What is the formula of median? What is mean median mode formula

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What is Median?The median is the middle value in a dataset when arranged in ascending or descending order. It divides the dataset into two equal halves and is a measure of central tendency that is less affected by extreme values (outliers) compared to the mean.Complete Median Formulas Reference TableFormula TypeFormulaWhen to UseExplanationExampleBasic Median (Odd n)Median = Value at position (n+1)/2When total observations (n) is oddThe middle value after arranging data in orderFor 5, 7, 9, 12, 15: n=5, Position = (5+1)/2 = 3rd value = 9Basic Median (Even n)Median = [Value at n/2 + Value at (n/2)+1]/2When total observations (n) is evenAverage of two middle valuesFor 4, 6, 8, 10: n=4, Median = (6+8)/2 = 7Median for Grouped Data (Continuous)Median = L + [(n/2 - CF)/f] × hFor continuous frequency distributionL = Lower boundary of median class, CF = Cumulative frequency before median class, f = frequency of median class, h = class intervalSee detailed example belowMedian for Discrete SeriesMedian = Value corresponding to (n+1)/2th observationFor discrete data with frequenciesFind cumulative frequency and locate middle positionIf n=20, find value at 10.5th positionPosition FormulaPosition of Median = (n+1)/2To find position before calculating medianDetermines which observation is the medianFor n=9, position = (9+1)/2 = 5th observationMedian Class Formulan/2To identify median class in grouped dataCompare with cumulative frequencies to find median classIf n=40, find class containing 20th observationMean-Median-Mode RelationshipMode = 3 Median - 2 MeanFor moderately skewed distributionEmpirical relationship between measures of central tendencyIf Mean=10, Median=12, then Mode = 3(12) - 2(10) = 16Alternative Mean-Mode FormulaMean = (Mode + 2 Median)/3Rearranged empirical formulaCalculate mean when mode and median are knownIf Mode=8, Median=10, then Mean = (8 + 2×10)/3 = 9.33Median from Mode and MeanMedian = (2 Mean + Mode)/3Another rearrangement of empirical formulaCalculate median when mean and mode are knownIf Mean=15, Mode=18, then Median = (2×15 + 18)/3 = 16Detailed Explanations by Category1. Ungrouped Data (Raw Data)For Odd Number of Observations:Formula: Median = (n+1)/2th termStep 1: Arrange data in ascending orderStep 2: Find position using (n+1)/2Step 3: The value at that position is the medianFor Even Number of Observations:Formula: Median = [n/2th term + (n/2+1)th term]/2Step 1: Arrange data in ascending orderStep 2: Find two middle positionsStep 3: Take average of values at these positions2. Grouped Data (Frequency Distribution)For Continuous Series:Median = L + [(n/2 - CF)/f] × h
Where:
- L = Lower boundary of median class
- n = Total number of observations
- CF = Cumulative frequency of class before median class
- f = Frequency of median class
- h = Class interval (width)
Steps to Calculate:Find n/2Create cumulative frequency columnIdentify median class (where cumulative frequency ≥ n/2)Apply the formula3. Relationship FormulasThe empirical relationship between mean, median, and mode:In normal distribution: Mean = Median = ModeIn skewed distribution: Mode = 3 Median - 2 MeanPractical ExamplesExample 1: Ungrouped Data (Odd n)Data: 12, 15, 18, 22, 25, 28, 30n = 7 (odd)Position = (7+1)/2 = 4th termMedian = 22Example 2: Ungrouped Data (Even n)Data: 10, 14, 16, 20, 24, 28n = 6 (even)Middle positions: 3rd and 4th termsValues: 16 and 20Median = (16+20)/2 = 18Example 3: Grouped DataClassFrequencyCumulative Frequency0-105510-2081320-30122530-40103540-50540n = 40, n/2 = 20Median class: 20-30 (CF = 25 ≥ 20)L = 20, CF = 13, f = 12, h = 10Median = 20 + [(20-13)/12] × 10 = 20 + 5.83 = 25.83Key Points to RememberAlways arrange data in order before finding median for ungrouped dataMedian is not affected by extreme values, making it useful for skewed distributionsFor grouped data, ensure you identify the correct median classThe empirical relationship between mean, median, and mode works best for moderately skewed, unimodal distributionsMedian divides the dataset into two equal halves by frequency, not by value rangeCommon ApplicationsStatistics Class 10-12: Basic median calculations and relationshipsCompetitive Exams: Quick calculation techniquesResearch & Analysis: When dealing with skewed data or outliersBusiness Analytics: Salary analysis, market researchQuality Control: Process monitoring and control chartsFormula Quick Reference CardData TypeQuick FormulaRaw Data (Odd n)(n+1)/2th valueRaw Data (Even n)Average of n/2th and (n/2+1)th valuesGrouped DataL + [(n/2-CF)/f] × hFrom Mean & Mode(2×Mean + Mode)/3Position Only(n+1)/2

What is the formula of median? What is mean median mode formula

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Formula Type	Formula	When to Use	Explanation	Example
Basic Median (Odd n)	Median = Value at position (n+1)/2	When total observations (n) is odd	The middle value after arranging data in order	For 5, 7, 9, 12, 15: n=5, Position = (5+1)/2 = 3rd value = 9
Basic Median (Even n)	Median = [Value at n/2 + Value at (n/2)+1]/2	When total observations (n) is even	Average of two middle values	For 4, 6, 8, 10: n=4, Median = (6+8)/2 = 7
Median for Grouped Data (Continuous)	Median = L + [(n/2 - CF)/f] × h	For continuous frequency distribution	L = Lower boundary of median class, CF = Cumulative frequency before median class, f = frequency of median class, h = class interval	See detailed example below
Median for Discrete Series	Median = Value corresponding to (n+1)/2th observation	For discrete data with frequencies	Find cumulative frequency and locate middle position	If n=20, find value at 10.5th position
Position Formula	Position of Median = (n+1)/2	To find position before calculating median	Determines which observation is the median	For n=9, position = (9+1)/2 = 5th observation
Median Class Formula	n/2	To identify median class in grouped data	Compare with cumulative frequencies to find median class	If n=40, find class containing 20th observation
Mean-Median-Mode Relationship	Mode = 3 Median - 2 Mean	For moderately skewed distribution	Empirical relationship between measures of central tendency	If Mean=10, Median=12, then Mode = 3(12) - 2(10) = 16
Alternative Mean-Mode Formula	Mean = (Mode + 2 Median)/3	Rearranged empirical formula	Calculate mean when mode and median are known	If Mode=8, Median=10, then Mean = (8 + 2×10)/3 = 9.33
Median from Mode and Mean	Median = (2 Mean + Mode)/3	Another rearrangement of empirical formula	Calculate median when mean and mode are known	If Mean=15, Mode=18, then Median = (2×15 + 18)/3 = 16

Class	Frequency	Cumulative Frequency
0-10	5	5
10-20	8	13
20-30	12	25
30-40	10	35
40-50	5	40

Data Type	Quick Formula
Raw Data (Odd n)	(n+1)/2th value
Raw Data (Even n)	Average of n/2th and (n/2+1)th values
Grouped Data	L + [(n/2-CF)/f] × h
From Mean & Mode	(2×Mean + Mode)/3
Position Only	(n+1)/2