Descriptive Statistics: Definition, Overview, Types, and Examples

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Descriptive statistics are numerical and graphical summaries that describe the main features of a dataset, such as its typical value, how widely the values spread out and how often each value occurs, without drawing conclusions about anything beyond the data itself. The dataset can be an entire population or a sample. A mean, a standard deviation or a frequency count tells you what the collected numbers look like, not what will happen next or what is true of a larger group.

What Descriptive Statistics Measure

A raw dataset of hundreds or thousands of values is hard to read. Descriptive statistics compress it into a handful of figures that answer three questions: where is the center, how spread out are the values, and how are they distributed?

Measures of central tendency

Central tendency describes what is typical in a dataset.

• Mean (arithmetic average): add all values and divide by the number of values. For data x₁ through xₙ, mean = (Σx) / n.
• Median: the middle value once the data are sorted. With an even number of values, it is the average of the two middle values. Half the observations sit above it and half below.
• Mode: the value that appears most often. A dataset can have one mode, several, or none.

Take the dataset (2, 2, 3, 5, 8). The sum is 20, so the mean is 20 / 5 = 4. The median is 3, the third of five sorted values. The mode is 2, the only repeated value. Three different “centers” from the same five numbers is normal: which one best represents the data depends on how the values are distributed. In a symmetric distribution the three tend to sit close together. In a skewed one they separate, and the median usually describes the typical observation better than the mean.

Measures of variability

Two datasets can share the same mean and look nothing alike. An average test score of 65 out of 100 could come from scores tightly packed between 60 and 70, or from scores scattered from 1 to 100. Measures of variability (also called measures of spread or dispersion) capture that difference.

• Range: maximum minus minimum. For (5, 19, 24, 62, 91, 100), the range is 100 − 5 = 95. It is quick to compute but depends entirely on the two most extreme values.
• Quartiles and interquartile range (IQR): quartiles split sorted data into four equal parts. Q1 marks the 25th percentile, Q2 the median and Q3 the 75th percentile. The IQR, Q3 − Q1, measures the spread of the middle half of the data and ignores the extremes.
• Mean absolute deviation: the average distance of each value from the mean, ignoring sign.
• Variance: the average of the squared deviations from the mean. Population variance divides the sum of squared deviations by n. Sample variance divides by n − 1, because a sample’s values tend to sit closer to its own mean than to the true population mean, so dividing by n would understate the spread. The adjustment matters most for small samples; with large datasets the two results converge.
• Standard deviation: the square root of the variance. Because it is in the same units as the data, it is easier to interpret than variance.
• Coefficient of variation: standard deviation divided by the mean, which allows spread to be compared across datasets with different scales.

Frequency distribution

A frequency distribution records how often each value or category occurs. It works for numbers grouped into intervals and for categorical data that cannot be averaged at all.

Consider a list of six account types: checking, checking, savings, savings, savings, brokerage. The distribution is:

• Checking: 2 (33.3% of the total)
• Savings: 3 (50%)
• Brokerage: 1 (16.7%)
• Not checking: 4 (66.7%)

The percentages are relative frequencies, or proportions: each count divided by the total. Proportions let you compare categories across datasets of different sizes. For categorical data like this, counts, proportions and the mode (savings here) are the meaningful summaries, while a mean or standard deviation would make no sense.

Shape of the distribution

Two further measures describe shape:

• Skewness measures asymmetry. Positive (right) skew means a long tail of high values pulls the mean above the median. Negative (left) skew means the opposite.
• Kurtosis measures how heavy the tails are relative to a normal distribution. Higher kurtosis means extreme values occur more often than a bell curve would suggest.

Worked Example: Summarizing Monthly Returns

Hypothetical example for illustration only. Suppose a portfolio posts these six monthly returns: 2%, −1%, 3%, 1%, 4%, −3%.

Mean. The sum is 2 − 1 + 3 + 1 + 4 − 3 = 6, so the mean is 6 / 6 = 1% per month. This is the arithmetic mean of the monthly figures; it is not the compounded (geometric) return over the six months, which would be slightly different.

Median. Sorted, the returns are −3, −1, 1, 2, 3, 4. The two middle values are 1 and 2, so the median is 1.5%.

Mode. No value repeats, so there is no mode.

Range. 4 − (−3) = 7 percentage points.

Variance and standard deviation. The deviations from the 1% mean are 1, −2, 2, 0, 3 and −4. Squared, they are 1, 4, 4, 0, 9 and 16, which sum to 34.

• Treating the six months as the whole population: variance = 34 / 6 ≈ 5.67, standard deviation ≈ 2.38%.
• Treating them as a sample of the portfolio’s longer history: variance = 34 / 5 = 6.8, standard deviation ≈ 2.61%.

Z-score. A z-score expresses how many standard deviations a value sits from the mean: z = (x − mean) / standard deviation. Using the sample figure, the −3% month has z = (−3 − 1) / 2.61 ≈ −1.53, meaning it was about one and a half standard deviations below the average month.

Read together, these figures say the portfolio averaged a modest gain with monthly swings of a few percentage points, and the median sitting above the mean hints that the losing months pulled the average down. They describe the six months; they do not forecast the seventh.

