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Descriptive Statistics — Definition, Formula & Examples

Descriptive statistics is the branch of statistics that summarizes, organizes, and displays data from a sample or population without drawing conclusions beyond the data itself. Common descriptive measures include the mean, median, standard deviation, and range.

Descriptive statistics encompasses the numerical measures (such as measures of central tendency, variability, and position) and graphical representations (such as histograms, boxplots, and stem-and-leaf plots) used to characterize the distribution of a dataset. Unlike inferential statistics, descriptive statistics makes no generalizations to a larger population or probability claims; it strictly quantifies and portrays the observed data.

Key Formula

xˉ=∑i=1nxins=∑i=1n(xi−xˉ)2n−1\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \qquad s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}}
Where:
  • xˉ\bar{x} = Sample mean
  • xix_i = Each individual data value
  • nn = Number of data values in the sample
  • ss = Sample standard deviation

How It Works

Descriptive statistics works in three layers. First, you measure the center of your data — typically with the mean or median — to capture a "typical" value. Second, you measure the spread using the range, interquartile range, or standard deviation to show how much values vary. Third, you describe the shape of the distribution: is it symmetric, skewed left, or skewed right? Are there outliers? Graphs like histograms and boxplots let you see all three layers at once. In practice, you almost always compute descriptive statistics before attempting any inference, because understanding what you have is a prerequisite to drawing valid conclusions.

Worked Example

Problem: A teacher records quiz scores for 7 students: 60, 70, 75, 80, 85, 90, 100. Compute the key descriptive statistics: mean, median, range, and standard deviation.
Step 1 — Mean: Add all values and divide by the number of values.
xˉ=60+70+75+80+85+90+1007=5607=80\bar{x} = \frac{60+70+75+80+85+90+100}{7} = \frac{560}{7} = 80
Step 2 — Median: With 7 values already in order, the median is the 4th value.
Median=80\text{Median} = 80
Step 3 — Range: Subtract the minimum from the maximum.
Range=100−60=40\text{Range} = 100 - 60 = 40
Step 4 — Standard Deviation: Find each squared deviation from the mean, sum them, divide by n − 1, and take the square root.
s=(60−80)2+(70−80)2+(75−80)2+(80−80)2+(85−80)2+(90−80)2+(100−80)26=400+100+25+0+25+100+4006=10506=175≈13.23s = \sqrt{\frac{(60-80)^2+(70-80)^2+(75-80)^2+(80-80)^2+(85-80)^2+(90-80)^2+(100-80)^2}{6}} = \sqrt{\frac{400+100+25+0+25+100+400}{6}} = \sqrt{\frac{1050}{6}} = \sqrt{175} \approx 13.23
Answer: Mean = 80, Median = 80, Range = 40, Standard Deviation ≈ 13.23. Together these four values summarize the center and spread of the quiz scores.

Visualization

Why It Matters

In AP Statistics, nearly every free-response question begins with "describe the distribution," which requires descriptive statistics. Data analysts, medical researchers, and economists rely on descriptive measures to audit data quality and spot patterns before running any hypothesis test. Mastering these basics ensures you can communicate findings clearly in any field that uses data.

Common Mistakes

Mistake: Using the mean to describe a skewed dataset without mentioning the median.
Correction: When a distribution is skewed or has outliers, the median and IQR are more resistant measures of center and spread. Always report the median alongside the mean for skewed data, and note which better represents a typical value.
Mistake: Dividing by n instead of n − 1 when computing the sample standard deviation.
Correction: The formula with n in the denominator gives the population standard deviation (σ). For a sample, divide by n − 1 to get an unbiased estimate (s). AP Stats problems almost always use n − 1.