Mean Median Mode Calculator

Mean, median, and mode are the three primary measures of central tendency in statistics. They each describe the center of a data set in a different way.

Mean (Arithmetic Average)

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

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

The mean is sensitive to outliers — a single very large or small value can shift the mean significantly.

Median

The median is the middle value when data is sorted in ascending order. For $n$ data points:

• If $n$ is odd: median $= x_{\frac{n+1}{2}}$
• If $n$ is even: median $= \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2}$

The median is robust to outliers and is preferred for skewed distributions.

Mode

The mode is the value that appears most frequently. A data set can be:

• Unimodal — one mode
• Bimodal — two modes
• Multimodal — more than two modes
• No mode — all values appear equally often

These three measures together give a comprehensive picture of where the "center" of a data set lies.

Calculating the Mean

1. Add all data values together: $\sum x_i$
2. Divide by the total count $n$
3. Result: $\bar{x} = \frac{\sum x_i}{n}$

Weighted Mean: When values have different weights:

$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$

Calculating the Median

1. Sort the data in ascending order
2. Count the number of values $n$
3. If $n$ is odd: the median is the value at position $\frac{n+1}{2}$
4. If $n$ is even: the median is the average of values at positions $\frac{n}{2}$ and $\frac{n}{2}+1$

Calculating the Mode

1. Count the frequency of each value
2. Identify the value(s) with the highest frequency
3. If all values appear once, there is no mode

Comparison Table

When to Use Each Measure

• Mean: Use for normally distributed data without extreme outliers (e.g., test scores in a large class).
• Median: Use for skewed data or when outliers are present (e.g., household income).
• Mode: Use for categorical data or to find the most common value (e.g., most popular shoe size).

Relationship Between Mean, Median, and Mode

For a perfectly symmetric distribution: mean $=$ median $=$ mode.

For a right-skewed distribution: mean $>$ median $>$ mode.

For a left-skewed distribution: mean $<$ median $<$ mode.

• Forgetting to sort data before finding the median — the median requires ordered data; using unsorted data gives an incorrect result.
• Confusing mean and median for skewed data — the mean is pulled toward outliers, so for skewed distributions the median is a better measure of center.
• Claiming "no mode" when there are tied frequencies — if multiple values share the highest frequency, they are all modes (bimodal or multimodal).
• Dividing by the wrong count — ensure you divide by the total number of data points, not the number of distinct values.
• Including outliers without consideration — always check for extreme values that might make the mean misleading.