Univariate, Bivariate and Multivariate Data

Univariate analysis looks at one variable at a time. The average age of everyone in a room, or the standard deviation of a single stock’s daily returns, are univariate statistics.

Bivariate analysis examines two variables together to see whether they move in relation to each other, for example a stock’s returns compared with those of the S&P 500. Common bivariate descriptive measures include:

• Covariance: whether two variables tend to rise and fall together (positive) or in opposite directions (negative).
• Correlation coefficient: covariance scaled to a range of −1 to +1, so the strength of a linear relationship can be compared across pairs.
• Cross-tabulation: a table of joint frequencies for two categorical variables.

When more than two variables are analyzed at once, the analysis is multivariate.

Visualizing Descriptive Statistics

• Histogram: groups numeric data into ranges called bins and draws a bar for the count in each, showing whether values cluster in the center, spread evenly or lean to one side.
• Box plot (box-and-whisker plot): draws a box from Q1 to Q3 with a line at the median, whiskers extending toward the extremes, and individual points for outliers. Side-by-side box plots compare distributions across groups.
• Scatter plot: plots two variables against each other and is the standard picture of a bivariate relationship.
• Line graph: tracks a variable over time, such as a price or U.S. inflation rates by year.
• Bar and pie charts: show counts or proportions for categories.
• Stem-and-leaf display: a text-based layout that sorts values while keeping each individual number visible, useful for small datasets.

Outliers and How They Distort the Summary

An outlier is an observation that differs markedly from the rest of the dataset. It might be a data-entry error, a measurement problem or a genuine but rare event.

Outliers affect some statistics far more than others. In the dataset (1, 1, 1, 997), the mean is 1,000 / 4 = 250, a number that describes none of the observations. The median is 1, which matches three of the four values. The mean, range, variance and standard deviation are all sensitive to extremes; the median and IQR are robust, meaning a single extreme value barely moves them. That is why income and home-price data are typically reported as medians.

Two common ways to flag outliers:

• IQR method: a value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is treated as an outlier. Box plots use this rule to place their whiskers.
• Z-score method: values with an absolute z-score above a chosen cutoff, often 3, are flagged. Because the mean and standard deviation are themselves pulled by outliers, this method can understate how unusual an extreme point is in a small dataset.

Whether to remove an outlier depends on why it exists. A typo should be corrected or dropped. A genuine extreme, such as a market crash day in a return series, may be the most informative observation in the set, and deleting it would make the data look safer than it was. Reporting results with and without the outlier is one way to show its effect.

Descriptive vs. Inferential Statistics

Descriptive statistics summarize what was observed. A retailer that reports its number of sales, average order size and average revenue by day of the week is describing what happened.

Inferential statistics use a sample to draw conclusions about a larger population or about the future. If the same retailer uses its sales history to estimate demand for a new product, or tests whether a price change raised revenue by more than chance would explain, it has moved into inference. Inference relies on probability tools such as confidence intervals, hypothesis tests, p-values and significance levels.

Descriptive analysis uses no significance level because it makes no claim beyond the data. A sample mean is descriptive when reported as “the average of these 200 survey responses” and becomes an input to inference when used to estimate the average for all customers.

Financial reporting offers a clean illustration. Preparing an income statement, with total revenue, net income and earnings per share, is descriptive. Using those figures to project next year’s earnings or value the company is inferential.

How Descriptive Statistics Are Used in Finance

Much of everyday financial data is descriptive statistics under another name:

• Averages and growth rates: average daily trading volume, average revenue growth, and year-over-year changes summarize performance over a defined period.
• Volatility: the standard deviation of returns is the standard measure of how much an asset’s price has fluctuated. Annualized historical volatility is a descriptive statistic of past returns.
• Valuation comparisons: the median P/E ratio of a sector is a reference point for judging whether a company trades rich or cheap relative to peers. The median is preferred because a few companies with very high or negative earnings can distort the mean.
• Distribution of returns: skewness and kurtosis describe whether losses or gains have tended to come in rare large bursts.
• Screening: a stock screener applies descriptive cutoffs, such as percentile, average or range on dividend yield or market cap, to a universe of stocks.

In research reports and academic papers, a “descriptive statistics” table usually appears before any modeling, listing the count, mean, standard deviation, minimum, maximum and sometimes median for each variable so readers can see what the data look like.

Limitations

• No conclusions beyond the data. Treating historical averages as forecasts is a step into inference without the tools that quantify uncertainty.
• Summaries hide detail. Very different datasets can share the same mean and standard deviation. Without a chart or shape measures, bimodal or heavily skewed data can look ordinary.
• Dependence on choices. The period selected, the bin width of a histogram, sample versus population formulas and the treatment of outliers all change the output. Two analysts can describe the same raw data differently.
• Correlation is not causation. Bivariate measures describe co-movement, not cause and effect.
• Data quality. A summary is only as accurate as the data behind it. Errors, missing values and unrepresentative samples carry straight through into every statistic.

FAQs

The Bottom Line

Investors, analysts and anyone reading financial data rely on descriptive statistics through measures like volatility, median P/E ratios and average trading volume. These figures show what past data looked like, so treating them as forecasts requires inferential tools that quantify uncertainty. Pairing summaries with charts, choosing robust measures when outliers exist and stating sample or population formulas gives a more honest picture.