Numerical Descriptives: Zero to Hero Tutorial

This comprehensive tutorial takes you from the foundational concepts of summarising continuous and discrete numerical data all the way through advanced interpretation, reporting, visualisation, assumption checking, and practical usage within the DataStatPro application. Whether you are encountering numerical descriptive statistics for the first time or deepening your understanding of how to characterise, display, and communicate the distribution of numerical variables, this guide builds your knowledge systematically from the ground up.

Table of Contents

1. Prerequisites and Background Concepts
2. What are Numerical Descriptives?
3. The Mathematics Behind Numerical Descriptives
4. Considerations and Data Quality Checks
5. Types of Numerical Descriptive Measures
6. Using the Numerical Descriptives Calculator Component
7. Step-by-Step Procedure
8. Interpreting the Output
9. Visualising Numerical Data
10. Confidence Intervals for Numerical Descriptives
11. Advanced Topics
12. Worked Examples
13. Common Mistakes and How to Avoid Them
14. Troubleshooting
15. Quick Reference Cheat Sheet

1. Prerequisites and Background Concepts

Before diving into numerical descriptive statistics, it is essential to be comfortable with the following foundational statistical and mathematical concepts. Each is briefly reviewed below.

1.1 Variables and Observations

A variable is any measurable characteristic that can take on different values across observations. A numerical variable (also called a quantitative variable) records values on a numeric scale that has an inherent magnitude, enabling arithmetic operations such as addition, subtraction, and averaging.

• Observation: A single unit of study (one person, one trial, one time point).
• Dataset: A rectangular array of observations (rows) and variables (columns).
• Value: The specific numeric measurement recorded for a variable on a given observation.

1.2 Scales of Measurement: Interval and Ratio

Numerical descriptive statistics are appropriate for variables measured on interval or ratio scales:

⚠️ For interval scales, differences are meaningful but ratios are not — saying 40°C is "twice as hot" as 20°C is meaningless. For ratio scales, both differences and ratios are meaningful — a person weighing 80 kg is genuinely twice as heavy as one weighing 40 kg. Most numerical descriptives apply to both scales, but ratios such as the coefficient of variation require ratio-scale data.

1.3 Continuous vs. Discrete Numerical Variables

Numerical variables are further classified by the set of values they can take:

• Continuous variables: Can take any real value within a range; the precision is limited only by the measurement instrument. Examples: height, blood pressure, temperature, time.
• Discrete variables: Take only countable, distinct values (often non-negative integers). Examples: number of children, number of hospital admissions, count of errors.

Most numerical descriptives apply equally to both types, though visualisation choices (histograms vs. bar charts) and some distribution assumptions differ.

1.4 The Concept of a Distribution

The distribution of a numerical variable describes how its values are spread across the number line. Understanding a distribution requires characterising:

1. Location (central tendency): Where the centre or typical value lies.
2. Spread (dispersion): How much values vary around the centre.
3. Shape: Whether the distribution is symmetric, skewed, peaked, or flat.
4. Outliers: Whether extreme values are present.

Numerical descriptive statistics provide compact summaries of each of these four distributional properties.

1.5 The Normal Distribution

The normal (Gaussian) distribution is the most important continuous probability distribution in statistics. It is parameterised by its mean $\mu$ and standard deviation $\sigma$:

$X \sim \mathcal{N}(\mu,\; \sigma^2)$

Key properties:

• Perfectly symmetric and bell-shaped around $\mu$.
• Mean, median, and mode are all equal to $\mu$.
• Approximately 68% of observations fall within $\pm 1\sigma$ of $\mu$.
• Approximately 95% of observations fall within $\pm 1.96\sigma$ of $\mu$.
• Approximately 99.7% of observations fall within $\pm 3\sigma$ of $\mu$.
• Skewness = 0; excess kurtosis = 0.

Many numerical descriptives (mean, standard deviation, standard error) are most meaningful and interpretable when the underlying distribution is approximately normal.

1.6 Robustness

A statistical measure is robust if it is relatively unaffected by outliers or departures from assumed distributional shapes. This is a critical concept for choosing between competing descriptive measures:

• Non-robust measures: Mean, variance, standard deviation — one extreme outlier can substantially shift these measures.
• Robust measures: Median, interquartile range, median absolute deviation — designed to be resistant to the influence of extreme values.

1.7 Population Parameters vs. Sample Statistics

All numerical descriptives computed from data are sample statistics — estimates of unknown population parameters:

Sample statistics carry sampling variability; confidence intervals (Section 10) quantify the precision of these estimates.

1.8 Summation Notation

Numerical descriptives are expressed using summation notation. For a variable $X$ with $n$ observations $x_1, x_2, \ldots, x_n$:

$\sum_{i=1}^{n} x_i = x_1 + x_2 + \cdots + x_n$

$\sum_{i=1}^{n} x_i^2 = x_1^2 + x_2^2 + \cdots + x_n^2$

$\sum_{i=1}^{n} (x_i - \bar{x})^2 = \text{Sum of squared deviations from the mean}$

Understanding this notation is essential for interpreting the mathematical formulae throughout this tutorial.

2. What are Numerical Descriptives?

2.1 The Core Purpose

Numerical descriptive statistics are mathematical summaries that characterise the distribution of a continuous or discrete numerical variable. Their collective purpose is to replace a raw list of $n$ numbers with a small, interpretable set of values that faithfully conveys the essential features of the data — location, spread, shape, and extremes — without requiring the reader to examine every individual observation.

2.2 The Four Pillars of a Numerical Description

Every complete numerical description addresses four fundamental questions:

No single number captures all four pillars. A complete description always reports at least one measure from each category.

