Mean Median Mode

Mean, median, and mode are the three classic measures of central tendency in statistics. Each one answers the question ‘what’s the typical value in this data?’ but they answer it differently. Mean is the arithmetic average. Median is the middle value when sorted. Mode is the most frequent value. For a perfectly symmetric distribution they all coincide. For real-world data — which is rarely symmetric — they diverge, and the difference between them reveals important things about the data’s shape and what is going on inside it.

Mean — The Arithmetic Average

The mean is the sum of all values divided by the count of values:

$$ \bar{x} = \dfrac{x_1 + x_2 + \cdots + x_n}{n} = \dfrac{1}{n} \sum_{i=1}^{n} x_i $$

Example: for the dataset {2, 3, 3, 4, 7, 7, 7, 8, 9}, sum = 50, count = 9, mean = 50/9 ≈ 5.56.

The mean is the most-used measure of central tendency and the basis of nearly all parametric statistics. It uses every data point. Its main weakness: it is heavily distorted by outliers. A single billionaire in a sample of 100 people from a small town can shift the ‘mean income’ by an absurd amount, while every other person’s life is unchanged.

Median — The Middle Value

The median is the value at the center of the sorted dataset.

• If \( n \) is odd, the median is the \( \dfrac{n+1}{2} \)-th value.
• If \( n \) is even, the median is the average of the two middle values: the \( n/2 \)-th and the \( (n/2)+1 \)-th.

For our dataset {2, 3, 3, 4, 7, 7, 7, 8, 9} (already sorted, n = 9), the median is the 5th value: 7.

The median is the right measure when the data has outliers or is heavily skewed. House prices, household incomes, medical lab values that follow a long-tail distribution — these are reported as medians, not means, because the median resists outliers. Half the data is below, half above, no matter what the extremes do.

Mode — The Most Frequent Value

The mode is the value that appears most often in the dataset.

For {2, 3, 3, 4, 7, 7, 7, 8, 9}, the mode is 7 — it appears three times.

A dataset can have:

• One mode (unimodal) — the simplest case.
• Two modes (bimodal) — two values tied for most frequent. Often a hint that the data combines two underlying populations.
• More than two modes (multimodal) — same logic, more groups.
• No mode — every value appears equally often.

The mode is the only measure of central tendency that works for categorical data (e.g., the most common eye color in a survey is ‘brown’ — you cannot take a ‘mean’ of eye colors). For numeric data, the mode is less useful than mean or median unless the data is grouped or you specifically want to know the most popular value.

When the Three Measures Differ

The relative position of mean, median, and mode tells you about the shape of the distribution:

• Symmetric distribution (normal/bell-shaped): mean = median = mode. All three coincide at the center.
• Right-skewed (positively skewed): a tail extending to the right (a few high outliers). Mode < median < mean. The outliers pull the mean to the right of the median.
• Left-skewed (negatively skewed): a tail extending to the left (a few low outliers). Mean < median < mode. The outliers pull the mean to the left.

Quick check: if the mean is much higher than the median, the data has high-value outliers. If lower, low-value outliers. The gap between mean and median is one of the simplest, most useful diagnostic numbers in statistics.

Which One Should You Report?

Rules of thumb that hold in most situations:

• Symmetric, no outliers, numeric data: report the mean. It is the most powerful statistic, uses every data point, and underlies most further analysis.
• Skewed data or data with outliers: report the median. It represents the typical value without being thrown off by extremes. Always report median income, median house price, median survival time.
• Categorical data or interested in most popular value: report the mode. ‘The most common eye color in our sample is brown.’
• Best practice: when in doubt, report all three plus the standard deviation. Readers can interpret the data themselves.

Weighted Mean

Sometimes data points have different weights. A student’s GPA is a weighted mean of course grades, with weights equal to credit hours. A weighted mean is calculated as:

$$ \bar{x}_w = \dfrac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} $$

Where each \( x_i \) is multiplied by its weight \( w_i \) before summing. If all weights are equal, the weighted mean reduces to the ordinary mean.

Related study notes: Normal Distribution, Variance and Standard Deviation, Expected Value, Central Limit Theorem.

Frequently Asked Questions