Descriptive statistics #

The Descriptive statistics module summarises each selected variable in one row of a table: the measures of location, spread, shape, counts, quantiles and diversity you tick, with standard errors and confidence intervals where they exist, for numeric and categorical variables alike.

How to use #

1. Select your variables
2. Open Descriptive statistics from the menu
3. Tick the statistics you want under Configuration, or apply a preset
4. Click Generate descriptive statistics and read the results

Presets #

• Apply preset – two presets set the checkboxes for the common cases; applying one clears every other statistic first.
  • Parametric – sample size, mean, standard deviation, minimum, maximum
  • Nonparametric – sample size, median, minimum, maximum, quartiles (25%, 75%)

Both presets keep Sample size. The Use sample statistics (n-1 denominator) and Report as excess kurtosis toggles are presentation settings rather than statistics, and a preset leaves them as they are.

Configuration #

Every statistic is a checkbox, grouped as the panel groups them, and a box that needs a setting reveals it beneath. A statistic a variable cannot have – a geometric mean where a value is zero, a proportion where there are three levels – prints N/A in its cell, and hovering the cell says why.

Central tendency #

• Mean – the arithmetic average: the right summary of a roughly symmetric variable, and the quantity most tests compare – see mean.
• Sum – the total of the values; meaningful for additive quantities such as counts or revenue, not for rates, ratios or indices.
• Median – the middle value of the sorted data, unmoved by outliers or skew – see median.

Mean vs. median? When the two diverge, a tail or an outlier is pulling the mean – see when the mean and the median disagree.

• Mode – the most frequent value; several values sharing the highest count are all listed, and a variable in which no value repeats reads No mode – see mode. Adds a Mode frequency column.
• Trimmed mean – the mean of what remains after the Trim percentage is cut from each end of the sorted values, a compromise between the mean and the median; at a 50% trim it is the median – see trimmed mean. The column header carries the percentage.
• Trim percentage – the share cut from each end, 0–50%, 10% by default; one value serves the trimmed mean, its standard error and its confidence interval, and the box shows whenever any of the three is ticked. A cleared or unreadable box counts as 10%.
• Geometric mean – the nth root of the product of the values, for multiplicative quantities such as growth rates or ratios; N/A when any value is zero or negative.
• Harmonic mean – the reciprocal of the mean of the reciprocals, for averaging rates such as speeds; N/A when any value is zero or negative.
• Hodges-Lehmann pseudomedian – the median of all pairwise averages $(x_i+x_j)/2$: nearly as resistant to outliers as the median and nearly as efficient as the mean when the data is symmetric, reported together with its confidence interval from the signed-rank distribution – see Hodges–Lehmann estimate. The one statistic here computed in R.

Median or pseudomedian? The pseudomedian suits data that is roughly symmetric but not trusted to be normal; for a strongly skewed variable the median is still the more interpretable – see Hodges–Lehmann estimate.

Dispersion #

• Minimum – the smallest value, headed Min; a −999 here is a missing-value code that was never declared – see min.
• Maximum – the largest value, headed Max; a 999 here is the same kind of code – see max.
• Range – maximum minus minimum, set entirely by the two extremes – see range.
• Variance – the average squared deviation from the mean, in squared units; s² under sample statistics and σ² under population statistics, and the header says which – see variance.
• Standard deviation – the square root of the variance, in the variable’s own units, headed Std dev (s) or Std dev (σ) by the sample toggle – see standard deviation.

Rule of thumb? In a roughly normal distribution about 68% of the values fall within one SD of the mean and about 95% within two – see normal distribution.

• Winsorized standard deviation – the SD after the extreme Winsorization percentage of values at each tail is replaced by the boundary value rather than dropped; the spread that belongs with a trimmed mean – see winsorizing. The header carries the percentage and the denominator, as Winsorized SD (10%, s); N/A when the percentage leaves no central data (50% on an even n).
• Winsorization percentage – the share replaced at each end, 0–50%, 10% by default, set independently of the trim percentage; a cleared or unreadable box counts as 10%.
• Interquartile range – Q3 minus Q1, the width of the middle half of the data, headed IQR; unmoved by outliers, so the spread to report beside a median – see IQR.
• Mean absolute deviation – the average absolute distance from the mean, headed Mean AD; less sensitive to extremes than the SD because nothing is squared, in the same units as the mean.
• Median absolute deviation – the median of |x − median|, unscaled, headed Median AD; multiplied by 1.4826 it estimates the SD of a normal variable, and the Modified Z outliers rule is built on it – see median absolute deviation.
• Coefficient of variation – the SD as a percentage of the mean, for comparing spread across variables measured on different scales, headed CV (%, s) or CV (%, σ) by the sample toggle; defined only for non-negative ratio-scale data, so N/A when any value is negative or the mean is zero.