2.3 When to Use Numerical Descriptives

2.4 Real-World Applications

2.5 Distinguishing Numerical Descriptives from Related Analyses

3. The Mathematics Behind Numerical Descriptives

3.1 Notation

Consider a numerical variable $X$ with $n$ valid observations $x_1, x_2, \ldots, x_n$, arranged in ascending order as the order statistics $x_{(1)} \leq x_{(2)} \leq \cdots \leq x_{(n)}$.

3.2 Measures of Central Tendency

3.2.1 Arithmetic Mean

The arithmetic mean (simply "the mean") is the sum of all values divided by the number of observations:

$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \cdots + x_n}{n}$

The mean is the centre of gravity of the data — the point at which the distribution balances. It uses all observations equally and is the most efficient estimator of the population mean $\mu$ when the distribution is normal. However, it is sensitive to outliers.

3.2.2 Median

The median $\tilde{x}$ is the middle value when observations are arranged in order. It divides the distribution into two equal halves:

$\tilde{x} = \begin{cases} x_{((n+1)/2)} & \text{if } n \text{ is odd} \\ \dfrac{x_{(n/2)} + x_{(n/2 + 1)}}{2} & \text{if } n \text{ is even} \end{cases}$

The median is the 50th percentile of the distribution. It is robust to outliers and is preferred over the mean when the distribution is skewed or when extreme values are present.

3.2.3 Mode

For numerical data, the mode is the value (or range of values) that appears most frequently. For continuous variables, the mode is typically identified from a histogram as the peak(s) of the distribution rather than a specific repeated value. For discrete data with genuine ties, the mode is the most frequently occurring value.

3.2.4 Trimmed Mean

The trimmed mean (or truncated mean) removes a fixed proportion $\alpha$ of the most extreme observations from each tail before computing the mean:

$\bar{x}_{\alpha} = \frac{1}{n - 2\lfloor \alpha n \rfloor} \sum_{i = \lfloor \alpha n \rfloor + 1}^{n - \lfloor \alpha n \rfloor} x_{(i)}$

Common choices: $\alpha = 0.05$ (5% trimmed mean) or $\alpha = 0.10$ (10% trimmed mean). The trimmed mean is more robust than the mean while being more efficient than the median, making it a useful middle ground.

3.2.5 Geometric Mean

The geometric mean is appropriate for positively skewed, multiplicative, or ratio- scale data (e.g., concentration values, growth rates, fold changes):

$\bar{x}_{geom} = \left(\prod_{i=1}^n x_i\right)^{1/n} = \exp\!\left(\frac{1}{n}\sum_{i=1}^n \ln x_i\right)$

The geometric mean is defined only for strictly positive values ($x_i > 0$). It is the antilog of the mean of the log-transformed values and is always $\leq$ the arithmetic mean (equality holds when all values are identical).

3.2.6 Harmonic Mean

The harmonic mean is appropriate for averaging rates or ratios (e.g., speeds, price-to-earnings ratios):

$\bar{x}_{harm} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}}$

The harmonic mean is always $\leq$ the geometric mean $\leq$ the arithmetic mean (the AM-GM-HM inequality), with equality when all values are identical.

3.3 Measures of Dispersion

3.3.1 Range

The range is the simplest measure of spread:

$\text{Range} = x_{(n)} - x_{(1)} = \max(x) - \min(x)$

The range is easy to compute and interpret but is maximally non-robust — it is determined entirely by the two most extreme observations and increases without bound as $n$ grows.

3.3.2 Interquartile Range

The interquartile range (IQR) is the range of the middle 50% of the data:

$IQR = Q_3 - Q_1$

Where $Q_1$ is the 25th percentile and $Q_3$ is the 75th percentile. The IQR is the most widely used robust measure of dispersion. It is unaffected by the values of the most extreme observations and provides direct information about the spread of the central bulk of the data.

3.3.3 Percentiles and Quartiles

The $p$-th percentile $P_p$ is the value below which $p\%$ of observations fall. For $n$ ordered observations, the percentile is computed using linear interpolation:

$L = \frac{p(n-1)}{100} + 1$

If $L$ is an integer, $P_p = x_{(L)}$. Otherwise, $P_p = x_{(\lfloor L \rfloor)} + (L - \lfloor L \rfloor)(x_{(\lceil L \rceil)} - x_{(\lfloor L \rfloor)})$.

Key percentiles:

⚠️ Multiple methods exist for computing percentiles (e.g., Type 7 in R, Excel's PERCENTILE.INC). These differ in how they handle boundary cases and interpolation. DataStatPro uses the standard linear interpolation method (Type 7 / inclusive method) by default. The method is stated in the output footnote.

3.3.4 Variance

The sample variance $s^2$ measures the average squared deviation from the mean. It uses $n - 1$ in the denominator (Bessel's correction) to produce an unbiased estimate of the population variance $\sigma^2$:

$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 = \frac{\sum_{i=1}^n x_i^2 - n\bar{x}^2}{n-1}$

The divisor $n - 1$ reflects the one degree of freedom consumed in estimating $\bar{x}$. Using $n$ instead of $n-1$ yields the population variance formula (maximum likelihood estimator), which is biased for small samples.

3.3.5 Standard Deviation

The sample standard deviation $s$ is the square root of the sample variance:

$s = \sqrt{s^2} = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$

The standard deviation is expressed in the same units as the original variable, making it directly interpretable as a typical distance from the mean. Under normality, approximately 68% of observations fall within $\bar{x} \pm s$.

3.3.6 Standard Error of the Mean

The standard error of the mean (SEM or SE) quantifies the precision of $\bar{x}$ as an estimate of $\mu$ — it is the standard deviation of the sampling distribution of the mean:

$SE_{\bar{x}} = \frac{s}{\sqrt{n}}$

As $n$ increases, $SE_{\bar{x}}$ decreases proportionally to $1/\sqrt{n}$. The SEM is not a measure of the spread of individual observations (that role belongs to $s$).

