The most common numerical summaries describe a dataset's center. The mean is the arithmetic average — the sum of all values divided by the count — while the median is the middle value of an ordered dataset (or the average of the two middle values when the count is even). The mode is the value or values that occur most frequently. When data are skewed or contain extreme outliers, the median is preferred because it is robust: a single unusually large or small value can pull the mean dramatically but barely affects the median. To describe spread, statisticians use the range (maximum minus minimum), variance (the average of squared deviations from the mean), and standard deviation (the square root of the variance, which is expressed in the original units of the data). The interquartile range, defined as \(\text{IQR} = Q_3 - Q_1\), captures the spread of the middle 50% of the data, where the quartiles divide ordered data into four equal parts. Percentiles generalize this idea: the k-th percentile is the value below which k% of the observations fall. The z-score, computed as \(z = (x - \mu) / \sigma\), expresses how many standard deviations a value lies from the mean and is a useful way to standardize observations across different scales.
The shape of a distribution also matters. A positively skewed distribution has a long right tail and a mean greater than the median, while a negatively skewed distribution has a long left tail and a mean less than the median. Kurtosis measures the "tailedness" or peakedness of a distribution relative to the normal. These numerical summaries are paired with visual displays: histograms show the frequency distribution of a continuous variable with adjacent bars, boxplots summarize a dataset using its minimum, quartiles, and maximum while flagging outliers, and scatterplots display pairs of numerical values to reveal relationships between two variables. Choosing the right summary and the right plot depends on the type of variable at hand: categorical variables take values from named categories, numerical variables take numeric values, and numerical variables are further split into discrete (countable, such as the number of children) and continuous (any value in an interval, such as weight).