Mean, Median, Mode Calculator
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Understanding Measures of Central Tendency
What Are Mean, Median, and Mode?
Mean, median, and mode are the three main measures of central tendency in statistics. They each describe the center or typical value of a data set, but they do so in different ways and are useful in different situations.
Mean (Average)
The sum of all values divided by the count. Most commonly used measure of center.
Median
The middle value when data is sorted. Not affected by extreme values (outliers).
Mode
The most frequently occurring value. Can have no mode, one mode, or multiple modes.
How to Calculate the Mean
The mean (or arithmetic mean) is calculated by adding all values together and dividing by the number of values. It's the most commonly used average.
Example: Data set: 10, 15, 20, 25, 30
How to Calculate the Median
To find the median, first sort the data from smallest to largest. For an odd number of values, the median is the middle value. For an even number, it's the average of the two middle values.
Odd Count Example:
Even Count Example:
How to Calculate the Mode
The mode is the value that appears most frequently in a data set. A set can have no mode (all values unique), one mode (unimodal), two modes (bimodal), or more (multimodal).
Example: Data set: 5, 10, 10, 15, 20, 20, 20, 25
When to Use Each Measure
| Measure | Best For | Avoid When |
|---|---|---|
| Mean | Symmetric data without outliers | Data has extreme outliers |
| Median | Skewed data, income, home prices | Rarely inappropriate |
| Mode | Categorical data, finding popular choices | All values are unique |
Variance and Standard Deviation
While mean, median, and mode describe the center of data, variance and standard deviation describe how spread out the data is from the center.
Formulas
A larger standard deviation means data is more spread out from the mean.
Real-World Applications
Business & Economics
- • Median income (not skewed by billionaires)
- • Average sales per customer
- • Most popular product (mode)
- • Quality control (standard deviation)
Education & Science
- • Class average scores (mean)
- • Median test scores
- • Scientific measurement analysis
- • Research data summarization
The Effect of Outliers
Consider salaries: $40k, $45k, $50k, $55k, $1 million. The mean is $238k, but the median is $50k. The median better represents the "typical" salary because it's not distorted by the outlier. This is why median is often used for income and housing statistics.
Frequently Asked Questions
Which measure of central tendency should I use?
Use mean for symmetric data without outliers (like test scores). Use median when data has outliers or is skewed (like income, home prices). Use mode for categorical data or finding the most common value (like shoe sizes sold). When unsure, reporting all three gives the most complete picture.
Can a data set have more than one mode?
Yes! A data set can be unimodal (one mode), bimodal (two modes), multimodal (multiple modes), or have no mode at all (when all values appear equally often). For example, [1, 2, 2, 3, 3, 4] is bimodal with modes 2 and 3.
What does it mean if mean is greater than median?
When mean > median, the data is right-skewed (positively skewed), meaning there are some high values pulling the mean up. This is common in income data where a few high earners increase the average but don't affect the median. When mean < median, data is left-skewed.
How do variance and standard deviation relate to mean?
Variance measures how spread out data is from the mean by averaging squared differences. Standard deviation is the square root of variance, giving a measure in the same units as your data. About 68% of data falls within one standard deviation of the mean in a normal distribution.