⚠️ A common and serious error is reporting the SEM instead of the SD as the measure of variability in individual observations. The SEM is always smaller than the SD and can give a misleading impression of how variable the data are. Use SD to describe variability in the data; use SEM to describe precision of the mean estimate.

3.3.7 Coefficient of Variation

The coefficient of variation (CV) expresses the standard deviation as a percentage of the mean, enabling comparison of variability across variables measured on different scales or with different units:

$CV = \frac{s}{\bar{x}} \times 100\%$

The CV is a dimensionless measure of relative variability. It is only meaningful for ratio-scale variables (true zero exists) and when $\bar{x} > 0$. CV is widely used in clinical chemistry, analytical measurement science, and quality control.

3.3.8 Mean Absolute Deviation

The mean absolute deviation (MAD from mean) is a robust alternative to the standard deviation:

$MAD_{mean} = \frac{1}{n} \sum_{i=1}^n |x_i - \bar{x}|$

Unlike the variance, which squares deviations (over-weighting outliers), the MAD from mean uses absolute deviations. Under normality, $MAD_{mean} \approx 0.798 \times s$.

3.3.9 Median Absolute Deviation

The median absolute deviation (MAD from median) is the most robust common measure of spread:

$MAD_{median} = \text{median}\!\left(|x_i - \tilde{x}|\right)$

To use $MAD_{median}$ as a robust estimator of the standard deviation under normality, apply the consistency factor:

$\hat{\sigma}_{robust} = 1.4826 \times MAD_{median}$

$MAD_{median}$ has a 50% breakdown point — it remains stable even when up to 50% of observations are outliers. DataStatPro reports both MAD measures, labelling them clearly as "MAD (from mean)" and "MAD (from median)".

3.4 Measures of Shape

3.4.1 Skewness

Skewness measures the degree and direction of asymmetry in the distribution:

$g_1 = \frac{\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^3}{\left(\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2\right)^{3/2}}$

The bias-corrected sample skewness (Fisher's formula, used by most software including DataStatPro):

$G_1 = \frac{n}{(n-1)(n-2)} \sum_{i=1}^n \left(\frac{x_i - \bar{x}}{s}\right)^3$

3.4.2 Kurtosis

Kurtosis measures the heaviness of the tails and the peakedness of the distribution relative to a normal distribution:

$g_2 = \frac{\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^4}{\left(\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2\right)^{2}}$

The bias-corrected excess kurtosis (subtracting 3 so that a normal distribution has excess kurtosis = 0) is:

$G_2 = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum_{i=1}^n \left(\frac{x_i - \bar{x}}{s}\right)^4 - \frac{3(n-1)^2}{(n-2)(n-3)}$

⚠️ Both skewness and kurtosis are sensitive to sample size and outliers. With $n < 50$, these estimates are highly unstable. Use formal normality tests (Shapiro-Wilk) to complement visual inspection for normality assessment.

3.5 Measures of Position

3.5.1 Z-Scores (Standardised Values)

The z-score of observation $x_i$ expresses its distance from the mean in standard deviation units:

$z_i = \frac{x_i - \bar{x}}{s}$

Z-scores enable comparison of observations from different variables or different populations. Under normality, $|z_i| > 3$ is a widely used criterion for flagging potential outliers.

3.5.2 Percentile Rank

The percentile rank of observation $x_i$ is the percentage of observations in the sample that are less than or equal to $x_i$:

$PR(x_i) = \frac{\#\{j : x_j \leq x_i\}}{n} \times 100\%$

Percentile ranks are used for norm-referenced scoring (e.g., standardised test reporting) and for identifying the relative standing of individual observations.

3.6 The Five-Number Summary

The five-number summary provides a compact, robust distributional snapshot:

$\{\text{Min},\; Q_1,\; \text{Median},\; Q_3,\; \text{Max}\}$

This summary is the basis for the box plot (Section 9.3) and directly communicates the range, central tendency, spread (IQR), and potential skewness of the distribution. It is resistant to outliers (except for the min and max).

3.7 Normality Assessment Statistics

3.7.1 Shapiro-Wilk Test

The Shapiro-Wilk test is the most powerful test for normality for $n \leq 2000$. It computes the correlation between the ordered sample values and the expected order statistics under normality:

$W = \frac{\left(\sum_{i=1}^n a_i x_{(i)}\right)^2}{\sum_{i=1}^n (x_i - \bar{x})^2}$

Where $a_i$ are the Shapiro-Wilk coefficients derived from the expected normal order statistics. $W$ ranges from 0 to 1; values close to 1 indicate normality.

$H_0$: The data are drawn from a normal distribution.

If $p < \alpha$, reject normality.

3.7.2 Kolmogorov-Smirnov Test (Lilliefors Correction)

The Kolmogorov-Smirnov test (with Lilliefors correction for estimated parameters) computes the maximum absolute difference between the empirical CDF and the normal CDF with estimated $\mu$ and $\sigma$:

$D = \sup_x |F_n(x) - F(x;\hat{\mu}, \hat{\sigma})|$

The Lilliefors test is less powerful than Shapiro-Wilk for detecting non-normality but is useful as a supplementary check, especially for larger samples.

3.7.3 Skewness and Kurtosis Tests

Formal tests for normality using the skewness and kurtosis statistics:

Standard errors under normality:

$SE_{G_1} \approx \sqrt{\frac{6n(n-1)}{(n-2)(n+1)(n+3)}}, \qquad SE_{G_2} \approx 2 \times SE_{G_1} \times \sqrt{\frac{n^2 - 1}{(n-3)(n+5)}}$

Test statistics:

$z_{G_1} = \frac{G_1}{SE_{G_1}}, \qquad z_{G_2} = \frac{G_2}{SE_{G_2}}$

Values $|z| > 1.96$ suggest significant departure from normality at $\alpha = .05$.

