Descriptive statistics #

The Descriptive statistics module computes summary statistics for your selected variables. Pick the statistics you need, click the button, and get a results table — one row per variable.

How to use #

1. Select your variables
2. Open Descriptive statistics from the menu
3. Check the statistics you want (or apply a preset)
4. Click Generate descriptive statistics

Results appear in up to two tables: one for numeric variables, one for categorical variables.

Presets #

Two presets configure the checkboxes for common use cases:

• Parametric — mean, standard deviation, minimum, maximum
• Nonparametric — median, minimum, maximum, quartiles (25%, 75%)

Applying a preset clears all other checkboxes first. The Use sample statistics (n-1 denominator) toggle is treated as a global setting and is left untouched by presets.

Available statistics #

Central tendency #

Measures of where the “center” of your data lies.

• Mean — the arithmetic average. Most useful when data is roughly symmetric without extreme outliers.

• Median — the middle value when data is sorted. More robust than the mean when data is skewed or contains outliers.

Mean vs. median: when these two are close, your data is roughly symmetric. When they diverge, something is pulling the mean away — usually outliers or skew. For example, if the mean salary is $75,000 but the median is $55,000, a few very high salaries are inflating the average. In such cases, the median better represents the “typical” value.

• Mode — the most frequently occurring value. Can be meaningful for any variable type, but especially for categorical data. A variable can have multiple modes if several values share the highest frequency. If every value is unique (no value repeats), the cell shows No mode — i.e. no value repeats, rather than “every value is a mode”.

• Trimmed mean — the mean after removing a percentage of extreme values from both ends. The trim percentage (0–50%, default 10%) controls how much is cut. A 10% trim removes the lowest 10% and the highest 10% of values before averaging. This gives a compromise between the mean (sensitive to outliers) and the median (ignores all but the middle value). At the maximum 50% trim everything outside the centre is removed, so the result equals the median (matching R’s mean(x, trim = 0.5)).

• Geometric mean — the nth root of the product of values. Appropriate for data that is multiplicative in nature, such as growth rates or ratios. Reported as N/A when any value is zero or negative.

• Harmonic mean — the reciprocal of the mean of reciprocals. Useful for averaging rates (e.g. speed, efficiency). Reported as N/A when any value is zero or negative.

• Hodges-Lehmann pseudomedian — the median of all pairwise averages (xᵢ + xⱼ)/2. A robust location estimator that combines two desirable properties: it ignores outliers almost as well as the median, while being nearly as efficient as the mean when the data really is symmetric. Reported alongside its confidence interval, which inverts the Wilcoxon signed-rank distribution. The estimator and CI are computed via R’s stats::wilcox.test(x, conf.int = TRUE) — see Reproducibility.

When to prefer HL: when your data is roughly symmetric but you don’t trust it to be normal, or when you want a robust location estimate that’s more informative than a plain median. For very skewed data, the median is still more interpretable.

Dispersion #

Measures of how spread out your data is.

• Minimum and Maximum — the smallest and largest values. Always worth checking — an unexpected minimum (like -999) or maximum (like 999) often signals a data entry error or a missing value code that wasn’t handled.

• Range — the difference between maximum and minimum. Easy to understand, but highly sensitive to outliers — a single extreme value changes the range dramatically.

• Variance — the average squared deviation from the mean. Expressed in squared units of the original variable, so if you’re measuring height in centimeters, variance is in cm². This makes it hard to interpret directly — standard deviation is usually more practical.

• Standard deviation (SD) — the square root of variance. Expressed in the same units as the original variable, making it the most commonly reported measure of spread.

Rule of thumb: in a roughly normal distribution, about 68% of values fall within ±1 SD of the mean, and about 95% within ±2 SD.

• Winsorized standard deviation — the SD computed after replacing the extreme values at each tail with the cutoff value at the boundary (rather than discarding them, as with the trimmed mean). The winsorization percentage (0–50%, default 10%) is set independently of the trimmed-mean percentage. Less sensitive to outliers than the plain SD; the natural companion to the trimmed mean. Reported as N/A when the chosen percentage would leave no central data (e.g. 50% on an even-n sample of 4 or 6 observations).