Shape #

• Skewness – asymmetry: 0 is symmetric, a positive value a longer right tail, a negative one a longer left tail – see skewness. Always the bias-corrected sample estimator G₁, whatever the sample toggle; N/A below three observations or when every value is identical.
• Kurtosis – the weight of the tails relative to a normal distribution, reported as excess kurtosis (normal = 0) unless Report as excess kurtosis is unticked, and headed Excess kurtosis or Kurtosis accordingly – see kurtosis. Always the bias-corrected G₂; N/A below four observations or when every value is identical.
• Report as excess kurtosis – on by default: raw kurtosis minus 3, so a normal distribution scores 0. Shown whenever kurtosis, its standard error or its confidence interval is ticked, so the form can be set for an interval requested on its own.
• SE/CI method – how the standard errors and confidence intervals of skewness and kurtosis are built; shown when any of the four is ticked, and named in their column headers as normal or bootstrap.
  • Analytical (normal-theory) – closed-form standard errors derived under a normal distribution, and intervals of estimate ± z × SE; honest only near normality.
  • Bootstrap (BCa) – resamples the data over the bootstrap replications setting and takes the bias-corrected and accelerated interval, with the standard error the SD of the replicates – see BCa. A cell built on too few usable replicates is flagged, and hovering it says which of the two floors was missed; the point estimate stays valid either way. At the shipped 100 replications the interval is always flagged – raise the setting to 1,000 for a quotable interval.

Summary counts #

• Sample size – the number of non-missing observations, headed N – see sample size.
• Count of distinct values – how many different values occur, missing excluded, headed Distinct; five distinct values in a variable meant to be binary point at inconsistent coding, which a frequency table then shows.
• Missing value count – how many cells are empty, as a count and a percentage of all rows, headed Missing.
• Zero count – how many values equal zero, as a count and a percentage of the non-missing values, headed Zeroes.
• Mild outliers (1.5·IQR) – the count and percentage of values outside $[Q_1-1.5\text{IQR},Q_3+1.5\text{IQR}]$, the points a box plot draws beyond its whiskers – see Tukey’s fences. Adds an Inlier range column. N/A when the IQR is 0.
• Extreme outliers (3·IQR) – the same rule with the fences at 3 IQR, always a subset of the mild outliers. Adds its own inlier-range column; N/A when the IQR is 0.
• Modified Z outliers (|M| > 3.5) – the count and percentage of values whose modified Z-score $M=0.6745\cdot (x-\text{median})/\text{MAD}$ exceeds 3.5 in absolute value; built on the median and the median AD, so one extreme value cannot mask another – see modified z-score. Adds its own inlier-range column; N/A when the median AD is 0.

Which rule? The 1.5·IQR fences match a box plot and suit any shape; 3·IQR isolates the unambiguous cases; the modified Z rule gives a Z-style threshold without the masking of the classical one – see finding one variable’s outliers.

Quantiles #

• Quartiles (25%, 75%) – Q1 and Q3, the values below which a quarter and three quarters of the data fall, headed Q1 (25%) and Q3 (75%) and computed by linear interpolation (R’s type 7) – see Q1 and Q3.
• Custom percentiles – any percentiles you name under Values (comma separated), one column each, headed P and the number – see percentile. A box with no valid entry adds no columns and raises a warning.
• Values (comma separated) – the percentiles to compute, 0–100, as 10, 90; unreadable entries and duplicates are dropped and the rest sorted.

Diversity #

Three measures of how evenly the values are spread across the distinct levels, computed for categorical and numeric variables alike – a numeric variable’s distinct values are its levels. Once 90% or more of a numeric variable’s values are distinct the measures only restate that, and hovering the cells says so.

