Descriptive Statistics
Descriptive statistics summarize and organize data so you can understand its main features at a glance. This includes measures of central tendency like mean, median, and mode, as well as measures of spread such as range, interquartile range, variance, and standard deviation. Mastering descriptive statistics is the foundation for every other topic in the discipline.
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Study Tips
- ✓Always plot your data before computing summary statistics; a histogram or box plot can reveal outliers and skewness that a single number cannot.
- ✓Remember that the mean is sensitive to outliers while the median is resistant. Choose your measure of center based on the shape of the distribution.
- ✓Practice converting between variance and standard deviation fluently. Variance is the square of the standard deviation, and working in squared units is essential for ANOVA and regression.
- ✓Use real datasets from sports, economics, or public health to practice computing descriptive statistics by hand before relying on software.
Common Mistakes to Avoid
A common mistake is confusing population parameters with sample statistics. Students often use n instead of n-1 in the denominator when calculating sample variance, which leads to a biased estimate. Another frequent error is reporting the mean for heavily skewed distributions when the median would be more appropriate. Additionally, many students forget that standard deviation has the same units as the data, while variance is in squared units, leading to misinterpretation of spread.
Descriptive Statistics FAQs
Common questions about descriptive statistics
Use the mean when your data are roughly symmetric and free of extreme outliers, because it incorporates every data point. Use the median when your data are skewed or contain outliers, because the median is resistant to extreme values. For example, household income data are typically right-skewed, so the median income is a more representative measure of the typical household than the mean.
Population standard deviation divides by N (the total number of values in the population), while sample standard deviation divides by n-1 (Bessel's correction). We use n-1 for samples because it produces an unbiased estimate of the population variance. In practice, most datasets are samples, so you should use n-1 unless you are certain you have the entire population.
A z-score tells you how many standard deviations a data point is from the mean. A z-score of 0 means the value equals the mean, a z-score of +2 means the value is two standard deviations above the mean, and a z-score of -1.5 means it is 1.5 standard deviations below. Z-scores let you compare values from different distributions on the same scale.