Median Formulas for K-12 Students

A comprehensive guide to understanding median calculations across grade levels

Elementary School (K-5)

Introduction to Median

The median is the middle value in a set of numbers when arranged in order.

Finding the Median with Odd Number of Values:

1. Arrange all numbers in order
2. Find the middle number

Example: Find the median of 3, 8, 5, 1, 7

Ordered: 1, 3, 5, 7, 8

Median: 5

Finding the Median with Even Number of Values:

1. Arrange all numbers in order
2. Find the two middle numbers
3. Add them and divide by 2

Example: Find the median of 4, 2, 6, 8, 1, 3

Ordered: 1, 2, 3, 4, 6, 8

Median = (3 + 4) ÷ 2 = 3.5

Middle School (6-8)

Median Formula

For Odd Number of Values (n):

Median = Value at position \(\frac{n+1}{2}\)

Example: Find the median of {12, 5, 22, 30, 7}

n = 5 (odd)

Position = \(\frac{5+1}{2} = 3\)

Ordered: 5, 7, 12, 22, 30

Median = 12

For Even Number of Values (n):

Median = \(\frac{\text{Value at position } \frac{n}{2} + \text{Value at position } \frac{n}{2}+1}{2}\)

Example: Find the median of {15, 25, 35, 45, 55, 65}

n = 6 (even)

Positions = \(\frac{6}{2} = 3\) and \(\frac{6}{2}+1 = 4\)

Ordered: 15, 25, 35, 45, 55, 65

Median = \(\frac{35 + 45}{2} = 40\)

Median vs. Mean and Mode

Comparison of Measures of Central Tendency

Measure	Definition	When to Use
Mean	Average of all values	When data is normally distributed
Median	Middle value	When data has outliers
Mode	Most frequent value	When frequency is important

High School (9-12)

Median for Grouped Data

Median Formula for Grouped Data:

Median = L + \(\frac{\frac{n}{2} - CF}{f} \times w\)

L = Lower boundary of the median class
n = Total number of observations
CF = Cumulative frequency before the median class
f = Frequency of the median class
w = Class width

Example: Finding Median from Grouped Data

Class Interval	Frequency (f)	Cumulative Frequency (CF)
10-20	5	5
20-30	12	17
30-40	22	39
40-50	15	54
50-60	8	62

Total number of observations (n) = 62

n/2 = 31

Median class = 40-50 (where 31st observation falls)

L = 40 (lower boundary)

CF = 39 (cumulative frequency before median class)

f = 15 (frequency of median class)

w = 10 (class width)

Median = 40 + ((31 - 39) / 15) × 10

Median = 40 + ((-8) / 15) × 10 = 40 - 5.33 = 34.67

Quartiles and Percentiles

Quartiles:

First Quartile (Q₁): The median of the lower half of the data
Second Quartile (Q₂): The median of the entire data set
Third Quartile (Q₃): The median of the upper half of the data

Q₁ = Value at position \(\frac{n+1}{4}\)

Q₃ = Value at position \(\frac{3(n+1)}{4}\)

Interquartile Range (IQR):

IQR = Q₃ - Q₁

The IQR is a measure of statistical dispersion and is used to identify outliers.

Outliers: Values < Q₁ - 1.5(IQR) or > Q₃ + 1.5(IQR)

Advanced Applications (Grade 11-12)

Statistical Applications of Median

Median Absolute Deviation (MAD):

A robust measure of variability that uses the median instead of the mean.

Find the median of the data set
Calculate the absolute deviation of each data point from the median
Find the median of these absolute deviations

MAD = median(|X_i - median(X)|)

Box and Whisker Plots:

A visual representation of the data distribution based on the five-number summary:

Minimum value
First quartile (Q₁)
Median (Q₂)
Third quartile (Q₃)
Maximum value

Min

Q₁

Median

Q₃

Max

Practical Applications

Real-World Uses of Median

When to Use Median Instead of Mean:

Skewed Distributions: When data is not normally distributed
Presence of Outliers: When extreme values would significantly affect the mean
Income and Housing Data: Median income and median home prices are more representative than means
Ordinal Data: When data represents rankings rather than precise measurements

Examples in Various Fields:

Field	Application of Median
Economics	Median household income, median home prices
Health	Median survival time in clinical studies
Education	Median test scores to evaluate performance
Computer Science	Median filter for noise reduction in image processing
Environmental Science	Median pollution levels over time