Descriptive Statistics with Python

Descriptive statistics help us summarize and understand data characteristics. These methods transform raw data into useful summaries that show patterns, typical values, and variability. They are a practical starting point for statistical analysis and machine learning because they help reveal structure, anomalies, and questionable assumptions before modeling [1].

Before moving far into modeling or inference, it is good practice to examine the basic properties of your data. Descriptive statistics provide this early view by showing where distributions are centered, how widely values vary, and what shape they take. These summaries help us assess data quality, spot potential issues such as outliers or skewness, and communicate findings to readers who may not need the underlying technical detail.

Descriptive statistics provide different types of summary information about data, organized into several key categories:

• Measures of central tendency (e.g., mean, median, mode): Identify typical or representative values in a dataset.
• Measures of variability (e.g., variance, standard deviation, range, interquartile range): Quantify how spread out the data points are.
• Measures of distribution shape (e.g., skewness, kurtosis): Describe the asymmetry and tail behavior of the data.

Understanding these descriptive measures matters for exploratory data analysis and for diagnosing potential problems in modeling. For instance, high skewness might suggest the need for data transformation, while extreme kurtosis could indicate the presence of outliers that warrant investigation. This chapter explores each of these fundamental concepts, giving the statistical basics needed for effective data analysis and modeling.

Central tendency measures answer a key question: what is a typical value in this dataset? The mean and median answer this question for numerical data, while the mode identifies the most frequent value. The right choice depends on the data's properties and the analysis goals [2].

The arithmetic mean, commonly called the average, is calculated by summing all values and dividing by the count of observations:

Where:

• $\bar{x}$: Sample mean
• $x_i$: Individual observation
• $n$: Number of observations

The mean represents the balance point of a distribution and incorporates information from every data point. Under a normal model it is an efficient estimator of location, but symmetry alone is not enough to guarantee that property [2]. Its sensitivity to all values also makes it vulnerable to outliers: a single extremely large or small value can pull the mean far from where most observations cluster.

For example, consider household income in a neighborhood where most families earn between $40,000 and$80,000 annually, but one household earns $5 million. The mean income might be$150,000, far above what most residents earn, while the median would remain around $60,000, better representing the typical household.

The median is the middle value when observations are arranged in order. For datasets with an odd number of observations, it is simply the center value; for even counts, it is the average of the two middle values:

Where $x_{(i)}$ denotes the $i$-th value in the sorted dataset.

For example, consider the sorted dataset: 12, 15, 18, 22, 25, 28, 31. With seven observations (odd count), the median is the fourth value: 22. If we add an eighth observation to get 12, 15, 18, 22, 25, 28, 31, 45, the median becomes the average of the fourth and fifth values: (22 + 25) / 2 = 23.5.

The median's main advantage is its resistance to the magnitude of extreme tail values. Moving an already extreme observation farther into the tail does not change its rank and usually leaves the median unchanged; changing observations near the middle can still change it. This makes the median particularly useful for skewed distributions or data with outliers.

For a discrete sample, the mode is any value with the highest observed frequency. Two values tied for that maximum give two sample modes, as when exam scores of 75 and 90 each occur eight times and no other score occurs more than five times. For a continuous distribution, modes are peaks or local maxima of its density. Multiple peaks may suggest a mixture of subgroups, but they do not prove one, and the peaks seen in a histogram or KDE can change with bin width or smoothing bandwidth [3]. If every observed value has the same frequency, all values formally tie; many introductory summaries report that situation as having no unique mode.

The mode is especially useful for nominal categorical data, where mean and median are undefined, such as the most common color preference in a survey or the most frequent diagnosis in a medical dataset. Ordered categorical data may also support a median. For continuous numerical data, estimated density modes can help identify peaks, although their locations depend on the estimation method.

In practice, for continuous data, we often examine the mode through histograms or kernel density estimates rather than computing the exact most frequent value, especially when data have many unique values.

While central tendency measures locate the center of a distribution, measures of spread quantify how much data points vary around that center. Two datasets can have identical means but very different spreads, so center and scale should be examined together [5].

Population variance is the mean squared deviation from the population mean. For a random sample, the following corrected sample variance divides the sum of squared deviations by $n-1$:

Where:

• $s^2$: Sample variance
• $x_i$: Individual observation
• $\bar{x}$: Sample mean
• $n$: Number of observations

For $n \geq 2$, division by $n-1$ is commonly called Bessel's correction. Under an independent, identically distributed sampling model with finite variance, it makes $s^2$ an unbiased estimator of the population variance; it does not make $s$ an unbiased estimator of the population standard deviation [6]. For independent variables with finite variances, the variance of their sum equals the sum of their variances. Because variance has squared units, such as dollars squared, direct interpretation can be awkward.

