Understand the arithmetic mean formula, including mathematical notation and step-by-step explanation for ungrouped and grouped data.
Mean Median Mode Calculator
Median = Middle value after sorting data
If even:
Median = (Middle1 + Middle2) / 2
The median formula is used to calculate the median, a positional measure of central tendency in statistics.
The median represents the middle value of an ordered dataset and divides the data distribution into two equal halves.
This page explains:
Median formula for ungrouped data
Median position formula
Median formula for grouped data
Statistical symbols used in median calculations
All formulas are presented using standard statistical notation.
The median formula determines the middle value of a dataset after arranging the observations in ascending or descending order.
The formula depends on the number of observations is odd or even.
Ungrouped data refers to raw numerical values that are not organized into class intervals.
Before applying the formula:
Arrange the dataset in ascending order
Count the total number of observations (n)
When the total number of observations (n) is odd:
Median = Value at position (n + 1) / 2
Where:
n = Total number of observations
Data must be arranged in order
This formula directly identifies the middle observation.
When the total number of observations (n) is even:
Median = [Value at (n/2) + Value at (n/2 + 1)] / 2
Where:
n = Total number of observations
The median is the average of the two middle values
This ensures the dataset is divided into two equal parts.
The median position formula helps determine the location of the median in an ordered dataset.
Median Position = (n + 1) / 2
Where:
n = Total number of observations
This formula is especially useful when identifying the middle value in discrete datasets.
When data is organized into class intervals with corresponding frequencies, the grouped median formula is used:
Median = L + [(N/2 − cf) / f] × h
Where:
L = Lower boundary of the median class
N = Total frequency (Σf)
cf = Cumulative frequency before the median class
f = Frequency of the median class
h = Class width
This formula is applied to continuous frequency distributions and statistical tables.
To apply the grouped median formula:
Calculate N/2
Construct the cumulative frequency column
Locate the class interval where the cumulative frequency ≥ N/2
That interval is the median class
The lower boundary of this class is used as L in the formula.
For discrete ordered values, use the positional formulas:
(n + 1) / 2 for odd datasets
Average of the middle two values for even datasets
For continuous grouped data, use:
Median = L + [(N/2 − cf) / f] × h
This method accounts for class intervals and cumulative frequencies.
For ungrouped data: Median = Value at position (n + 1) / 2 (odd case)
For grouped data: Median = L + [(N/2 − cf) / f] × h
N/2 represents half of the total frequency and is used to locate the median class in grouped data.
Cumulative frequency helps identify the class interval that contains the median value in a grouped frequency distribution.
Median Position = (n + 1) / 2
It determines the location of the median in an ordered dataset.
Yes. Observations must be arranged in ascending or descending order before applying positional median formulas.
The median formula is a positional statistical method used to determine the central value of ordered quantitative data. It is applied in both ungrouped datasets and grouped frequency distributions using standard statistical notation.
All formulas presented follow conventional mathematical and statistical principles.