Measures of centre and spread
A few well-chosen numbers can capture where data are centred and how spread out they are. Choosing the right ones depends on the shape of the data.
The mean
The mean is the sum of the values divided by how many there are: \(\bar{x} = \frac{\sum x_i}{n}\). It is the balance point of the data. Because every value counts, the mean is sensitive to outliers: one billionaire moving into a small town raises the mean income dramatically.
\bar{x} = \frac{1}{n}\sum_{i=1}^n x_iThe median and mode
The median is the middle value when the data are sorted (the average of the two middle values if \(n\) is even). It is resistant to outliers. The mode is the most frequent value and is the only measure of centre for categorical data. For skewed data, report the median; for roughly symmetric data, the mean works well.
Range and interquartile range
The range is maximum minus minimum, but it depends only on the two most extreme values. The quartiles \(Q_1\) and \(Q_3\) mark the 25th and 75th percentiles, and the interquartile range \(\text{IQR} = Q_3 - Q_1\) measures the spread of the middle half of the data. The \(1.5 \times \text{IQR}\) rule flags outliers below \(Q_1 - 1.5\,\text{IQR}\) or above \(Q_3 + 1.5\,\text{IQR}\).
Variance and standard deviation
The standard deviation measures typical distance from the mean. Square each deviation \(x_i - \bar{x}\), add them, divide by \(n - 1\) (for a sample), then take the square root. Dividing by \(n - 1\) rather than \(n\) corrects for the fact that a sample tends to underestimate the population's spread.
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}Worked example
For the data 2, 4, 4, 5, 10: the mean is \(25/5 = 5\); the median is 4; the mode is 4. Deviations from the mean are \(-3, -1, -1, 0, 5\); their squares sum to \(9 + 1 + 1 + 0 + 25 = 36\). So \(s^2 = 36/4 = 9\) and \(s = 3\). The value 10 pulls the mean above the median.
The five-number summary and boxplots
The five-number summary is minimum, \(Q_1\), median, \(Q_3\), maximum. A boxplot draws a box from \(Q_1\) to \(Q_3\) with a line at the median, and whiskers to the most extreme non-outlier values. Outliers are plotted as separate points. Boxplots are ideal for comparing groups, such as test scores by class.
z-scores
A z-score tells how many standard deviations a value is from the mean: \(z = \frac{x - \bar{x}}{s}\). A score of 85 in a class with mean 70 and standard deviation 10 has \(z = 1.5\). z-scores let you compare values on different scales, such as an SAT score and an ACT score.
z = \frac{x - \mu}{\sigma}Choosing summaries
For symmetric data without outliers, report the mean and standard deviation. For skewed data or data with outliers, report the median and IQR. Always pair a measure of centre with a measure of spread: knowing two cities both average 20°C says nothing about whether one swings from 0 to 40.
Key takeaways
The mean is the balance point but sensitive to outliers; the median is resistant. Spread is measured by the range, IQR, and standard deviation. The \(1.5 \times\) IQR rule flags outliers. z-scores standardise values so they can be compared across scales.
Flashcard
Sample mean
Check your understanding
For the data 1, 2, 3, 4, 100, which is the best measure of a typical value?