# Lecture 3: Descriptive Statistics - Measures of Central Tendency

## Statistics / Biostatistics

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Define and calculate the mean, median, and mode
2. Explain when each measure of central tendency is most appropriate
3. Describe the effect of skewness and outliers on the mean vs. median
4. Calculate a weighted mean and a trimmed mean
5. Apply measures of central tendency to clinical data scenarios

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## Lecture Content

### I. Purpose of Measures of Central Tendency

Central tendency describes the "center" or "typical value" of a dataset. It provides a single summary number to represent an entire distribution and is an essential first step in any data analysis. The choice of which measure to use depends on the type of data and the shape of the distribution, so understanding the properties of each measure is fundamental to good statistical practice.

### II. The Arithmetic Mean

The arithmetic mean is defined as the sum of all values divided by the number of values, expressed by the formula x-bar = (sum of x_i) / n. It is the most commonly used measure of central tendency for continuous data. The mean uses every data point in its calculation, which makes it sensitive to extreme values (outliers). Two important mathematical properties are that the sum of deviations from the mean always equals zero and that the mean minimizes the sum of squared deviations. The mean is best used when data are approximately symmetric with no major outliers.

As an example, consider systolic blood pressures of five patients: 120, 130, 125, 140, and 135 mmHg. The mean is (120 + 130 + 125 + 140 + 135) / 5 = 130 mmHg. A related measure is the **weighted mean**, which is used when observations carry different weights or frequencies. Its formula is x-bar_w = (sum of w_i * x_i) / (sum of w_i). Weighted means appear in meta-analysis, GPA calculations, and composite scores.

### III. The Median

The median is the middle value when data are arranged in ascending order. If n is odd, the median is the (n+1)/2 th value; if n is even, it is the average of the n/2 th and (n/2 + 1) th values. The median is not affected by extreme values, making it a resistant measure. It depends only on the middle position(s), not on the actual magnitude of all values, and is therefore preferred for skewed data or data with outliers.

Consider hospital lengths of stay (in days): 2, 3, 3, 5, 42. The mean is 11.0 days, heavily distorted by the outlier of 42, while the median is 3 days, which better represents the typical patient. The median is commonly used for income data, hospital charges, length of stay, and survival times -- all of which tend to have right-skewed distributions.

<image>Two side-by-side dot plots of patient hospital stay durations. Panel A: A symmetric dataset where mean and median coincide at approximately the same point, both marked with vertical lines. Panel B: A right-skewed dataset with one extreme outlier; the mean is pulled far to the right while the median remains near the bulk of the data. Annotations show the numerical values of mean and median for each panel.</image>

### IV. The Mode

The mode is the most frequently occurring value in a dataset. It can be used with any type of data, including nominal data, making it the only measure of central tendency applicable to categories. A dataset can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes), and a mode may not exist if all values are unique. The mode is less commonly used for continuous data, where exact repetition is rare, but it is very useful for categorical data. For example, the most common blood type in a population might be O+, making O+ the mode.

Bimodal distributions can signal the presence of two subpopulations. For instance, the age distribution at a mixed pediatric-adult clinic might show two peaks, one centered on children and another on adults.

### V. Comparing Mean, Median, and Mode

In a perfectly symmetric distribution, the mean, median, and mode all coincide. In a right-skewed (positively skewed) distribution, the mode is less than the median, which is less than the mean, because the mean is pulled toward the right tail. In a left-skewed (negatively skewed) distribution, the order reverses: mean is less than median, which is less than mode, because the mean is pulled toward the left tail.

A useful rule of thumb for choosing the right measure is as follows: use the mean for symmetric data with no outliers, use the median for skewed data or data with outliers, and use the mode for categorical data.

<image>Three distribution curves overlaid on a single axis. Panel A: A symmetric bell curve with mean, median, and mode all at the same central point. Panel B: A right-skewed curve with mode marked at the peak, median slightly right, and mean furthest right. Panel C: A left-skewed curve with mean furthest left, median slightly left of center, and mode at the peak. Each measure is marked with a distinctly colored vertical dashed line and labeled.</image>

### VI. Other Measures of Central Tendency

The **trimmed mean** removes a fixed percentage of the smallest and largest values before computing the mean. For example, a 10% trimmed mean discards the bottom 10% and top 10% of values. This provides a compromise between the mean, which uses all data, and the median, which ignores extremes.

The **geometric mean** is the nth root of the product of n values. It is appropriate for data that are multiplicative or log-normally distributed and is commonly used in microbiology (for example, antibody titers) and environmental health (pollutant concentrations). The geometric mean is always less than or equal to the arithmetic mean.

The **harmonic mean** is the reciprocal of the arithmetic mean of the reciprocals. It is used in specific contexts, such as averaging rates. While less common in clinical practice, it has important applications in certain specialized analyses.

### VII. Clinical Applications

In clinical research, the choice of central tendency measure carries real consequences. Researchers report the mean blood pressure in a clinical trial arm when the distribution is symmetric. Oncology studies typically use the median survival time because survival data are right-skewed. Drug safety reports might identify the modal category of adverse events to highlight the most common side effect. Vaccine immunogenicity studies calculate the geometric mean titer (GMT) because antibody titers follow a log-normal distribution.

It is always important to report which measure of central tendency was used and to justify that choice, so that readers can properly interpret the findings.

<image>A clinical data summary table for a hypothetical drug trial. The table includes columns for Variable, Mean, Median, and Mode. Rows include: Age (years), Serum creatinine (mg/dL), Number of prior hospitalizations, and Pain score (ordinal 0-10). Next to the table, a box highlights which measure of central tendency is most appropriate for each variable and why (e.g., median for creatinine due to right skew, mode for pain score if ordinal).</image>

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