4. Considerations and Data Quality Checks

4.1 Identifying Outliers

An outlier is an observation that appears inconsistent with the bulk of the data. Outliers can arise from data entry errors, measurement malfunctions, genuine extreme values, or distributional heavy tails. It is critical to identify, investigate, and make a principled decision about outliers before computing and reporting descriptives.

Common outlier detection methods:

⚠️ Outlier removal must be justified on scientific or methodological grounds, not statistical convenience. An outlier is not grounds for deletion merely because it is extreme — it may represent the most scientifically interesting observation. Document all outlier decisions transparently. DataStatPro flags outliers but never removes them automatically.

4.2 Missing Data Assessment

Before any numerical summary is computed, the extent and pattern of missing data must be evaluated. Report $n_{miss}$ and the missing rate $n_{miss}/n_{total}$. Investigate the missing data mechanism (MCAR, MAR, or MNAR) as in any statistical analysis.

Impact of missing data on numerical descriptives:

• Complete case analysis (default): Compute descriptives on valid observations only. Unbiased under MCAR; potentially biased under MAR or MNAR.
• Mean/median imputation: Replace missing values with the sample mean or median. Reduces apparent variability; distorts distributional shape. Not recommended.
• Multiple imputation: Statistically principled but complex; appropriate for inference, less so for purely descriptive purposes.

DataStatPro reports $n_{valid}$ and $n_{miss}$ in all output tables.

4.3 Checking Variable Type and Scale

Before computing numerical descriptives, confirm that the variable is genuinely measured on an interval or ratio scale. Common pitfalls:

4.4 Distributional Assumptions

Many uses of numerical descriptives (confidence intervals for the mean, standard error interpretation, power calculations) assume approximate normality. Check this assumption using:

1. Histograms: Assess overall shape visually.
2. Q-Q plot: Plot sample quantiles against theoretical normal quantiles; departures from the diagonal indicate non-normality.
3. Shapiro-Wilk test: Formal test of normality.
4. Skewness and kurtosis: Numerical shape indicators.

When data are non-normal, prefer robust measures (median, IQR, MAD) over mean-based summaries, especially with small samples.

4.5 Sample Size Adequacy

The stability and interpretability of numerical descriptives depend on $n$:

4.6 Significant Figures and Rounding

Numerical descriptives should be reported to a level of precision consistent with the measurement instrument:

• Report the mean and standard deviation to one more decimal place than the raw data.
• Report the median and IQR to the same precision as the raw data.
• Report test statistics ($z$, $t$, $W$) to two decimal places.
• Report p-values to two or three decimal places (or as $< .001$).
• Do not report spurious precision (e.g., mean $= 3.14159265$ for data measured to the nearest integer).

4.7 Weighted Data

For survey data with design weights, apply weights when computing all numerical descriptives to produce population-representative estimates:

$\bar{x}^{(w)} = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}, \qquad s^{2(w)} = \frac{\sum_{i=1}^n w_i (x_i - \bar{x}^{(w)})^2}{\sum_{i=1}^n w_i - 1}$

DataStatPro supports weighted numerical descriptives when a weight variable is specified, reporting both unweighted and weighted estimates side by side.

5. Types of Numerical Descriptive Measures

5.1 Measures of Central Tendency

5.2 Measures of Dispersion

5.3 Measures of Shape

5.4 Measures of Position

5.5 Five-Number Summary

$\{x_{(1)},\; Q_1,\; \tilde{x},\; Q_3,\; x_{(n)}\}$

Compact, robust distributional description; forms the basis for box plots.

5.6 Normality Diagnostics

6. Using the Numerical Descriptives Calculator Component

The Numerical Descriptives Calculator in DataStatPro provides a fully featured tool for computing, diagnosing, visualising, and reporting descriptive statistics for numerical variables.

Step-by-Step Guide

Step 1 — Navigate to the Component

Go to Descriptive Statistics → Numerical Descriptives.

Step 2 — Input Method

Choose how to provide your data:

• Raw data: Upload a CSV/Excel file or paste a column of numeric values. DataStatPro automatically detects the variable type, identifies non-numeric entries, and flags missing values.
• Multiple variables: Select two or more numeric columns to run batch descriptives across all selected variables simultaneously, producing a comparative summary table.
• Grouped analysis: Designate a categorical grouping variable to compute descriptives separately within each group, enabling direct comparison of distributional summaries across subgroups.

Step 3 — Variable Configuration

• Assign a meaningful variable name and unit of measurement for display.
• Specify the measurement scale (interval or ratio) to unlock scale-appropriate measures (e.g., CV requires ratio scale).
• Specify a grouping variable (optional) to produce stratified descriptives.
• Specify a weight variable (optional) for survey-weighted estimates.
• Designate whether log-transformation should be applied for geometric mean reporting (appropriate for log-normal data).

Step 4 — Missing Data Handling

Select one of the following:

• Exclude missing (valid $n$ only): All summaries computed on valid observations. $n_{miss}$ reported separately.
• Flag and exclude: Missing values flagged and listed; summaries exclude missing.

Step 5 — Outlier Handling

• Select outlier detection method: Tukey IQR fence (default), Z-score ($|z| > 3$), or Modified Z-score (Iglewicz-Hoaglin).
• Choose whether to flag only or to compute descriptives both with and without flagged outliers (DataStatPro never removes outliers automatically).

