Statistics begins with the task of summarizing data, and the first step is choosing meaningful measures of central tendency. The mean, calculated as \(\bar{x} = \sum x_i / n\), captures the arithmetic average and is the most widely used summary, but it is sensitive to extreme values. The median, the middle value of sorted data (or the average of the two middle values when the count is even), is more robust to outliers and is therefore preferred when distributions are skewed. The mode identifies the most frequently occurring value, and a dataset may be unimodal, bimodal, or multimodal, or have no mode at all if no value repeats.
Beyond central tendency, we measure how spread out the data is. The range, equal to the maximum minus the minimum, is the simplest measure of dispersion, but it captures only the extremes and ignores how data is distributed between them. The variance \(\sigma^2 = \sum (x_i - \bar{x})^2 / n\) and its square root, the standard deviation \(\sigma\), capture the average deviation from the mean and are the most informative measures of spread. Standard deviation is expressed in the original units of the data, making it easier to interpret than variance, which is in squared units. When estimating population variance from a sample, dividing by \(n - 1\) instead of \(n\), a correction known as Bessel's correction, yields an unbiased estimate.
Quartiles offer a more nuanced view of spread by dividing sorted data into four equal parts: Q1 at the 25th percentile, Q2 at the median, and Q3 at the 75th percentile. The interquartile range, calculated as \(\text{IQR} = Q_3 - Q_1\), describes the spread of the middle 50% of the data and is robust to outliers. This property makes the IQR the basis for a common outlier detection rule: any value falling below \(Q_1 - 1.5 \times \text{IQR}\) or above \(Q_3 + 1.5 \times \text{IQR}\) is flagged as unusual, a method used in box plots. The shape of the distribution also matters: a distribution is skewed when it is asymmetric. In a right-skewed (positive) distribution the tail extends to the right and the mean exceeds the median, while in a left-skewed (negative) distribution the tail extends to the left and the mean is less than the median. Recognizing skewness guides whether the mean or median better represents the typical value.