Trimming vs. winsorizing: trimming removes the outermost values; winsorizing keeps the sample size and instead pulls the extremes inward. Winsorized SD is what robust inference methods (e.g. Yuen’s t-test) use to pair with a trimmed mean.

• Interquartile range (IQR) — the difference between the 75th and 25th percentiles. Captures the spread of the middle 50% of the data — essentially the range of “typical” values, ignoring the extremes on both ends. Unlike SD, it is not affected by outliers — a single extreme value won’t change the IQR.

Practical use: if IQR is much smaller than the range, it means your data has a compact core with a few far-flung values. This is a quick way to gauge whether outliers are inflating your spread statistics.

• Mean absolute deviation — the average absolute distance from the mean. Like IQR, it is less sensitive to outliers than SD, because it doesn’t square the deviations (squaring amplifies the impact of extreme values). A good companion to the mean when you want a spread measure with the same units but less sensitivity to extremes.

• Median absolute deviation — the median of |x − median(x)|. The robust counterpart to mean AD: by replacing both the centring point (mean → median) and the aggregation (mean → median), it stays stable even when a substantial fraction of the data is contaminated. Multiplied by 1.4826 it estimates the SD of a normal distribution; the Modified Z outliers rule uses this internally.

SD vs. mean AD vs. median AD: for clean normal data they tell a similar story. Heavy-tailed data inflates SD the most; the mean AD stays stable longer; the median AD is the most resistant of the three. If your SD is noticeably larger than your mean AD, that’s a sign a few extreme values are driving the spread.

• Coefficient of variation (CV) — the standard deviation divided by the mean, expressed as a percentage. Useful for comparing variability between variables measured on different scales — for example, comparing the variability of reaction times (measured in milliseconds) with the variability of accuracy scores (measured in percent). The sample-vs-population toggle (see below) controls which SD is used in the numerator; the column header reflects the choice as CV (%, s) or CV (%, σ). Only defined for non-negative, ratio-scale data; reported as N/A when the data contains any negative value or the mean is zero.

Shape #

How the distribution of values looks beyond center and spread.

• Skewness — measures asymmetry. A value near 0 indicates a symmetric distribution. Positive skewness means a longer right tail; negative means a longer left tail.

Example: income data is typically positively skewed — most people earn moderate amounts, with a long tail of high earners pulling the distribution to the right.

• Kurtosis — measures how heavy the tails are relative to a normal distribution. By default, reported as excess kurtosis (raw kurtosis minus 3), so a normal distribution has a value of 0. Positive values indicate heavier tails; negative values indicate lighter tails.

Heavy vs. light tails: a distribution with heavy tails (positive kurtosis) produces more extreme values than you’d expect from a normal distribution — more outliers, more “surprising” data points. A distribution with light tails (negative kurtosis) is the opposite — values cluster closer together with fewer extremes. For example, exam scores that bunch in the middle with few very high or very low scores would have negative kurtosis.

Formulas: skewness and kurtosis always use the bias-corrected sample estimators G₁ and G₂ — the same formulas Excel, SPSS, SAS, and R’s e1071::skewness(type = 2) report by default. The sample-vs-population toggle does not affect them. Skewness needs at least 3 observations; kurtosis needs at least 4 (and SE/CI for kurtosis needs 5). Both are undefined (reported as N/A) when the standard deviation is zero — i.e. all values are identical.

Counts #

• Sample size (N) — the number of non-missing observations.

• Distinct values — how many unique values the variable has. Helpful for spotting coding errors or verifying categorical variables. For example, a “Dominant hand” variable with 5 distinct values when you expected 2 might indicate inconsistent coding (“Left”, “left”, “L”, “RIGHT”, “Right”).

• Missing value count — how many observations have no value, shown as both a count and a percentage of the total.

• Zero count — how many observations equal zero, shown as both a count and a percentage.

• Mild outliers (1.5·IQR) — count (and percent) of values falling outside [Q1 − 1.5·IQR, Q3 + 1.5·IQR]. These are the values that appear as individual points beyond the whiskers in a standard box plot. Inlier range is N/A when the IQR is zero (at least half the data sits at a single value), since the rule loses its meaning.