Step 6 — Set Display Options

• ✅ $n_{valid}$, $n_{miss}$, and total $n$.
• ✅ Mean, median, mode, trimmed mean (selectable $\alpha$), geometric mean, harmonic mean.
• ✅ Minimum, maximum, range.
• ✅ $Q_1$, $Q_3$, IQR, and full percentile table (selectable percentile set).
• ✅ Variance ($s^2$), standard deviation ($s$), SEM ($SE_{\bar{x}}$).
• ✅ Coefficient of variation (CV).
• ✅ MAD (from mean and from median), with robust $\hat{\sigma}$.
• ✅ Skewness ($G_1$) and excess kurtosis ($G_2$) with standard errors and z-tests.
• ✅ Five-number summary table.
• ✅ Shapiro-Wilk and Lilliefors normality tests with p-values.
• ✅ Outlier table with flagging method, z-score, and modified z-score.
• ✅ 95% confidence intervals for the mean (t-based), median (bootstrap), and SD.
• ✅ Histogram with optional normal curve overlay, density curve, and rug plot.
• ✅ Box plot (standard and notched).
• ✅ Violin plot.
• ✅ Q-Q plot with confidence band.
• ✅ Empirical cumulative distribution function (ECDF) plot.
• ✅ Dot plot / strip chart.
• ✅ Stem-and-leaf display (for $n \leq 200$).
• ✅ Grouped comparison plots (when grouping variable specified).
• ✅ APA 7th edition results paragraph (auto-generated).
• ✅ Publication-ready descriptives table.

Step 7 — Run the Analysis

Click "Compute Numerical Descriptives". DataStatPro will:

1. Validate data: check variable type, identify non-numeric entries, count missing values.
2. Compute the complete set of central tendency, dispersion, shape, and position measures.
3. Detect outliers using the selected method and produce an outlier report.
4. Run Shapiro-Wilk and Lilliefors normality tests.
5. Compute 95% CIs for the mean (t-distribution), median (bootstrap), and SD (chi-square).
6. Generate all selected visualisations with customisable formatting.
7. Produce the APA-compliant results paragraph and formatted descriptives table.

7. Step-by-Step Procedure

7.1 Full Manual Procedure

Step 1 — Define the Variable

State the variable name, unit of measurement, scale (interval or ratio), and the population of observations. Confirm that the variable is genuinely numerical.

Step 2 — Count Total and Missing Observations

$n_{total} = n_{valid} + n_{miss}$

Report $n_{miss}$ and the missing rate $n_{miss}/n_{total}$. Apply the chosen missing data strategy before proceeding.

Step 3 — Sort the Data

Sort the $n_{valid}$ observations in ascending order to obtain the order statistics $x_{(1)} \leq x_{(2)} \leq \cdots \leq x_{(n)}$.

Step 4 — Compute Central Tendency Measures

Mean:

$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$

Median:

$\tilde{x} = \begin{cases} x_{((n+1)/2)} & n \text{ odd} \\ (x_{(n/2)} + x_{(n/2+1)})/2 & n \text{ even} \end{cases}$

Mode: Identify the most frequently occurring value(s).

Geometric mean (if applicable):

$\bar{x}_{geom} = \exp\!\left(\frac{1}{n}\sum_{i=1}^n \ln x_i\right)$

Step 5 — Compute the Five-Number Summary

Identify: $\{x_{(1)},\; Q_1,\; \tilde{x},\; Q_3,\; x_{(n)}\}$ using linear interpolation for $Q_1$ and $Q_3$.

$IQR = Q_3 - Q_1$

$\text{Range} = x_{(n)} - x_{(1)}$

Step 6 — Compute Dispersion Measures

Variance:

$s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2$

Standard deviation:

$s = \sqrt{s^2}$

Standard error of the mean:

$SE_{\bar{x}} = \frac{s}{\sqrt{n}}$

Coefficient of variation (ratio scale only):

$CV = \frac{s}{\bar{x}} \times 100\%$

Median absolute deviation:

$MAD_{median} = \text{median}(|x_i - \tilde{x}|)$

Step 7 — Compute Shape Measures

Skewness ($G_1$) and excess kurtosis ($G_2$) using bias-corrected formulae (Section 3.4).

Compute $z_{G_1} = G_1/SE_{G_1}$ and $z_{G_2} = G_2/SE_{G_2}$ to formally test departure from normality.

Step 8 — Detect Outliers

Apply Tukey's IQR fence:

$\text{Lower fence} = Q_1 - 1.5 \times IQR$

$\text{Upper fence} = Q_3 + 1.5 \times IQR$

Flag any $x_i$ outside the fence as a potential outlier. Investigate each flagged observation individually.

Step 9 — Assess Normality

Compute the Shapiro-Wilk statistic $W$ (for $n \leq 2000$). Inspect the histogram and Q-Q plot. Report skewness and kurtosis. Conclude whether the normality assumption is tenable.

Step 10 — Compute Confidence Intervals

95% CI for the mean (t-based):

$\bar{x} \pm t_{\alpha/2,\; n-1} \times SE_{\bar{x}}$

Where $t_{\alpha/2,\; n-1}$ is the critical value from the t-distribution with $n - 1$ degrees of freedom ($t \approx 1.960$ for $n > 120$; $t \approx 2.306$ for $n = 9$).

95% CI for the standard deviation (chi-square based):

$\left[\sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2,\; n-1}}},\;\; \sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2,\; n-1}}}\right]$

95% CI for the median (bootstrap): Use $B = 2000$ bootstrap resamples to compute the percentile bootstrap CI.

Step 11 — Produce Visualisations

Select appropriate chart types (see Section 9), annotate with key descriptive values (mean, median, SD), and ensure axes are labelled with variable name and units.

Step 12 — Interpret and Report

Use APA reporting guidelines (Section 15). Always report $n_{valid}$, $n_{miss}$, mean, SD, median, IQR, min, max, skewness, and the result of the normality assessment. Report CIs for the mean. For non-normal distributions, emphasise the median and IQR.