Standard deviation addresses this interpretation issue by taking the square root of variance, returning to the original units:

Standard deviation expresses dispersion in the data's original units. Its distance interpretation is clearest for distributions that are roughly symmetric and not heavy-tailed. For a normal distribution, about 68% of probability lies within one population standard deviation of the mean, 95% within two, and 99.7% within three [7]. These percentages are approximations for data whose distribution is only approximately normal, not a distribution-free outlier rule.

The range is the simplest measure of spread, calculated as the difference between the maximum and minimum values:

While easy to compute and interpret, the range uses only two data points and is extremely sensitive to outliers. A single erroneous or extreme value can inflate the range dramatically, making it an unreliable measure of typical spread.

The interquartile range (IQR) is less sensitive to extreme values because it measures the spread of the middle 50% of the data:

Where:

• $Q_1$: First quartile (25th percentile)
• $Q_3$: Third quartile (75th percentile)

Because the IQR focuses on the central half of the ordered values, changing an already extreme tail value usually leaves it unchanged. The conventional inner-fence rule flags values below $Q_1 - 1.5 \times \text{IQR}$ or above $Q_3 + 1.5 \times \text{IQR}$ as potential outliers. A flag is a prompt for investigation, not a reason to delete an observation automatically [8].

For example, consider a dataset of daily website visits over a week: 120, 135, 142, 128, 155, 130, 138. We can calculate several descriptive measures of center and spread:

• Mean: (120 + 135 + 142 + 128 + 155 + 130 + 138) / 7 = 135.43
• Median (middle sorted value): 135
• Mode: No unique mode (all values occur once)
• Range: 155 - 120 = 35
• IQR: Q3 - Q1 = 142 - 128 = 14 (exclusive median-of-halves convention, omitting the overall median)
• Standard Deviation: 11.22

The mean and median are close, although that alone does not establish symmetry. The range is 35, the IQR is 14 under the exclusive median-of-halves convention, and the sample standard deviation is 11.22. Whether that spread is large or small depends on the scale and application.

Beyond center and spread, distribution shape gives useful information about data behavior. Two common measures, skewness and kurtosis, summarize asymmetry and tail weight, respectively [9].

One common sample measure of skewness uses central moments computed with a consistent $1/n$ denominator:

For nonconstant data, $m_2>0$ and the statistic $g_1$ is defined and unitless. Some software instead reports a finite-sample bias-adjusted estimator, so comparisons should use the same convention.

A nondegenerate symmetric distribution with a finite third moment has skewness zero, but zero skewness does not by itself prove symmetry. Positive skewness, or right skew, often accompanies a longer or heavier right tail. In many familiar unimodal cases the mean then exceeds the median, but that ordering is not guaranteed for every distribution.

Negative skewness, or left skew, often accompanies a longer or heavier left tail. Test scores on an easy exam can show this pattern when many students score near the upper boundary and a smaller number receive much lower scores.

There is no universal cutoff that turns a skewness value into “moderate” or “high.” Interpret its magnitude relative to the estimator, sample size, application, and the accompanying distribution plot.

Moment kurtosis standardizes the fourth central moment:

For nonconstant data, $m_2>0$ and $g_2$ is defined. Because fourth powers amplify large standardized deviations, kurtosis is sensitive to extreme observations. It is not a direct partition of probability between the center and tails, nor does it determine whether a density has a sharp or flat peak [10]. A normal distribution has population kurtosis 3. Many software packages instead report excess kurtosis, $g_2-3$, and may also apply a finite-sample bias correction; under the excess convention the normal reference is zero.

High kurtosis means that extreme standardized deviations contribute strongly to the fourth moment. It can be a warning that a normal model understates important extremes, but it does not guarantee a larger exceedance probability at every threshold. Risk analysis should therefore examine relevant tail probabilities and quantiles directly rather than rely on kurtosis alone.

Low kurtosis means that extreme standardized deviations make a smaller fourth-moment contribution than under the chosen reference. A continuous uniform distribution, for example, has raw kurtosis 1.8. This value does not by itself characterize the height or flatness of its density peak.

Kurtosis is one diagnostic for departures from a normal model, not a method-selection rule. When extremes matter, inspect distribution plots, tail quantiles, and model residuals and then choose methods whose assumptions fit the task.

Visualizing descriptive statistics makes these concepts clearer, helping us see patterns more easily. The following examples show how distributions can differ in their centers and spreads as well as in their overall shapes.

Example: Comparing Distributions with Different Properties

This visualization compares three distributions: a symmetric normal distribution, a right-skewed distribution, and a high-kurtosis distribution with heavy tails. Each plot includes reference lines showing the mean with a solid line and the median with a dashed line to illustrate how these measures respond to distributional shape.

Which descriptive statistics to report depends on the data's characteristics and your analysis goals. Consider these guidelines:

• For roughly symmetric, unimodal distributions without influential extremes: Mean and standard deviation are often useful summaries. Their adequacy still depends on the analytical question and the distribution's tails.
• For skewed distributions or data with influential extremes: Median and IQR are useful resistant summaries. Retain the mean when the target is an expectation, total, or cost, and consider reporting both conventional and resistant summaries.
• For nominal categorical data: Report frequencies or proportions and, when useful, the mode. For discrete counts, the mean, median, quantiles, frequencies, or modes may be appropriate depending on the question. When a plot has several stable peaks, report and investigate them rather than compressing the distribution into one center.