• Extreme outliers (3·IQR) — count (and percent) of values falling outside the wider band [Q1 − 3·IQR, Q3 + 3·IQR]. Always a subset of the mild outliers — these are the truly far-from-typical values. Same IQR=0 caveat as above.

• Modified Z outliers (|M| > 3.5) — count (and percent) of values whose modified Z-score M = 0.6745·(x − median)/MAD exceeds 3.5 in absolute value (Iglewicz & Hoaglin 1993). Here MAD is the median absolute deviation, not the mean form. Unlike a classical Z-score, this rule uses the median and median AD, so the centre and scale used to flag outliers are themselves resistant to outliers — a single extreme value can no longer “mask” others. When the median AD is zero (more than half the data is identical to the median), the rule loses its meaning — same as the IQR=0 case above.

Each outlier rule that is enabled also produces an Inlier range column showing the cutoff pair [lower, upper]. Pasted directly into a filter as value BETWEEN lower AND upper, this keeps the inliers; inverting the condition keeps the outliers. In the degenerate cases noted above (IQR=0, MAD=0), both the count and the inlier-range cells report N/A; hovering the cell shows an explanation of why the rule could not be applied.

Picking a rule: for general use, the mild outliers (1.5·IQR) rule matches what a box plot shows and works on any distribution shape. The extreme rule isolates the most unambiguous outliers. The modified Z rule is the right choice when you want a Z-style threshold without the masking problem of the classical mean/SD version — it agrees with the IQR rules on heavy-tailed data but uses a sharper, distance-based cutoff.

Quantiles #

• Quartiles (25%, 75%) — the values below which 25% and 75% of the data falls. Together with the median (50th percentile), these define the “box” in a box plot. The 25th percentile (Q1) means “25% of participants scored below this value.”

• Custom percentiles — enter comma-separated values (e.g. “10, 90” or “5, 25, 50, 75, 95”) to compute any percentiles you need.

Standard errors #

The standard error estimates how much a statistic would vary if you repeated the study with a different sample from the same population. A smaller SE means the statistic is more precisely estimated.

Standard deviation vs. standard error: SD describes the spread of individual values in your data. SE describes the precision of a computed statistic (like the mean). SD stays roughly the same as you collect more data; SE shrinks, because larger samples give more precise estimates.

• SE of mean — standard error of the arithmetic mean
• SE of median — bootstrap standard error of the median: the empirical SD of the median across resamples with replacement, using the resample count from your global Bootstrap replications setting. No distributional assumption.
• SE of proportion — for binary categorical variables only (exactly two categories)
• SE of skewness — standard error of the sample skewness
• SE of kurtosis — standard error of the sample kurtosis

Method for skewness/kurtosis SE and CI: a SE/CI method dropdown appears in the Shape section when any of the four corresponding stats are enabled. Choose Analytical (normal-theory) for closed-form SEs derived under the assumption of a normal underlying distribution, or Bootstrap for a distribution-free alternative that resamples the data — the CI uses the bias-corrected and accelerated (BCa) method (Efron 1987), which adjusts the percentile bounds for bias and skewness in the sampling distribution. Bootstrap uses the resample count from your global Bootstrap replications setting; entering an integer in Bootstrap seed (instead of leaving it empty) makes the resamples reproducible across runs. The method that was used appears in the column header itself — SE (Skewness, normal) vs SE (Skewness, bootstrap), and likewise for the CI columns — so the source is unambiguous when the table is exported or shared. The header reads “bootstrap” rather than “BCa” because the small-sample fallback to the percentile interval (described below) leaves the column still labelled honestly. When the sample is too small for the bias/acceleration corrections to be estimated reliably, the SE and CI cells still show values (with the CI falling back to the percentile interval) and a hover tooltip flags them as degraded; the point estimate in the Skewness / Kurtosis column remains valid.

Where to find the “excess kurtosis” toggle: the Report as excess kurtosis checkbox lives in the Shape section and appears whenever kurtosis itself or its SE/CI is selected — so you can request the CI alone and still control the excess-vs-raw form.

