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CDS Mathematics · Statistics

Frequency Tables and Cumulative Frequency

A cumulative table gives class frequencies by subtraction; a discrete x–f table gives the mean by Σfx ÷ Σf, the median from the cumulative column, and the mode as the most frequent x.

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Why this matters

Fifteen PYQs, most of them EASY or MODERATE and many in sets of three to five items on one table. Solve for any missing frequency first; then mean, median and mode each take one line.

Concept 1 of 2: Cumulative frequency tables

A 'less than' table is a running total. The number in a class is the difference of two running totals, and 'more than' tables work the same way from the other end.

Definition

  • From a 'less than' table: frequency of aa–bb = (less than bb) − (less than aa).
  • 'More than aa' = total − (less than aa).
  • From a table with a cumulative column, each frequency is the rise in the cumulative column.
  • Read cut-offs carefully: 'at least 60 but less than 80' includes 6060 and excludes 8080; '50% marks' of 8080 is 4040.

Frequency from a cumulative table

f(a–b)=F(b)−F(a)f(a\text{–}b) = F(b) - F(a)

Worked example

Less than 20: 12; less than 40: 30; less than 60: 55; less than 80: 70. How many scored from 40 to 60, and how many scored 40 or more?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (I) 2025 — Elementary Mathematics · Q91Easy

Example 1 · Statistics · Frequency Tables and Cumulative Frequency

Consider the following for the items that follow : The frequency distribution of marks obtained by students in an English examination is given below :
Marks obtainedNumber of Students
Below 4050
Below 50125
Below 60210
Below 70315
Below 80350
What is the number of students who scored between 60 and 70 marks ?

Percent of the maximum, not marks

'Less than or equal to 50%50\% marks' in a test of 8080 means 4040 marks, not 5050. The two readings usually both land on class boundaries, so both are printed as options.
Drill 4 more on cumulative frequency tables

Concept 2 of 2: Mean, median and mode of an x–f table

Each value xx appears ff times. The mean weights each value by its frequency; the median is found by counting down the cumulative column to the middle position; the mode is the value with the largest ff.

Definition

  • Missing frequencies: the total gives one equation, a stated mean gives another.
  • Mean: xˉ=∑fx∑f\bar x = \dfrac{\sum fx}{\sum f} (the order of the rows does not matter).
  • Median: sort the xx values, build cumulative frequencies, and find the value covering position N+12\dfrac{N + 1}{2} (or the two middle positions when NN is even).
  • Mode: the xx with the largest frequency.

Mean of a frequency table

xˉ=∑fx∑f\bar x = \dfrac{\sum f x}{\sum f}

Worked example

xx: 1,2,3,41, 2, 3, 4 with ff: 5,9,4,25, 9, 4, 2. Find the mean, median and mode.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q62Moderate

Example 2 · Statistics · Frequency Tables and Cumulative Frequency

Consider the following frequency distribution :
xf
86
54
65
108
99
46
74
What is the median for the distribution ?

Sort the x values first

Tables often list xx out of order. The mean does not care, but the median does: build the cumulative column only after putting xx in increasing order.
Drill 9 more on mean, median and mode of an x–f table

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Watch out for (2)

Test yourself on Statistics

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