Reporting several compatible statistics usually gives a more useful picture than relying on one measure. A difference between mean and median can suggest asymmetry, while a large difference between standard deviation and IQR can prompt investigation of tails or outliers; neither comparison proves the cause. Skewness and kurtosis add compact shape information, but plots and context are still needed to avoid misleading simplifications.

Descriptive statistics are widely useful across data science and analytics:

• Quality Control Manufacturing: Control charts monitor process statistics over time and flag patterns that warrant investigation [12]. Their control limits describe expected process variation and are not the same as engineering specification limits, so a statistically stable process can still produce items outside specification [13].
• Healthcare Research: Survival quantiles can summarize treatment outcomes, but censoring must be handled with a survival estimator such as Kaplan–Meier rather than an ordinary sample median. A median is estimable only if the fitted survival curve falls to 0.5 or below during follow-up; restricted mean survival time may better match some clinical questions.
• Financial Analysis: Return distributions are examined for asymmetry, heavy tails, and other departures from normal models. Skewness and kurtosis can summarize some of those features, but risk decisions also require horizon-specific tail probabilities, quantiles, and loss measures [14].
• Marketing Analytics: Purchase amounts, time on site, and conversion rates can have quite different distributions. Analysts should inspect each metric rather than assume a common shape; when purchase amounts are strongly right-skewed, median and upper quantiles can complement the mean.
• Climate Science: Temperature distributions are described using centers, spreads, and tail measures across stated locations, seasons, and reference periods. A trend in the mean can indicate a location shift. Changes in exceedance rates above fixed, reference-period thresholds address whether defined extremes have become more common, while changes in tail quantiles address whether their magnitudes have shifted. Variance alone cannot establish either conclusion [15].

These examples lead to several practical guidelines for working with descriptive statistics effectively.

Use visualization alongside summary statistics, iterating between them as the analysis develops. Histograms can expose multimodality, box plots can flag potential outliers, and scatter plots can reveal gaps and relationships. Anscombe's quartet demonstrates that datasets can share the same means, variances, correlation, and fitted regression line while having visibly different structures [16].

Report uncertainty alongside inferential point estimates when possible. A sample mean is more informative about a population mean when accompanied by a standard error or confidence interval; a standard deviation instead describes variation among observations [17]. For small samples, acknowledge that estimates can change substantially with additional data.

Document your choices about handling missing values and outliers. Different approaches, such as excluding missing data, imputing values, or trimming extremes, can produce substantially different summaries. Transparency about these decisions allows others to evaluate and reproduce your analysis.

Choose the target estimand before combining percentages or ratios. If the goal is the pooled item-level return rate and one store has a 10% rate on 1,000 items while another has a 20% rate on 100 items, denominator weighting gives $(100+20)/(1000+100) \approx 10.9\%$, not 15%. An equal-store average answers a different question and may be appropriate when stores, rather than items, are the units of interest.

Standard deviation retains its mathematical meaning as the square root of variance, but a symmetric “mean ± one standard deviation” interpretation can mislead for strongly skewed data. The interval is neither a general coverage guarantee nor a bound on possible values. For nonnegative, right-skewed quantities such as many income measures, report the median and selected quantiles to show asymmetry and coverage; the IQR adds a resistant summary of central spread.

Beware of Simpson's paradox, where an association in aggregated data reverses or disappears after stratification by a third variable [18]. The grouping variable is not automatically a causal confounder; that interpretation needs a causal argument [19]. Computing summaries at relevant levels of aggregation helps reveal such reversals.

Descriptive statistics give us tools to understand and communicate data characteristics. The mean and median summarize numerical location, while frequencies and the mode summarize categorical prevalence. Variance and standard deviation quantify variability using every observation; the range and interquartile range describe it through ordered endpoints or quartiles. Skewness and kurtosis summarize aspects of asymmetry and extreme-deviation behavior, but they do not fully determine distribution shape or model suitability.

A single descriptive statistic rarely captures every feature relevant to an analysis. Effective summarization means choosing statistics that match the data and question, then pairing them with appropriate plots. Roughly symmetric, unimodal data without influential extremes are often summarized well by mean and standard deviation; skewed or outlier-prone data often benefit from median and IQR. Skewness and kurtosis are diagnostics, not automatic selectors of parametric methods or transformations.

Descriptive statistics involves calculation, but its real purpose is to develop the judgment needed to select appropriate summaries, recognize the patterns they reveal, and communicate findings clearly. These skills form the foundation for subsequent statistical analysis and machine learning, including data-driven decisions.

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Ready to test your understanding of descriptive statistics? Take this quiz to reinforce what you've learned about measures of central tendency, spread, and distribution shape.