4 Descriptive Statistics

What descriptive statistics show

Descriptive statistics summarize observed data to help characterize typical values, variation, relative position, and relationships between variables. A numerical summary describes the data at hand; by itself, it does not show that a pattern applies to a broader population or explain what caused it.

For a quantitative variable, begin with a graph such as a histogram or box plot. Consider the distribution’s shape, center, and spread, as well as any gaps, clusters, or unusual values. A few statistics cannot capture every feature, so interpret graphs and numerical summaries together.

Center and spread

The mean is the arithmetic average: add the values and divide by their number. Since the mean uses every observation, extreme values can pull it toward a tail. The median is the middle value after ordering the observations; with an even number of observations, it is the average of the two middle values. It is less affected by extremes. The mode is the most frequent value, and a distribution may have more than one mode or none that is useful.

For the data $2, 3, 3, 4, 8$, the mean is $4$, the median is $3$, and the mode is $3$. The larger value $8$ raises the mean above the median. For a roughly symmetric distribution without influential outliers, the mean is often a useful measure of center. For a skewed distribution or one with extreme values, the median may better represent a typical observation.

The range is the maximum minus the minimum, so it depends only on the two extremes. The interquartile range (IQR) is $Q_3-Q_1$, the difference between the third and first quartiles. It describes the span of the middle half of the data and is less sensitive to extremes than the range or standard deviation.

Variance is the average squared distance from the mean. The usual sample variance divides the sum of squared deviations by $n-1$; population variance divides by $N$. Standard deviation is the square root of the variance, expressed in the original units of the data. It describes spread around the mean and can be strongly affected by extreme observations.

Match center and spread measures according to the distribution. Use the mean and standard deviation together when the distribution is reasonably symmetric and has no influential outliers. Use the median and IQR when the distribution is skewed or has outliers; these pairs have similar sensitivity to extreme values.

Position in a distribution

Quartiles divide ordered data into four parts: $Q_1$ is the 25th percentile, the median is the 50th percentile, and $Q_3$ is the 75th percentile. A percentile describes relative standing: approximately the $p$th percentile is at or above about $p\%$ of observations, subject to the convention used to calculate percentiles. Software and textbooks may use different percentile algorithms, so check the method when exact results matter.

A z-score expresses a value’s distance from the mean in standard-deviation units:

A z-score of $-1$ means the value is one standard deviation below the mean. This comparison is most informative when the mean and standard deviation are appropriate summaries for the distribution. A z-score alone does not establish that a value is erroneous or unusually rare.

Summarizing relationships

Choose a summary suited to the variable types, and inspect a suitable graph before interpreting a number.

• One quantitative and one categorical variable: Compare the quantitative variable across groups. Side-by-side box plots can show differences in medians, spread, and possible outliers; report group-specific summaries as well. A difference in group averages is descriptive and does not, on its own, show why the groups differ.

• Two quantitative variables: Use a scatterplot to examine direction, form, strength, clusters, and unusual points. The correlation coefficient $r$ summarizes the direction and strength of a linear association and ranges from $-1$ to $1$, with its sign indicating direction. Correlation may miss a curved pattern, and outliers can substantially affect it. A positive correlation does not mean that one variable causes the other to increase.

• Two categorical variables: Compare counts or proportions in a table or suitable chart. State the groups and denominators clearly, since the same count can imply different proportions when group sizes differ.

For example, a positive correlation between study time and exam score means that, in the observed data, students who study more tend to have higher scores. It does not establish that study time alone caused the scores; other factors or the study design may matter. Because correlation summarizes linear association, use a scatterplot to check whether it fits the actual pattern.

Interpreting summaries responsibly

Choose statistics that suit the data. Means and standard deviations require quantitative values; a mean of category labels such as colors is not meaningful. For ordered categories, medians or proportions may be more suitable.

Keep units and the population in view. A statement such as “Median = 3” is incomplete unless it says what was measured and in what units. Also distinguish a sample statistic from a population value.

Look beyond any single number: a mean can hide skew, multiple clusters, or outliers, and a correlation can hide curvature. Check graphs and relevant subgroup summaries. Descriptive statistics report patterns in the data; causal or population-wide conclusions require suitable study design and, often, inferential methods.