• Shannon entropy (H) – $H=-{\sum }_i{p}_i\ln p_i$, in nats, headed Shannon H (nats): 0 when every case has the same value and ln k when the k levels are equally used, so it grows with the number of levels – see entropy.
• Pielou’s evenness (J) – $J=H/\ln k$, the entropy scaled to 0–1 by the number of levels, headed Pielou’s J: 1 is perfectly even, 0 one level holding everything, and comparable across variables where H is not – see Pielou’s evenness. N/A with a single level.
• Gini-Simpson (1−D) – the probability that two random observations fall in different levels, from 0 to 1 – see Gini-Simpson index.

Standard errors #

A standard error says how much a statistic would vary from sample to sample – the precision of the statistic, where the SD is the spread of the values – see standard error.

• Standard error of mean – the SD divided by √n, headed SE (mean).
• Standard error of median – a bootstrap: the SD of the median over resamples with replacement, as many as the bootstrap replications setting says, headed SE (median) – see bootstrap.
• Standard error of trimmed mean – the Tukey–McLaughlin standard error: the winsorized spread at the same trim, over the observations the trim left in place, headed SE (trimmed mean, {pct}%); the plain SE of the mean is not valid for a trimmed mean. Shares the Trim percentage box; N/A when the trim leaves fewer than two observations.
• Standard error of proportion – $\sqrt{p(1-p)/n}$ for the proportion of a binary variable, headed SE (proportion).
• Standard error of skewness – by the SE/CI method, headed SE (skewness, {method}).
• Standard error of kurtosis – by the SE/CI method, headed SE (kurtosis, {method}).

Confidence intervals #

Every interval is at the confidence level set in Settings, which the column headers repeat, and prints as a lower and an upper column – see confidence interval.

• CI for mean – Student’s t interval at n − 1 degrees of freedom, headed Mean CI lower ({level}%) and Mean CI upper ({level}%).
• CI for trimmed mean – the Tukey–McLaughlin (Yuen) interval: trimmed mean ± t × its standard error at h − 1 degrees of freedom, h the untrimmed count; at a 0% trim it equals the CI for the mean. Headed Trimmed mean CI lower ({level}%, {pct}%) and Trimmed mean CI upper ({level}%, {pct}%), the trim repeated so the interval reads with the estimate it brackets.
• CI for median – distribution-free, built from the order statistics by the Median CI method it reveals, headed Median CI lower ({level}%) and Median CI upper ({level}%); hovering a header names the construction – see interval for a median.
• Median CI method – which of two constructions produces the bounds.
  • Exact (order statistics) – inverts the Binomial(n, 0.5) sign test; conservative, its coverage at least the nominal level. N/A when no pair of order statistics reaches the level (n = 5 at 95%), with the reason on hover.
  • Interpolated (Hettmansperger-Sheather) – interpolates between adjacent order statistics to land closer to the nominal level; the interval the median notch in box plots draws, so pick it when the table and the plot should agree.
• CI for proportion – the Wilson score interval, well behaved near 0 and 1 where the textbook interval is not, headed Proportion CI lower ({level}%) and Proportion CI upper ({level}%); binary variables only – see interval for a proportion.
• CI for standard deviation – Bonett’s kurtosis-adjusted interval, not the chi-square pivot, headed Std dev CI lower ({level}%, Bonett) and Std dev CI upper ({level}%, Bonett); hovering a header says why another package’s number differs – see interval for a standard deviation. Needs sample statistics – the box is disabled and cleared under population statistics – and at least five observations.
• CI for variance – the bounds of the SD interval, squared, headed Variance CI lower ({level}%, Bonett) and Variance CI upper ({level}%, Bonett); the same construction and the same sample-statistics requirement.
• CI for skewness – by the SE/CI method: skewness ± z × SE, or the BCa bootstrap interval, headed Skew CI lower ({level}%, {method}) and Skew CI upper ({level}%, {method}).
• CI for kurtosis – by the SE/CI method, in the form Report as excess kurtosis sets, headed Kurt CI lower ({level}%, {method}) and Kurt CI upper ({level}%, {method}).

Reading a 95% CI? Over many repetitions of the study, about 95% of the intervals computed this way would contain the population value – see 95% CI.

Sample vs. population statistics #

• Use sample statistics (n-1 denominator) – on by default. Governs four statistics – variance, standard deviation, winsorized standard deviation and the coefficient of variation – with n − 1 as the denominator when ticked and n when not; their headers read s or σ accordingly. Skewness, kurtosis and every standard error and confidence interval keep the sample form regardless, and unticking the box also disables and clears CI for standard deviation and CI for variance, which exist only for the sample form. Keep it ticked unless your data is the whole population of interest: n on a sample understates the variability.

