Mean Median Mode Calculator
Use this free mean median mode calculator to analyze any data set in seconds. Whether working with raw lists, grouped data, or frequency tables, get fast answers with clear steps, statistical range, and standard deviation.
What is the difference between mean, median, and mode?
The mean is the arithmetic average of a data set. The median is the middle value when data points are sorted in ascending order. The mode is the most frequently occurring value. Together with range, they measure statistical central tendency.
- Arithmetic Mean: Sum of all data points divided by the total sample size ($n$). Highly sensitive to outliers.
- Median: The central midpoint. It separates the top 50% from the bottom 50% and resists extreme values.
- Mode: The peak frequency value. A sample can be unimodal, bimodal, multimodal, or have no mode at all.
- Range: The spread between the highest and lowest values ($Max - Min$).
Formulas for Central Tendency and Spread
Our tool uses standard statistical definitions to compute your metrics:
x̄ = (Σx) / nPosition = (n + 1) / 2Range = Max - Min | s = √[ Σ(x - x̄)² / (n - 1) ]Mean Median Mode and Range Calculator with Steps
Follow this worked example using a sample data set: 12, 7, 19, 7, 15, 24.
Step 1: Sort the Data: Arrange the values in ascending order: 7, 7, 12, 15, 19, 24 ($n = 6$).
Step 2: Calculate the Mean: Add all values and divide by the count:
(7 + 7 + 12 + 15 + 19 + 24) / 6 = 84 / 6 = 14.
Step 3: Find the Median: With an even count ($n = 6$), average the 3rd and 4th values:
(12 + 15) / 2 = 27 / 2 = 13.5.
Step 4: Identify the Mode: The number 7 appears twice, while all other numbers appear once. Mode = 7.
Step 5: Determine the Range: Subtract the minimum value from the maximum value:
24 - 7 = 17.
How to Find Mean Median Mode From a Frequency Table
When working with grouped data or a frequency distribution, find the values using frequency weights:
- Grouped Mean: Multiply each class midpoint ($m$) by its frequency ($f$), sum them up, and divide by total frequency ($\Sigma f$):
x̄ = Σ(f · m) / Σf. - Grouped Median: Identify the median class containing the $(N / 2)$ position and apply linear interpolation.
- Grouped Mode: Locate the modal class with the highest frequency density.
Frequently Asked Questions
What is the difference between mean, median, and mode?
The mean is the numerical average of all data points, the median is the center point of an ordered list, and the mode is the most frequent value. Mean is ideal for balanced data, median handles outliers, and mode works well for recurring categories.
How do you find the median when there is an even number of values?
Sort your values in ascending order, find the two middle numbers, add them together, and divide by 2. For example, in the list [4, 8, 12, 16], the two middle values are 8 and 12. Their median is $(8 + 12) / 2 = 10$.
What if there are two modes in a data set?
When two values tie for the highest count, the distribution is bimodal. If three or more values tie, it is multimodal. If every number appears only once, the data set has no mode.
Which is better to use: mean, median, or mode?
Use the mean for symmetric data without severe outliers. Use the median for skewed data (like salaries or home prices) because extreme values will not distort it. Use the mode when identifying the most common choice or popular item.
How do you find the mean, median, and mode of a data set?
First, add all values and divide by the total count to get the mean. Next, arrange the values in ascending order to locate the middle median value. Finally, tally the occurrences of each number to spot the mode.
Central Tendency Analyzer
* Mean represents the arithmetic average ($x̄ = Σx / n$). Median is the central value of the ordered sequence. Sample standard deviation ($s$) applies Bessel's correction ($n - 1$) for unbiased estimator sampling.
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