8. Interpreting the Output

8.1 Central Tendency: Mean vs. Median

8.2 Interpreting the Standard Deviation

8.3 Interpreting the IQR and Five-Number Summary

8.4 Interpreting Skewness and Kurtosis

8.5 Interpreting Normality Tests

⚠️ Normality tests are extremely sensitive to sample size. With $n > 200$, trivial departures from normality will be statistically significant. With $n < 20$, even severe departures may not be detected. Always combine formal tests with visual inspection (histogram and Q-Q plot). The practical question is not whether the data are perfectly normal, but whether the departure is large enough to affect the validity of subsequent analyses.

8.6 The Mean-SD vs. Median-IQR Decision

This is one of the most frequently encountered reporting decisions in numerical descriptives:

⚠️ The notation "Mean ± SD" means mean plus or minus one standard deviation. "Mean ± SEM" means mean plus or minus one standard error of the mean. These are very different quantities. Always specify which you are reporting. Most journals recommend reporting SD (not SEM) as the measure of variability when describing the sample.

8.7 Contextualising Descriptives: Reference Values

Numerical descriptives are most useful when compared against reference benchmarks:

9. Visualising Numerical Data

9.1 Histogram

The histogram is the most fundamental visualisation for continuous numerical data. It divides the value range into contiguous bins of equal width and displays the frequency (or density) of observations in each bin as a bar.

Construction choices:

• Number of bins: Too few bins obscure distributional shape; too many create noisy fluctuations. Common rules:
  • Sturges' rule: $K = \lceil 1 + 3.322 \log_{10} n \rceil$
  • Freedman-Diaconis rule: Bin width $= 2 \times IQR \times n^{-1/3}$ (robust, recommended)
  • Scott's rule: Bin width $= 3.49 \times s \times n^{-1/3}$ (assumes normality)
• Frequency vs. density: Use density on the y-axis when overlaying a theoretical probability distribution curve.

Reading a histogram:

• Overall shape (symmetric, skewed, bimodal, uniform).
• Location of the peak (mode).
• Spread (width of the bulk of the distribution).
• Outliers (isolated bars far from the main body).

Best practices:

• Overlay a normal curve to assess departures from normality visually.
• Optionally add a rug plot (tick marks below the x-axis for each observation).
• Label the x-axis with the variable name and units; label the y-axis as "Frequency" or "Density".

9.2 Density Plot (Kernel Density Estimate)

The kernel density estimate (KDE) is a smoothed, continuous version of the histogram, constructed by placing a smooth kernel function (typically Gaussian) at each observed value and summing:

$\hat{f}(x) = \frac{1}{nh} \sum_{i=1}^n K\!\left(\frac{x - x_i}{h}\right)$

Where $h$ is the bandwidth (smoothing parameter). A larger $h$ produces a smoother curve; a smaller $h$ reveals more local features. DataStatPro uses Silverman's rule of thumb for the default bandwidth:

$h = 0.9 \times \min\!\left(s,\; \frac{IQR}{1.34}\right) \times n^{-1/5}$

Advantages over histogram: Continuous, smooth, and not dependent on bin boundary choices. Excellent for comparing multiple distributions on the same plot.

9.3 Box Plot

The box plot (box-and-whisker plot) is a compact, robust visualisation of the five-number summary and outliers:

• Box: Spans $Q_1$ to $Q_3$; width represents the IQR.
• Median line: Horizontal line inside the box at $\tilde{x}$.
• Whiskers: Extend to the most extreme observations within the Tukey fences ($Q_1 - 1.5 \times IQR$ and $Q_3 + 1.5 \times IQR$).
• Outlier points: Individual observations beyond the whisker fences are plotted as dots or circles.
• Notched box plot: V-shaped notches around the median line indicate an approximate 95% CI for the median. Non-overlapping notches suggest the medians of two groups differ significantly.

Best practices:

• Include a jittered strip of individual data points overlaid on the box plot when $n < 100$ (to avoid hiding the raw data).
• For group comparisons, align box plots side by side with a common y-axis.
• Always specify whether whiskers represent 1.5×IQR (Tukey, standard), 2×IQR, or min/max.

9.4 Violin Plot

The violin plot combines the compact shape of a box plot with the distributional detail of a density plot. Each "violin" is a mirrored kernel density estimate, showing the full distributional shape. A box plot or five-number summary is often overlaid.

Advantages over box plots: Reveals distributional shape (bimodality, skewness, gaps) that box plots conceal. Particularly useful for comparing distributions across multiple groups.

9.5 Q-Q Plot (Quantile-Quantile Plot)

The Q-Q plot assesses normality by plotting the sample quantiles against the theoretical quantiles of a standard normal distribution:

• If the data are approximately normal, points fall on or near the diagonal reference line.
• Systematic deviations from the diagonal indicate non-normality:
  • S-shaped curve: Over-dispersed (heavier tails than normal).
  • Banana-shaped curve: Under-dispersed (lighter tails).
  • Points above the line at both ends: Positive skew.
  • Points below the line at both ends: Negative skew.
  • Points far off the line at one end: Outliers.

DataStatPro overlays a 95% confidence band (Kolmogorov-Smirnov envelope) on the Q-Q plot. Points outside the band indicate significant departures from normality.

9.6 Stem-and-Leaf Plot

The stem-and-leaf plot is a text-based display that shows the full distribution while preserving every individual value. Each observation is split into a "stem" (leading digits) and a "leaf" (trailing digit). The stems are listed vertically, and the leaves are arranged horizontally.

Appropriate for: $n \leq 200$; exploratory data analysis; teaching contexts. The back-to-back stem-and-leaf plot compares two groups simultaneously.

