoning. The task is to use those summaries to understand evidence, including distinctions a single summary can hide.2
Pattern: equal centers, different spread
The following small reaction-time samples are illustrative. Values are in milliseconds, and each listed value represents one observation. They are deliberately arranged to share a mean and median while differing in spread.
| Sample | Ordered observations (ms) | Mean (ms) | Median (ms) |
|---|---|---|---|
| A | 240, 250, 260, 270, 280, 290, 300 | 270 | 270 |
| B | 180, 230, 260, 270, 280, 310, 360 | 270 | 270 |
Each sample sums to 1,890 ms; dividing by seven gives 270 ms. Their fourth observations are also both 270 ms. Yet the claim that their distributions are identical is false. Sample B extends farther on each side and has fewer observations concentrated near the center. Equal averages do not mean that an individual observation is equally likely to be near the average.
Rule: match the spread measure to the feature you need
The range is maximum minus minimum. It is 60 ms for A and 180 ms for B. Range makes the extreme difference visible but depends only on two observations. It says nothing about where the remaining five values fall.
The interquartile range, or IQR, is the third quartile minus the first quartile. It summarizes the width of the middle portion of the data. For this lesson, calculate quartiles by excluding the overall median of an odd-sized dataset, then taking the median of each remaining half. Other software conventions can give different quartile values in small samples, so use a convention specified by the question.
For A, the lower half is 240, 250, 260 and the upper half is 280, 290, 300. Q1 = 250 ms, Q3 = 290 ms, and IQR = 40 ms. For B, the corresponding quartiles are 230 and 310 ms, giving an IQR of 80 ms. B’s middle half is wider as well as its full range.
Standard deviation uses each observation’s distance from the mean. For a sample of n observations, sample variance is the sum of squared deviations divided by n − 1; sample standard deviation s is its square root. Squaring prevents positive and negative deviations from canceling and gives large deviations more weight. Standard deviation returns the result to the original units; variance remains in squared units.3
For A, the deviations from 270 are −30, −20, −10, 0, 10, 20, and 30 ms. Their squares sum to 2,800 ms². The sample variance is 2,800/6 ≈ 466.7 ms², so s ≈ 21.6 ms. For B, the squared deviations sum to 19,600 ms², giving s ≈ √(19,600/6) ≈ 57.2 ms. The broader sample has the larger standard deviation even though its mean is unchanged.
| Sample | Range (ms) | IQR (ms) | Sample SD (ms) |
|---|---|---|---|
| A | 60 | 40 | 21.6 |
| B | 180 | 80 | 57.2 |
Application: add an extreme observation
Add an eighth value of 1,000 ms to Sample A. The new sum is 2,890 ms, so the mean becomes 2,890/8 = 361.25 ms. The two middle observations are 270 and 280 ms, so the median becomes 275 ms. One added observation moves the mean by 91.25 ms while moving the median by only 5 ms.
The median is less affected here because the size of the largest value does not determine the middle ranks. If that added value were 2,000 instead of 1,000 ms, the median would still be 275 ms, while the mean would rise further. This is why the median can better describe a typical observation when a few extreme values pull the mean away from most of the data.1
That does not make the extreme value irrelevant. It could represent a real lapse, a distinct subgroup, or a recording error. Removing it requires a reason grounded in the study and measurement process. A useful report can show the distribution and include both a robust center, such as the median, and information about the tail when the tail matters.
Separate variability from uncertainty
Standard deviation describes how dispersed observations are around their mean. It is not automatically the uncertainty in the estimated mean. For independent observations from a population with a finite variance, the usual estimated standard error of the mean is s/√n. A confidence interval requires an appropriate inferential method and assumptions in addition to a spread estimate.4
Do not read an unlabeled error bar as an SD or a confidence interval. Do not infer that two groups have equal distributions merely because their means coincide. And do not infer statistical significance from the descriptive table above: no hypothesis test was specified. First describe the observed pattern; then use the study’s stated uncertainty information to evaluate a population claim.
For a quick self-check, imagine changing every observation by the same amount. Adding 30 ms shifts the mean and median by 30 ms but preserves all pairwise distances and deviations from the new mean. Range, IQR, and standard deviation stay the same. Multiplying every observation by two instead doubles the center and these spread measures, while multiplying variance by four. The units and the operation explain the result.
ACTIVE PRACTICE · DESCRIPTIVE STATISTICS
Change One Observation and Recalculate the Center
The original ordered Sample A contains seven reaction times (ms): 240, 250, 260, 270, 280, 290, 300. The mean and median are both 270 ms. Add an eighth recorded observation, then compare their sensitivity.
New mean: ms. New median: ms.
Mean shift: ms. Median shift: ms.
These are actual example observations, not a theoretical sampling distribution. The median is calculated using the ordered eighth-value position; this slider only adds values at least as large as the existing maximum.
Check your reasoning
Choose an answer before opening its feedback. Each option explains the reasoning behind that choice.
Check 1
Five measured values are 10, 11, 12, 13, and 14 units. A corrected record replaces 14 with 100. What are the mean and median of the corrected dataset?
A. Mean 12 units; median 12 units.
Review this choice. The median remains 12, but the corrected value contributes to the sum. The new sum is 146, so the mean cannot remain 12.
B. Mean 29.2 units; median 12 units.
Correct. The sorted values are 10,11,12,13,100. Their sum is 146, giving 146/5=29.2. The middle, third value is 12.
C. Mean 29.2 units; median 29.2 units.
Review this choice. The mean is correct, but the median is determined by the central rank. The third observation is 12; there is no requirement that the mean and median coincide.
D. Mean 100 units; median 12 units.
Review this choice. 100 is the maximum, not the mean. The mean incorporates all five values and is 29.2.
The principle to carry forward
An extreme replacement can change the sum greatly while leaving the middle rank unchanged.
Check 2
A device records reaction times with a mean of 270 ms and a sample SD of 21.6 ms. Calibration reveals that every recorded value is exactly 30 ms too low. After adding 30 ms to each value, what are the corrected mean and sample SD?
A. Mean 270 ms; SD 21.6 ms.
Review this choice. The mean changes because every value increases. The spread is unchanged, but the corrected mean is 300 ms.
B. Mean 270 ms; SD 51.6 ms.
Review this choice. A common additive correction changes the center, not the deviations from the center. Both parts of this option apply the correction to the wrong summary.
C. Mean 300 ms; SD 21.6 ms.
Correct. Both each value and the mean rise by 30 ms, so each deviation x−mean remains the same. The SD remains 21.6 ms.
D. Mean 300 ms; SD 51.6 ms.
Review this choice. The corrected mean is 300 ms, but adding a constant does not add that constant to the SD. Distances between observations remain unchanged.
The principle to carry forward
A common additive shift changes location while preserving spread; a multiplicative rescaling changes both.
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Read a study and its figures
Follow a question through its design, graph, normalization and statistical description.
- Hypotheses, predictions, and competing explanations
- Experimental Controls: What a Comparison Can Show
- Observational and experimental studies: what can we conclude?
- Reading graphs: axes, scales, and the comparison that matters
- Normalized data, percent change, and fold change
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Related lessons
- Reading graphs: axes, scales, and the comparison that matters
- Normalized data, percent change, and fold change
- Observational and experimental studies: what can we conclude?
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Sources
1. NIST/SEMATECH e-Handbook: Measures of Location
2. AAMC: Scientific Inquiry and Reasoning Skill 4