Descriptive statistics summarize the data a researcher actually collected — nothing more. They describe the center, spread, and shape of a sample or dataset without making any claim about a broader population. That second job belongs to inferential statistics, which use sample data to estimate or test a claim beyond the sample itself. This guide covers the full toolkit of descriptive statistics — central tendency, dispersion, shape, and position — which one to use for which type of data, how to detect and handle outliers defensibly, and how to report the results in APA style.
Descriptive vs. inferential statistics: two different jobs
Descriptive statistics answer “what does this data look like?” — the mean age of enrolled participants, the proportion in each treatment arm, the standard deviation of a test score. Inferential statistics answer “what can I conclude about the population this sample was drawn from?” — a t-test, a confidence interval, a regression coefficient. A manuscript that reports “Group A scored higher than Group B” using only sample means, with no test statistic or confidence interval behind it, has made a descriptive observation and dressed it up as a generalizable finding. For the full treatment of that distinction, including where manuscripts get it wrong in a Results section, see Descriptive Statistics vs. Inferential Statistics. This guide focuses on the descriptive side: how each summary statistic is calculated, what it assumes, and when it is the right tool. Every one of those inferential tools ultimately produces a p value summarizing the evidence against the null hypothesis — see CASRAI’s guide on what a p value actually measures for how to interpret it correctly.
Measures of central tendency: mean, median, and mode
Central tendency measures identify a single “typical” value that represents a dataset.
- Mean — the arithmetic average (sum of values divided by count). It uses every value in the dataset, which makes it the most statistically efficient measure for symmetric, roughly normal data, but also the most sensitive to outliers and skew.
- Median — the middle value when data is ordered from smallest to largest (or the average of the two middle values for an even-numbered sample). The median is unaffected by the magnitude of extreme values, only their rank, which is why it is the preferred measure of central tendency for skewed distributions or data with outliers — household income, hospital length of stay, and reaction times are all classic examples of right-skewed data where the mean is pulled upward by a small number of extreme values and the median gives a more representative “typical” figure.
- Mode — the most frequently occurring value. It is the only measure of central tendency that applies to nominal (categorical) data — you cannot calculate a mean or median of eye colors, but you can report the modal category. A distribution can have more than one mode (bimodal, multimodal), which the mean and median cannot signal on their own.
A practical rule: report the mean for roughly symmetric interval/ratio data, switch to the median when the distribution is visibly skewed or contains outliers, and use the mode for categorical data or to flag multimodality that a mean would otherwise hide.
Measures of dispersion: why a mean alone is uninformative
Two datasets can share an identical mean and look nothing alike. A mean or median reported without a measure of spread tells a reader almost nothing about how representative that central value actually is — a sample tightly clustered around a mean of 50 and a sample ranging from 5 to 95 with the same mean of 50 describe very different underlying data. This is why reporting conventions (APA included) require a dispersion measure alongside any measure of central tendency, not as an optional extra.
- Range — the difference between the maximum and minimum value. Simple but entirely driven by the two most extreme observations, so it is highly sensitive to outliers and rarely used as the sole dispersion statistic.
- Interquartile range (IQR) — the range of the middle 50% of the data (Q3 minus Q1). Because it discards the top and bottom quarters, it is far more robust to outliers than the range and pairs naturally with the median as the skew-resistant summary pair. See CASRAI’s guide to the interquartile range and five-number summary for how to calculate it by hand and use it to flag outliers.
- Variance — the average of the squared deviations from the mean. Squaring the deviations makes variance mathematically convenient for further statistical work (it’s the building block for ANOVA and regression) but its units are squared, which makes it hard to interpret directly.
- Standard deviation (SD) — the square root of the variance, returning the measure to the original units of the data. SD is the standard way to report spread alongside a mean, and is the dispersion statistic most commonly seen in a manuscript’s results (“M = 23.4, SD = 4.1”).
- Coefficient of variation (CV) — the standard deviation expressed as a percentage of the mean (SD / mean × 100). Because it is unitless, the CV is useful for comparing the relative variability of two variables measured on different scales or in different units — comparing the variability of blood pressure (mmHg) against the variability of body weight (kg), for instance, where comparing raw SDs would not be meaningful. See CASRAI’s full guide to the coefficient of variation for worked examples, its critical limitations, and its role in assay-precision reporting.
- Mean absolute deviation (MAD) — the average of the absolute (unsquared) deviations from the mean, rather than the squared deviations variance uses. It is more intuitive to interpret and less sensitive to a single extreme outlier than SD, but lacks the mathematical properties that make variance the basis for inferential tests. A closely related but distinct statistic, median absolute deviation, uses the median instead of the mean at both steps and is the more robust choice for skewed or outlier-heavy data. See CASRAI’s guide to mean absolute deviation (average deviation from the mean) for the full comparison and worked examples.
