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Median – Calculation for Ungrouped & Grouped Data 📍📊

The Median is the value that divides the data into two equal halves. It is a widely used measure of central tendency when data contains extreme values, open-ended classes, or is skewed.

Median is a positional average, because it depends on the position of the middle item, not on all values.

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Definition

The Median is the value below which 50% of observations lie and above which the remaining 50% lie.

It is the middle value of a ranked (ordered) data set.

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Median for Ungrouped Data

1. For Odd Number of Observations

Median = Value of the (n + 1) / 2 th item

Example

Data: 5, 8, 10, 12, 15 n = 5 (odd)

Median = (5 + 1)/2 = 3rd item = 10

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2. For Even Number of Observations

Median = (Value of (n/2)th item + Value of (n/2 + 1)th item) / 2

Example

Data: 3, 6, 9, 12 n = 4 (even)

Median = (6 + 9) / 2 = 7.5

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Median for Discrete Series (Ungrouped Frequency Table)

Steps:

1. Compute cumulative frequency (C.F.)
2. Find the median position:

Median Position = (N + 1) / 2

3. Identify the value corresponding to that position.

Example

X	f	C.F.
10	2	2
20	3	5
30	4	9
40	1	10

N = 10
Median Position = (10+1)/2 = 5.5 → lies in C.F. group 20–30
Median = 30

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Median for Grouped (Continuous) Data

Formula

Median = L + [(N/2 – C.F_prev) / f] × h

Where:

• L = lower limit of the median class
• N = total frequency
• C.F_prev = cumulative frequency before median class
• f = frequency of median class
• h = class width

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Steps to Find Median (Continuous Series)

1. Compute cumulative frequencies.
2. Find N/2.
3. Identify median class (C.F. ≥ N/2).
4. Apply formula.

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Worked Example – Grouped Data

Example Table

Class Interval	f	C.F.
0–10	5	5
10–20	9	14
20–30	12	26
30–40	8	34
40–50	6	40

N = 40
N/2 = 20
Median class = 20–30 (because C.F. reaches ≥ 20 here)
L = 20
C.F_prev = 14
f = 12
h = 10

Apply formula:

Median = 20 + [(20 – 14) / 12] × 10
        = 20 + (6/12) × 10
        = 20 + 5
Median = 25

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Properties of Median ⭐

1. Not affected by extreme values

Example: Incomes 20,000; 22,000; 25,000; 4,00,000 Median remains stable vs. Mean.

2. Can be used for open-ended classes

Example: "Below 20", "Above 100".

3. Simple to understand and compute

Especially useful in qualitative and ordinal data.

4. Median is a positional average

Only depends on the middle position, not magnitude.

5. Better measure when distribution is skewed

Often used in economics and salary analysis.

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Advantages ✔️

• Unaffected by extreme values
• Suitable for skewed distributions
• Can handle open-ended classes
• Works well for qualitative/ordinal data

Limitations ❌

• Ignores all values except position
• Not algebraic (cannot be used for variance, correlation)
• Not suitable for further mathematical operations

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Choosing Median vs. Mean

• Data with outliers → choose Median
• Open-ended intervals → choose Median
• Need algebraic manipulation → choose Mean

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Quick Middle-Value Flow

Sorted Data → Find N → Compute N/2 → Identify median position → Median value.

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Summary ✨

• Median divides data into two equal halves.
• Use median formula for grouped data.
• Best average for skewed distributions and open-ended classes.
• Easy to compute and interpret.

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Quiz Time 🎯