Confidence intervals #

A confidence interval gives a range of plausible values for a population parameter. The width depends on the confidence level set in your settings (default: 95%).

• CI for mean — Student’s t critical value (df = n − 1)
• CI for median — distribution-free interval based on order statistics under a Binomial(n, 0.5) reference (Wilcoxon sign-test inversion). Reported as N/A when the sample is too small to achieve the requested coverage at exact discrete levels (e.g. n = 5 at 95%).
• CI for proportion — Wilson score interval. Better behaved than the textbook Wald interval, especially near 0 or 1. Binary categorical variables only.
• CI for standard deviation — chi-square based, with the critical values computed by iterative inversion of the regularised lower incomplete gamma. Available only when Use sample statistics (n-1) is checked, because the formula rests on (n−1)s²/σ² ~ χ²(n−1), which holds only for the sample SD; the checkbox is disabled and cleared when you switch to population statistics.
• CI for variance — derived by squaring the SD interval bounds. Same sample-statistics requirement as CI for standard deviation.
• CI for skewness and CI for kurtosis — see method note above. The HL pseudomedian CI is reported together with the estimate itself in the Central tendency section.

Interpreting a 95% CI: if you repeated the study many times, about 95% of the computed intervals would contain the true population value.

Diversity #

Information-theoretic and ecological measures of how spread-out values are across distinct levels. Computed for both categorical and numerical variables — in the numerical case, each distinct value is treated as its own level.

• Shannon entropy (H) — H = −Σ pᵢ · ln(pᵢ), in nats. An absolute measure of diversity: H = 0 when everything is the same value, and H = ln k when k levels are perfectly evenly used. Scales with the number of levels, so it isn’t directly comparable across variables with different k.

• Pielou’s evenness (J) — J = H / ln(k), where k is the number of distinct levels. Normalizes Shannon entropy to [0, 1] so it is comparable across variables: 1 = perfectly even, 0 = one level dominates everything. Undefined when there is only a single level.

• Gini-Simpson (1 − D) — probability that two random observations fall into different levels. Bounded in [0, 1]; higher values mean more diversity. The standard “diversity index” in ecology.

H vs. J: they answer different questions. H tells you how much diversity is here in absolute terms (so 2 levels at 50/50 give H ≈ 0.69, but 100 levels near-uniform give H ≈ 4.6). J tells you, given the levels present, how evenly they’re used — both of those examples give J ≈ 1. Report both when both are interesting; report only J when you want a cross-variable comparison.

For continuous numerical variables: if most values are unique, H ≈ ln n and J ≈ 1 — mathematically correct but not very informative. The diversity measures are most useful for categorical variables or discrete numericals (Likert items, counts).

Sample vs. population statistics #

The Use sample statistics (n-1 denominator) checkbox (on by default) controls whether variance, standard deviation, Winsorized SD, and coefficient of variation use sample-based formulas (n-1 denominator, bias-corrected) or their population counterparts. Skewness and kurtosis always use their bias-corrected sample forms (G₁, G₂) regardless of this toggle.

• Sample statistics (n-1) — use this when your data is a sample from a larger population, which is almost always the case in research. The result labels show s² and s.
• Population statistics (n) — use this only when your data represents the entire population of interest. The result labels show σ² and σ.

When in doubt, keep sample statistics (n-1) selected. Using n instead of n-1 on sample data underestimates the true variability.

Categorical variables #

Categorical variables get their own results table with a more limited set of statistics:

• Sample size, missing count, distinct values
• Mode (and its frequency)
• Diversity measures (Shannon H, Pielou’s J, Gini-Simpson)
• Proportion, SE of proportion, and CI for proportion — only for binary variables (exactly two non-missing categories). Each can be selected independently; the Category column identifies which of the two levels the proportion refers to (the more frequent one).