The mean/SD pairing and the median/IQR pairing are the two conventional reporting units in the literature — use whichever pairing matches the central-tendency measure you reported, never mix a median with an SD or a mean with an IQR without a stated reason.
Measures of shape: skewness and kurtosis
Central tendency and dispersion describe where a distribution sits and how spread out it is; shape statistics describe its form.
- Skewness quantifies asymmetry. A skewness of zero indicates a symmetric distribution; positive (right) skew means a longer tail toward higher values, with the mean typically pulled above the median; negative (left) skew is the mirror image. As a rough guide, skewness beyond roughly ±1 is often treated as substantial enough to affect which central-tendency and dispersion measures — and which statistical tests — are appropriate, though thresholds vary by field and sample size. See CASRAI’s guide to skewness (left-skewed vs. right-skewed distributions) for the full breakdown of what right- and left-skew look like and how they affect analysis choices.
- Kurtosis quantifies the “tailedness” of a distribution relative to a normal distribution — how much of the variance comes from infrequent extreme deviations. High kurtosis (“leptokurtic”) indicates heavier tails and a higher chance of outliers than a normal distribution predicts; low kurtosis (“platykurtic”) indicates lighter tails.
Both statistics are most often used as a check on whether parametric assumptions (like normality, required by many statistical tests) are reasonable for a given dataset, rather than reported as headline findings on their own. A histogram is usually the fastest way to see skew and kurtosis visually before calculating them numerically.
Measures of position: percentiles, quartiles, and z-scores
- Percentiles indicate the value below which a given percentage of observations fall — the 90th percentile is the value below which 90% of the data lies.
- Quartiles are the specific percentiles that divide data into four equal parts: Q1 (25th percentile), Q2 (50th percentile, the median), and Q3 (75th percentile). Q3 minus Q1 is the interquartile range described above.
- Z-scores (standard scores) express how many standard deviations a given observation falls above or below the mean: z = (x − mean) / SD. Z-scores put variables measured on different scales onto a common metric, which is why they are used both to compare an individual observation across different distributions and, as covered below, as one method of flagging outliers.
Everything above describes a single variable in isolation. When the question is how two variables move together, the bivariate analogue is a correlation coefficient — see CASRAI’s guide to the correlation coefficient for how direction and strength of association between two variables are summarized in a single number.
Frequency distributions
A frequency distribution shows how often each value, or each range of values (a “bin”), occurs in a dataset. For categorical data this is typically a simple table of counts and percentages by category, displayed as a bar chart or pie chart. For continuous data, values are grouped into bins and displayed as a histogram, which is the standard visual summary of a continuous variable’s shape, center, and spread — see that guide for how bin-width choice changes the apparent shape of a distribution and how histograms compare to box plots and density plots. Cumulative frequency distributions, which show the running total of observations up to each value, are the basis for reading off percentiles and quartiles directly from a chart.
Which statistic fits which measurement scale
Not every descriptive statistic is meaningful for every type of variable. Measurement scale determines which summary statistics are mathematically valid:
| Scale | What it allows | Central tendency | Dispersion | Example |
|---|---|---|---|---|
| Nominal | Categories with no order | Mode only | Frequencies/proportions | Sex, blood type, treatment arm |
| Ordinal | Ordered categories, unequal/unknown intervals | Median, mode | Range, IQR, percentiles | Likert-scale agreement, cancer stage |
| Interval | Ordered, equal intervals, no true zero | Mean, median, mode | Range, IQR, SD, variance | Temperature in °C, IQ score |
| Ratio | Ordered, equal intervals, true zero | Mean, median, mode | Range, IQR, SD, variance, CV | Age, weight, reaction time, income |
Reporting a mean and standard deviation for a nominal variable, or a mode alone for a ratio-scale variable with no accompanying spread, is a common and avoidable reporting error. When in doubt about which category a variable falls into, or how that classification determines which inferential test is appropriate downstream, see Types of Variables and Statistical Test.
Detecting and handling outliers
An outlier is an observation that falls unusually far from the rest of the data. Three common detection approaches:
- The IQR rule — flag any value below Q1 − 1.5×IQR or above Q3 + 1.5×IQR. This is the convention behind the “whiskers” on a box plot and is robust because it is based on quartiles rather than the mean.
- The z-score rule — flag any value with an absolute z-score beyond a chosen threshold, commonly 3 (i.e., more than 3 standard deviations from the mean). Because both the mean and SD used to calculate z-scores are themselves sensitive to outliers, this method can under-flag extreme values in small or already-skewed samples.