Reading results #

Results appear in a Descriptive statistics card with up to two tables, one row per variable. A variable with no valid value left under the active case filter is left out of the table and named in a warning above it; when no ticked statistic applies to any selected variable, no card appears and a notification says so. Each run adds a new card, so tables with different selections can sit side by side.

Numerical variables. One row per numeric variable and one column per ticked statistic, the columns in the order of the groups above. Every header carries the setting its numbers depend on – the trim, the confidence level, the SE/CI method, the s or σ denominator – so an exported table reads without the panel.

Categorical variables. One row per categorical variable, with the statistics that apply to categories: the sample size, missing count and distinct values, the mode and its frequency, the three diversity measures and, for a binary variable, the proportion with its standard error and interval.

The columns are the statistics’ own entries above; the ones no checkbox names:

• Variable – the variable’s display name.
• Mode frequency – how many observations share the modal value; N/A when there is no mode.
• HL CI lower ({level}%) – the lower bound of the pseudomedian’s interval, with HL CI upper ({level}%) its upper bound; computed with the estimate whether or not any CI box is ticked.
• Inlier range (1.5·IQR) – the cut-off pair [lower, upper] of the rule beside it, one column per enabled rule – Inlier range (3·IQR) and Inlier range (modified Z) likewise; feed the pair to a case filter, where Between keeps the inliers and Outside the outliers – see inlier range. N/A, with the reason on hover, when the rule’s spread is zero.
• Category – the level whose share the proportion columns describe: the more frequent of a binary variable’s two levels, the first met on a tie. For a variable with one level, or three or more, the proportion cells read N/A, and hovering one says why.
• Proportion – that level’s share of the non-missing observations. A binary numeric variable (0/1 dummies and the like) gets the same columns in the numeric table.

Reporting checklist #

Method:

• Which statistics were reported and why (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
• The construction behind any interval that has more than one: the Median CI method used, and that the SD and variance intervals are Bonett’s rather than the chi-square ones

Results:

• Central tendency – mean or median by the shape of the distribution, the HL pseudomedian for symmetric but non-normal data
• Dispersion – SD, winsorized SD, IQR or range as appropriate
• Sample size per variable, especially where missing data makes it vary
• Skewness and kurtosis where the shape matters to later analyses, with the SE/CI method – analytical or bootstrap
• Outlier counts where extreme values affect the interpretation – the rule used (1.5·IQR, 3·IQR, or modified Z with |M| > 3.5) and the inlier range, so readers know which values were flagged
• Any variable dropped from the table because every case was missing under the active filter

Reproducibility #

Most statistics are computed in the browser without R. The Hodges-Lehmann pseudomedian and its interval go through R’s stats::wilcox.test(x, conf.int = TRUE), shown in the R console, with an offset mu that keeps zero values in the estimate – a bare call can differ on data with zeros. Every run’s citation box lists the methods behind the ticked statistics and nothing else, with the stats package added when the pseudomedian ran. With Bootstrap seed empty the bootstrap paths draw fresh resamples on every run, so the median’s SE and the bootstrap intervals move in their last digits; an integer there makes them reproducible, each variable drawing its own stream keyed on its position in the selection. The method notes hold the reasoning behind these choices.

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 the 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. Check the missing counts before interpreting the other statistics – high missingness can bias every summary.

Comparing coefficients of variation across scales with different meanings. CV is only meaningful for ratio-scale variables with a true zero. Comparing the CV of a temperature in Celsius with that of a reaction time is misleading because 0 °C is not a true zero. The module refuses the most obvious misuse by reporting N/A when any value is negative, but a non-negative scale alone doesn’t make CV meaningful – interval scales such as years or dates still aren’t ratio scales.

Treating the distinct-value count as a quality check and stopping there. Five distinct values in a binary variable is a good start, but the frequency table in Distribution analysis shows which values are unexpected – far more actionable than the count alone.

Reading the mode of a continuous variable. The mode counts exact matches. For continuous measurements two values almost never coincide, so the result is either No mode or a near-arbitrary tie – neither is useful. Use the median or the HL pseudomedian as the typical value, and report the mode for categorical or discrete numeric variables (Likert items, counts, ordinal codes).

Reading N/A in the proportion columns as a failure. The proportion, its standard error and its interval exist only for a binary variable – exactly two distinct non-missing values, categorical or numeric. 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.

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 usual state.