Calculate Median
Online calculator to compute the statistical middle value (median) of a data series
Median Calculator
The Median
The median (also called central value) is the middle value of a sorted data series. It is a robust measure of location that is insensitive to outliers.
Median Concept
The median is the middle value of a sorted data series.
It divides the data into two equal halves.
● Data Points ● Median (middle value)
What is the Median?
The median (central value) is a fundamental measure of location in statistics:
- Definition: Middle value of a sorted data series
- Position: For odd count: middle value, for even count: average of the two middle values
- Quantile: The median corresponds to the 0.5-quantile (50th percentile)
- Robustness: Insensitive to outliers and extreme values
- Interpretation: Divides the data into two equal halves
- Application: Particularly suitable for skewed distributions
Calculating the Median
The calculation depends on the number of data values:
Odd Count
- 1. Sort the data in ascending order
- 2. Determine position: k = (n+1)/2
- 3. The k-th value is the median
- Example: With 7 values, position (7+1)/2 = 4
Even Count
- 1. Sort the data in ascending order
- 2. Find the two middle values (position n/2 and n/2+1)
- 3. Calculate the average of the two values
- Example: With 8 values: (Value₄ + Value₅)/2
Applications of the Median
The median is used in many fields:
Statistical Analysis
- Robust measure of location with outliers
- Descriptive statistics (with quartiles)
- Box plot representations
- Comparison with arithmetic mean to detect skewness
Practical Applications
- Real Estate: Median home price
- Income: Median income (more robust than average)
- Education: Median grades or test scores
- Medicine: Median survival time in studies
Formulas for the Median
Median (Odd Count)
For odd n: the middle value of the sorted list
Median (Even Count)
For even n: average of the two middle values
Median as Quantile
The median is the 0.5-quantile (50th percentile) of the distribution
General Form
Case distinction based on parity of n
Symbol Explanations
| \(\tilde{x}\) | Median (also Me or x̃) |
| \(x_{(k)}\) | k-th value of sorted list |
| \(n\) | Number of data points |
| \(Q(p)\) | Quantile function |
| \(F^{-1}\) | Inverse distribution function |
| 0.5 | 50th percentile (50%) |
Example Calculations for the Median
Example 1: Odd Count
Calculate: Median of 7 values
1. Sort Data
Sort data in ascending order
2. Determine Position
The 4th value of the sorted list is the median
Example 2: Even Count
Calculate: Median of 8 values
1. Sort Data
The two middle values are 7 and 8
2. Calculate Average
Average of values at position 4 and 5
Example 3: Even Count with Gap
Calculate: Median with larger gap between middle values
1. Already Sorted
The middle values are 2 and 6
2. Calculate Median
The median 4 lies between the data values
Interpretation
The median does not have to occur in the data. For even counts, it often lies between two values. In this case, median = 4 means: 50% of values are ≤ 4, 50% are ≥ 4.
Mathematical Foundations of the Median
The median is a robust measure of location with important properties that is preferable to the arithmetic mean, especially for skewed distributions.
Properties of the Median
The median has important mathematical properties:
- Robustness: Insensitive to outliers – extreme values have little impact on the median
- Existence: Always exists for any finite data set
- Uniqueness: The median is always uniquely determined (even for even counts by convention)
- Position Invariance: Invariant under monotone transformations
- Minimization: The median minimizes the sum of absolute deviations: ∑|xᵢ - Med| is minimal
Median vs. Arithmetic Mean
The choice between median and mean depends on the data:
When to Use Median?
- Skewed Distributions: For asymmetric data (e.g., income, real estate prices)
- Outliers: When extreme values are present
- Ordinal Data: For ranked data without metric distances
- Robustness: When resistance to extreme values is important
When to Use Mean?
- Symmetric Distributions: For normally distributed or approximately symmetric data
- No Outliers: When extreme values are meaningful
- Metric Data: For ratio or interval scales
- Mathematical Properties: When algebraic operations are important
Relationship to Quantiles
The median is closely related to the quantile concept:
The Median as Quantile
The median is the 0.5-quantile (50th percentile or second quartile Q₂). It divides the data so that 50% of values lie below and 50% above it.
Quartiles and Median
The median is one of three quartiles: Q₁ (25%), Q₂ = Median (50%), Q₃ (75%). Together they form the basis for box plots.
Applications in Different Fields
Economics and Social Sciences
- Income Statistics: Median income better than average for skewed data
- Real Estate: Median price more meaningful than average price
- Education: Median grades as robust measure
- Demography: Median age of population
Natural Sciences and Medicine
- Clinical Studies: Median survival time (robust to censoring)
- Lab Values: Reference values often given as median
- Environmental Data: Median pollutant concentrations
- Astronomy: Median brightness of stars
Skewness and Median
The relationship between median and mean reveals information about skewness:
Symmetric
Mean ≈ Median
Normal Distribution
Right Skewed
Mean > Median
e.g., Income
Left Skewed
Mean < Median
e.g., Exam Grades
Summary
The median is a robust and intuitively understandable measure of location that indicates the "typical" middle value of a distribution. Its insensitivity to outliers makes it particularly valuable in practical data analysis, especially for skewed distributions. As the 50th percentile, it is a special case of the quantile function and, together with the quartiles, forms the basis for important exploratory data analysis methods such as the box plot.
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