Pythagorist
Statistics

Cumulative Frequency Calculator

Build a cumulative frequency table and find the median, quartiles, and interquartile range. Enter your frequencies in order to get the cumulative totals and key statistics.

5
13
25
35
42
45

Median (n/2)

22.5

LQ (n/4)

11.25

UQ (3n/4)

33.75

IQR

22.5

How it works

  1. Build the cumulative frequency column

    Running totals: 5, 13, 25, 35, 42, 45.

    Result:5, 13, 25, 35, 42, 45

  2. Find the median position

    median position=n2=452=22.5\text{median position} = \frac{n}{2} = \frac{45}{2} = 22.5

    Read off the median from the cumulative frequency graph at position 22.5.

  3. Find the quartile positions

    Q1 at n4=11.25,Q3 at 3n4=33.75Q_1 \text{ at } \frac{n}{4} = 11.25, \quad Q_3 \text{ at } \frac{3n}{4} = 33.75

    Lower quartile (Q1) at position 11.25, upper quartile (Q3) at position 33.75.

  4. Calculate the interquartile range (IQR)

    IQR=Q3−Q1=33.75−11.25=22.5IQR = Q_3 - Q_1 = 33.75 - 11.25 = 22.5

    IQR = upper quartile − lower quartile = 33.75 − 11.25 = 22.5.

    Result:22.5

Cumulative Frequency & Quartiles

median at n2,Q1 at n4,Q3 at 3n4,IQR=Q3−Q1\text{median at } \tfrac{n}{2}, \quad Q_1 \text{ at } \tfrac{n}{4}, \quad Q_3 \text{ at } \tfrac{3n}{4}, \quad IQR = Q_3 - Q_1
nnTotal frequency
Q1Q1Lower quartile (25th percentile)
Q3Q3Upper quartile (75th percentile)
IQRIQRInterquartile range = Q3 − Q1

Example substitution

n=40⇒median at 20,Q1 at 10,Q3 at 30n = 40 \Rightarrow \text{median at } 20, \quad Q_1 \text{ at } 10, \quad Q_3 \text{ at } 30

With total frequency 40: read median from position 20, Q1 from position 10, Q3 from position 30.

Worked examples

Frequencies: 5, 8, 12, 10, 7, 3. Find the cumulative frequencies and median position.

  1. 1
    Build cumulative frequency column: 5, 13, 25, 35, 42, 45
  2. 2
    Total frequency (n): n = 45
  3. 3
    Median at n/2: Median at position 22.5
  4. 4
    Q1 at n/4, Q3 at 3n/4: Q1 at 11.25, Q3 at 33.75
  5. 5
    IQR = Q3 − Q1: 33.75 − 11.25 = 22.5
Answer: CF: 5, 13, 25, 35, 42, 45 | Median position: 22.5

Frequently asked questions

What is cumulative frequency?+

Cumulative frequency is a running total — each class's cumulative frequency is the sum of all frequencies up to and including that class.

How do you find the median from a cumulative frequency graph?+

Read off the value at the n/2 position on the cumulative frequency axis.

How do you find quartiles?+

Q1 is at n/4, Q2 (median) at n/2, Q3 at 3n/4 on the cumulative frequency scale.

What is the interquartile range (IQR)?+

IQR = Q3 − Q1. It measures the spread of the middle 50% of data.

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