- Visual inspection — a histogram, box plot, or scatter plot often makes an outlier obvious well before any numeric rule confirms it, and is a useful sanity check against a rule that is technically correct but substantively wrong for the data at hand (e.g., a genuinely bimodal distribution where the z-score rule flags real, valid observations as outliers).
The research-integrity point: detecting an outlier is not the same as having a justification for removing it. A decision to exclude, winsorize, or otherwise treat outliers must be pre-specified in the analysis plan and disclosed in the manuscript — which rule was used, what threshold, and how many observations were affected — rather than decided after looking at whether removing a given point changes the result in a favorable direction. Choosing an outlier-exclusion rule, or deciding case-by-case which points to drop, only after seeing how it affects the p-value or the direction of an effect is a form of undisclosed researcher degrees of freedom, closely related to the broader set of questionable research practices that undermine the credibility of a reported result. The defensible sequence is: define the outlier rule before analysis, apply it consistently, and report both the rule and its effect — including results with and without the excluded points, when feasible — rather than silently reporting only the version that supports the conclusion.
Reporting descriptive statistics: APA conventions
APA style conventions for reporting descriptive statistics are widely adopted well beyond psychology journals:
- Report the sample size (n) for every statistic, either in the text, a table, or both — a mean without an accompanying n cannot be evaluated for precision or compared across groups of different sizes.
- Report mean and SD together, not the mean alone: “M = 23.45, SD = 4.12.” Use the median and IQR (or range) together in the same way when the distribution is skewed: “Mdn = 19.0, IQR = 14.5–26.0.”
- Decimal precision is typically two decimal places for means, standard deviations, and most other descriptive statistics, unless the measurement itself requires more or fewer significant digits to be meaningful (e.g., a lab assay reported in different native units). Be consistent within a table.
- Percentages are conventionally reported to one decimal place, alongside the raw count they’re derived from: “42.3% (n = 38/90).”
- Use italics for statistical symbols (M, SD, Mdn, n, N) per APA convention, and do not italicize abbreviations that are not statistical symbols.
Worked example of a conventional sentence: “Participants (N = 142) had a mean age of 34.6 years (SD = 9.2). Symptom severity scores were positively skewed, so the median is reported: Mdn = 18.0 (IQR = 12.0–27.0).” This is an illustrative sentence, not a reported finding from any real study — it demonstrates the reporting convention, not a dataset.
For the companion question of how to report the inferential statistics that typically follow — p-values, confidence intervals, and effect sizes — see How to Report P Values.
Calculating descriptive statistics in R, Python, and SPSS
- R: base functions
mean(),median(),sd(),var(),range(),quantile(), andsummary()cover most needs directly; thepsychpackage’sdescribe()function additionally returns skewness and kurtosis in one call. - Python: pandas’
DataFrame.describe()returns count, mean, SD, min, quartiles, and max in one table;scipy.statsprovidesskew()andkurtosis()for shape statistics not included in pandas’ default output. - SPSS: Analyze → Descriptive Statistics → Descriptives (or Frequencies/Explore for a fuller breakdown including percentiles and skewness/kurtosis) generates a standard descriptive-statistics output table without writing syntax. See the full SPSS guide for how it fits into a broader analysis workflow.
Frequently asked questions
What is the difference between descriptive and inferential statistics?
Descriptive statistics summarize the data actually collected — mean, median, standard deviation, frequencies — without making a claim beyond the sample. Inferential statistics use that sample data to estimate or test a claim about the broader population the sample was drawn from, using tools like p-values, confidence intervals, and regression. See Descriptive Statistics vs. Inferential Statistics for the full distinction and how manuscripts commonly conflate the two.
Why is the median preferred over the mean for skewed data?
The mean is calculated from every value in a dataset, so extreme values pull it toward the tail of a skewed distribution. The median only depends on the rank of values, not their magnitude, so it stays close to the bulk of the data and better represents a “typical” observation when a distribution is skewed or contains outliers.
Do I always need to report a measure of dispersion alongside a measure of central tendency?
Yes. A mean or median without a standard deviation, IQR, or range gives no indication of how representative that central value is — two datasets can share an identical mean while looking completely different in spread. Convention pairs mean with SD and median with IQR or range.
Is it acceptable to remove outliers from a dataset?
Only when the removal criterion is defined before looking at how it affects the result, applied consistently, and disclosed in the manuscript — including the rule used and how many observations were affected. Deciding which points to exclude after seeing their effect on the outcome is a questionable research practice, not a routine data-cleaning step.
How many decimal places should descriptive statistics be reported to?
APA convention is typically two decimal places for means and standard deviations and one decimal place for percentages, unless the underlying measurement scale requires different precision to remain meaningful. Consistency within a table matters more than the specific number of places chosen.