Reporting checklist #

Key things to include when writing up descriptive statistics:

Method:

• Which statistics were reported and why (e.g. median and IQR for skewed data instead of mean and SD)
• Whether sample (n-1) or population (n) statistics were used
• How missing data were handled

Results:

• Central tendency (mean or median, depending on distribution shape; HL pseudomedian for symmetric-but-non-normal data)
• Dispersion (SD, Winsorized SD, IQR, or range as appropriate)
• Sample size per variable, especially if it varies due to missing data
• Skewness and kurtosis if distribution shape is relevant to subsequent analyses (state which SE/CI method was used — analytical or bootstrap)
• Outlier counts when extreme values affect interpretation — note the rule used (1.5·IQR, 3·IQR, or modified Z with |M|>3.5) and consider quoting the inlier range so readers know exactly which values were flagged

Reproducibility #

Most descriptive statistics are computed in the browser without R. The one exception is the Hodges-Lehmann pseudomedian and its confidence interval, which are computed via R’s base stats::wilcox.test(x, conf.int = TRUE, conf.level = ...) — this matches the CI rule (exact signed-rank inversion for small n, normal approximation for large n) that R uses across its own ecosystem. The call appears in the R console, and a citation for the stats package is added automatically when HL is selected.

The bootstrap paths (BCa CIs for skewness/kurtosis and SE of the median) use Math.random when Bootstrap seed is empty, so successive runs on the same data produce slightly different intervals. Enter any integer in that setting to get fully reproducible results — every bootstrap call in the run starts from that seed.

Common pitfalls #

Reporting mean and SD for skewed data. If a variable is heavily skewed, the mean is pulled toward the tail and the SD is inflated by extreme values. Report median and IQR instead — they describe the “typical” value and spread without being distorted by outliers.

Ignoring missing data patterns. A variable with 40% missing values tells a different story than one with 2% missing. Always check missing counts before interpreting the other statistics — high missingness can bias every summary measure.

Using the coefficient of variation across variables with different scales of meaning. CV is useful for comparing relative variability, but it is only meaningful for ratio-scale variables with a true zero. Comparing CVs of temperature in Celsius vs. reaction time in milliseconds is misleading because 0°C is not a true zero. The module guards against the most obvious misuse by reporting CV as N/A when any value is negative, but a non-negative scale alone doesn’t make CV meaningful — interval scales (years, dates) still aren’t ratio-scale.

Treating distinct value count as a quality check and stopping there. Spotting 5 distinct values in a binary variable is a good start, but the frequency table (Distribution analysis) shows you which values are unexpected — much more actionable than the count alone.

Treating the mode as informative for continuous numeric variables. Mode looks at exact-equality counts. For continuous measurements (heights, reaction times, sensor readings) two values almost never coincide, so the result is either “no mode” or a near-arbitrary tie — neither is useful. Use the median or HL pseudomedian as your “typical value” instead, and report the mode only for categorical or discrete numerical variables (Likert items, counts, ordinal codes).

Treating SE or CI of proportion as off when they don’t appear. These stats are computed only for binary categorical variables (exactly two non-missing levels). With one level the proportion is trivially 1; with three or more, a single proportion no longer summarises the variable — use the frequency table for the full per-category breakdown instead.

Reading “No mode” as zero observations. A No mode cell doesn’t mean the variable is empty — it means every observed value is unique, so no value is more frequent than any other. For continuous numeric data this is the typical state; the mode is usually only informative for discrete or categorical variables.

Notes #

• Geometric mean and harmonic mean show N/A if any value is zero or negative
• Coefficient of variation is omitted when the mean is zero or any value is negative — hovering an empty CV cell shows the explanation
• Proportion, SE of proportion, and CI for proportion are blank for non-binary categoricals — hovering an empty cell shows why
• Skewness and kurtosis are reported as N/A when all values are identical (zero variance)
• Mode is reported as No mode when every value is unique (no value repeats)
• The CI for the median is computed exactly from the sample order statistics; it reports N/A for small samples where no rank pair achieves the requested coverage (e.g. n = 5 at 95%) — hovering an empty cell shows the explanation
• The CI for proportion uses the Wilson score interval, which is naturally bounded in [0, 1] without artificial clamping
• Each run produces a new results card — you can generate multiple tables with different statistics selected and compare them side by side