Descriptive statistics summarize observed data. Use measures of center (mean, median), spread (range, IQR, standard deviation), and shape (skew, outliers) together, because one number alone can hide important patterns.
- Report center and spread together for a more honest summary.
- Median and IQR are usually better than mean and standard deviation for strongly skewed data.
- A graph plus a numeric summary is more reliable than either one alone.
Center and spread answer different questions
Center describes a typical value, while spread describes how far values vary around that center. Mean uses every value and reacts to outliers. Median is the middle ordered value and is more resistant to extreme observations.
For spread, range is max minus min, but it depends only on two points. IQR = Q3 - Q1 captures the middle 50 percent. Standard deviation measures the root-mean-square distance from the mean and is common when data are roughly symmetric.
Shape and outliers change interpretation
A right-skewed distribution has a long right tail, often making mean larger than median. In this case, reporting only mean can overstate what is typical.
A boxplot or histogram can show clusters, gaps, and outliers that summary numbers miss. A common outlier flag is values below Q1 - 1.5*IQR or above Q3 + 1.5*IQR.
- Use mean with standard deviation for near-symmetric data.
- Use median with IQR for skewed data.
- Note whether unusual values are data errors or real observations.
Summarize weekly study hours
A class records weekly study hours: 2, 3, 3, 4, 5, 6, 18.
- Order the values and compute median as the middle value, which is 4.
- Compute mean: (2+3+3+4+5+6+18)/7 = 41/7 = 5.86.
- Find quartiles: Q1 = 3, Q3 = 6, so IQR = 3.
- Compare mean and median and inspect the high value 18 as a potential outlier.
Common mistakes
- Reporting only the mean for skewed data.
- Using range as the only measure of spread.
- Treating every outlier as a mistake without checking context.
- Comparing groups without checking whether units and scales match.
Try one
When data are strongly right-skewed, which pair is usually better to report: mean and standard deviation, or median and IQR?
Median and IQR, because they are less distorted by extreme high values.
Sources
- OpenStax: Introductory StatisticsOpen textbook covering descriptive statistics, probability, inference, and regression fundamentals.
- Khan Academy: Statistics and probabilityClear instructional coverage of core statistics ideas with worked examples and practice.