9.7 Empirical Cumulative Distribution Function (ECDF)

The ECDF plots the proportion of observations less than or equal to each value $x$:

$\hat{F}(x) = \frac{1}{n}\#\{i : x_i \leq x\}$

The ECDF is a step function that rises by $1/n$ at each observed value. It is a non-parametric estimate of the true CDF and is used to:

• Identify percentiles visually (read the $y$-value corresponding to any $x$).
• Compare two distributions (Kolmogorov-Smirnov test is based on the maximum vertical distance between two ECDFs).
• Assess goodness-of-fit to a theoretical distribution.

9.8 Dot Plot / Strip Chart

A strip chart (also called a dot plot or jitter plot) displays every individual observation as a dot along a single axis, with a small random vertical jitter to prevent overplotting. It is the most information-rich plot for small to medium samples ($n \leq 200$).

Advantages: Shows the actual data; reveals gaps, clusters, and potential outliers not visible in summary plots. Highly recommended as a supplement to box plots.

9.9 Error Bar Plot

An error bar plot displays the mean (or median) as a point and uncertainty as symmetric bars. Common configurations:

⚠️ Always label the error bar type explicitly in figure captions. An unlabelled error bar is uninterpretable — ± SD, ± SEM, and ± 95% CI have very different meanings and widths. Many published figures fail to specify this information.

9.10 Visualisation Selection Guide

10. Confidence Intervals for Numerical Descriptives

10.1 Why Confidence Intervals Are Essential

Sample descriptive statistics are estimates of population parameters. A 95% confidence interval (CI) provides a range of plausible values for the true population parameter, given the observed sample statistic. CIs convey both the direction and the precision of an estimate, making them far more informative than a point estimate alone.

10.2 CI for the Mean: t-Distribution

For any numerical variable, the exact 95% CI for the population mean $\mu$ uses the t-distribution:

$CI_\mu = \bar{x} \pm t_{\alpha/2,\; n-1} \times \frac{s}{\sqrt{n}}$

Where $t_{\alpha/2,\; n-1}$ is the upper $\alpha/2$ critical value of the t-distribution with $n - 1$ degrees of freedom. This CI is exact when the population is normal and asymptotically valid for non-normal populations with sufficiently large $n$ (by the Central Limit Theorem).

Critical values for common $n$:

10.3 CI for the Mean: Bootstrap (Non-Normal Data)

When normality is violated and $n$ is small, a bootstrap CI for the mean provides better coverage. The percentile bootstrap proceeds as follows:

1. Draw $B = 2000$ (or more) bootstrap resamples of size $n$ with replacement.
2. Compute $\bar{x}^*_b$ for each resample $b = 1, \ldots, B$.
3. The 95% CI is the 2.5th and 97.5th percentiles of the $B$ bootstrap means: $CI_{boot} = [Q_{0.025}(\bar{x}^*), Q_{0.975}(\bar{x}^*)]$.

DataStatPro uses the bias-corrected and accelerated (BCa) bootstrap by default, which provides better coverage than the simple percentile bootstrap, especially for small samples.

10.4 CI for the Median: Bootstrap

No simple closed-form CI exists for the median. DataStatPro uses the BCa bootstrap CI:

1. Draw $B = 2000$ bootstrap resamples of size $n$ with replacement.
2. Compute $\tilde{x}^*_b$ for each resample.
3. The 95% CI is the BCa-corrected percentile interval of the $B$ bootstrap medians.

10.5 CI for the Standard Deviation: Chi-Square Distribution

The exact 95% CI for the population standard deviation $\sigma$ uses the chi-square distribution (assuming normality):

$CI_\sigma = \left[\sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2,\; n-1}}},\;\; \sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2,\; n-1}}}\right]$

⚠️ The chi-square CI for $\sigma$ is sensitive to departures from normality, more so than the t-CI for $\mu$. Use the bootstrap CI for $s$ when the distribution is clearly non-normal.

10.6 CI for the Variance

The exact 95% CI for the population variance $\sigma^2$:

$CI_{\sigma^2} = \left[\frac{(n-1)s^2}{\chi^2_{1-\alpha/2,\; n-1}},\;\; \frac{(n-1)s^2}{\chi^2_{\alpha/2,\; n-1}}\right]$

10.7 CI Width as a Function of $n$ and $s$

The width of the 95% CI for the mean is approximately $2 \times 1.96 \times s/\sqrt{n}$. For a given $s$, quadrupling $n$ halves the CI width:

Required $n$ to achieve target CI half-width $\delta$ (95% CI, $\alpha = .05$):

$n \approx \left(\frac{1.96 \times s}{\delta}\right)^2$

Example: If $s = 15$ (e.g., IQ scores), to achieve a CI half-width of ±3 points:

$n \approx \left(\frac{1.96 \times 15}{3}\right)^2 = (9.8)^2 \approx 96$

10.8 Confidence Intervals for the Geometric Mean

For log-normally distributed data, compute the CI on the log scale and transform back:

1. Compute $\bar{y} = \frac{1}{n}\sum_{i=1}^n \ln x_i$ and $s_y = SD(\ln x_i)$.
2. 95% CI on log scale: $\bar{y} \pm t_{\alpha/2,\; n-1} \times s_y/\sqrt{n}$.
3. Transform back: $CI_{geom} = [\exp(\bar{y} - t \cdot s_y/\sqrt{n}),\; \exp(\bar{y} + t \cdot s_y/\sqrt{n})]$.

11. Advanced Topics

11.1 Subgroup Comparisons and Stratified Descriptives

When the variable of interest is examined across levels of a grouping variable (e.g., treatment vs. control), stratified descriptives provide within-group summaries and enable direct comparison of central tendency, spread, and shape across groups.

A useful compact format is the comparative descriptives table:

⚠️ Descriptive comparisons between groups do not constitute formal hypothesis tests. A large apparent difference in means may not be statistically significant (underpowered study), and a small apparent difference may be highly significant (very large $n$). Always follow descriptive comparisons with appropriate inferential tests (t-test, Mann-Whitney U, ANOVA).

11.2 Data Transformations for Non-Normal Variables

When a variable is substantially non-normal, transforming it can improve the interpretability of mean-based descriptives and the validity of downstream parametric tests.

Common transformations:

⚠️ After transformation, descriptives are computed and reported on the transformed scale. The geometric mean on the original scale equals the antilog of the arithmetic mean on the log scale. Always state explicitly that transformed-scale descriptives are being reported and provide the back-transformed mean (geometric mean) for interpretability.

11.3 Robust Descriptive Statistics

When data contain outliers or are drawn from heavy-tailed distributions, robust estimators are preferred over classical mean-based measures:

The breakdown point is the proportion of data that can be replaced by arbitrarily large values before the estimator becomes unreliable.

11.4 Effect Size: Cohen's $d$ and Standardised Differences

When comparing the means of two groups, Cohen's $d$ standardises the mean difference by the pooled standard deviation, producing an interpretable, scale-free effect size:

$d = \frac{\bar{x}_1 - \bar{x}_2}{s_{pooled}}, \qquad s_{pooled} = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}$

Cohen's conventions for $|d|$:

| $|d|$ | Effect Size Label | | :---- | :---------------- | | $0.20$ | Small | | $0.50$ | Medium | | $0.80$ | Large |

11.5 Standardised Scores and Norm-Referencing

Z-scores standardise raw values to a common scale with mean 0 and SD 1, enabling cross-variable and cross-population comparisons. Many applied contexts use T-scores (mean 50, SD 10) or IQ-type scaled scores (mean 100, SD 15):

$T_i = 50 + 10 \times z_i = 50 + 10 \times \frac{x_i - \bar{x}}{s}$

$IQ_i = 100 + 15 \times z_i$

Percentile ranks from the z-score (under normality): $PR = \Phi(z_i) \times 100$, where $\Phi$ is the standard normal CDF.

11.6 The Central Limit Theorem and its Implications

The Central Limit Theorem (CLT) states that for any population with finite mean $\mu$ and variance $\sigma^2$, the sampling distribution of $\bar{x}$ approaches a normal distribution as $n \to \infty$:

$\bar{x} \sim \mathcal{N}\!\left(\mu,\; \frac{\sigma^2}{n}\right) \quad \text{approximately, for large } n$

Practical implications for descriptives:

• For $n \geq 30$ and mildly non-normal populations, the t-CI for $\mu$ is approximately valid.
• For highly skewed distributions or $n < 30$, bootstrap CIs are preferred.
• The CLT justifies reporting the mean and SD even for non-normal data, provided $n$ is sufficiently large and the goal is inference about $\mu$.

11.7 Detecting Bimodality

A bimodal distribution has two distinct peaks, often indicating the presence of two subpopulations. Indicators of bimodality:

• Histogram shows two visible humps.
• Large positive kurtosis alone does not imply bimodality.
• Hartigan's dip test formally tests for unimodality against multimodal alternatives.
• Bimodality coefficient $BC = (G_1^2 + 1) / (G_2 + 3(n-1)^2/((n-2)(n-3)))$; $BC > 0.555$ suggests bimodality (Pfister et al., 2013).
• Gaussian mixture models can formally decompose a bimodal distribution into component distributions.

When bimodality is detected, computing a single mean and SD is misleading — report descriptives separately for each subpopulation if they can be identified.

11.8 Process Capability Indices

In quality control and manufacturing, process capability indices relate the distribution of a measured variable to specification limits ($LSL$ = lower, $USL$ = upper):

$C_p = \frac{USL - LSL}{6s}, \qquad C_{pk} = \min\!\left(\frac{USL - \bar{x}}{3s},\; \frac{\bar{x} - LSL}{3s}\right)$

• $C_p \geq 1.33$: Process capable (if centred).
• $C_{pk} \geq 1.33$: Process capable and well-centred.
• $C_{pk} < 1$: Process produces unacceptable proportion of out-of-specification output.

DataStatPro computes and reports $C_p$, $C_{pk}$, $C_{pm}$ (Taguchi index), and the estimated proportion out-of-specification when specification limits are supplied.

11.9 Temporal Trends and Rolling Descriptives

When a numerical variable is measured repeatedly over time, rolling (moving window) descriptives track how central tendency and variability change:

$\bar{x}_{rolling,t} = \frac{1}{w}\sum_{i=t-w+1}^{t} x_i$

Where $w$ is the window width. Rolling means, medians, and standard deviations are plotted over time to reveal trends, seasonal patterns, and structural breaks. DataStatPro supports rolling descriptives with user-specified window width.

11.10 Sensitivity Analysis: Influence of Outliers

To assess the influence of potential outliers on key descriptives, a sensitivity analysis reports descriptives both including and excluding flagged outliers:

A large $\Delta\bar{x}$ with a small $\Delta\tilde{x}$ confirms that the outlier is exerting disproportionate leverage on the mean. Report both full-data and outlier-excluded descriptives when outliers are detected, along with a justification for any exclusions.

12. Worked Examples

Example 1: Symmetric Distribution — Exam Scores

A class of $n = 25$ students sits an exam (0–100 marks). Three students were absent (missing); valid responses: $n_{valid} = 22$.

Raw scores (sorted):

42, 51, 55, 58, 61, 63, 65, 67, 68, 70, 71, 72, 73, 74, 75, 76, 78, 80, 82, 85, 88, 93

Step 1 — Central Tendency:

$\bar{x} = \frac{42 + 51 + \cdots + 93}{22} = \frac{1527}{22} = 69.41$

Median ($n = 22$, even): Average of 11th and 12th values:

$\tilde{x} = \frac{71 + 72}{2} = 71.50$