NDA Maths · Statistics

Frequency Distributions and Graphical Representation

How to organise raw data into class intervals + frequencies, and which graph (histogram, polygon, ogive, pie chart) tells the story best.

All Statistics notes NDA Maths strategy

Why this matters

14 PYQs across 2017–2025 — small subtopic but reliable scoring territory. Three shapes dominate: picking the right graph for given data, computing a histogram class's relative height when widths are unequal, and reading values straight off a frequency table (mode, cumulative count, median).

Concept 1 of 3

Histograms, Frequency Polygons & Ogives

Intuition

A histogram is bars whose AREA represents frequency — so when class widths are unequal, the bar height must be frequency divided by class width (frequency density), not raw frequency. Connecting the midpoints of histogram tops gives a frequency polygon. Plotting cumulative frequency against class boundaries gives an ogive — used to read the median directly.

Definition

Histogram bars have width = class width and height = $\text{frequency density}$ . For equal class widths, density is proportional to raw frequency. An ogive is a cumulative-frequency curve; the value of $x$ at which the ogive equals $n/2$ is the median.

Frequency Density (for unequal class widths)

\text{Density} = \dfrac{\text{Frequency}}{\text{Class width}}

Class widthupper bound − lower bound of the class

Visualization · change the bin width, watch the shape change

Bin width:

Same 30 data points each time — only the bin width changes. At width 50 the shape looks almost uniform; at width 5 it looks jagged. Choosing bin width is part of the analysis, not the data.

Worked example

Three classes have widths 5, 10, 5 with frequencies 30, 50, 25. Find the height of each histogram bar (relative).

Density = frequency / class width.
Class 1: $30 / 5 = 6$ .
Class 2: $50 / 10 = 5$ .
Class 3: $25 / 5 = 5$ .
Heights are in ratio 6 : 5 : 5 — note class 2 has the highest frequency but NOT the tallest bar.

Answer:Heights = 6, 5, 5 (in density units)

Practice this conceptself-check · 4 quick reps

Try it yourself

Two classes have widths 5 and 10 with frequencies 25 and 40. Compare the heights of their histogram bars.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Frequency $20$ , class width $5$ . Density?
2.
Frequency $30$ , class width $10$ . Density?
3.
Two classes have equal frequency, widths $5$ and $10$ . Which bar is taller?
4.
Frequency $7$ , class width $1$ . Density?

From the bank · past-year question

Example 1StatisticsEASY

For a histogram based on a frequency distribution with unequal class intervals, the frequency of a class should be proportional to

[Q103 · Apr · 2021]

Bar height $\neq$ frequency when class widths differ

If two classes have the same frequency but different widths, the wider class has the SHORTER bar — because density (height) divides frequency by width. Students draw bars of equal height for equal frequencies; correct histograms make AREAS equal, not heights.

Drill 4 more on histograms, frequency polygons & ogives

Concept 2 of 3

Pie Charts

Intuition

A pie chart shows how a whole is split into parts. Each part's sector angle is proportional to its share of the total — and all the sector angles together must add to

360^\circ

Definition

For a category with frequency $f_i$ and total frequency $\sum f_i = N$ , the sector angle is $\theta_i = \dfrac{f_i}{N} \times 360^\circ$ . Equivalently, the angle is proportional to the frequency, with proportionality constant $360/N$ .

Sector Angle in a Pie Chart

\theta_i = \dfrac{f_i}{N} \times 360^\circ \qquad \sum_i \theta_i = 360^\circ

$f_i$ frequency / count of category $i$
$N$ total frequency $\sum f_i$

Diagram · pie sector = (f / total) × 360°

Walk8/20 → 144° (40%)
Cycle6/20 → 108° (30%)
Bus4/20 → 72° (20%)
Car2/20 → 36° (10%)

Each slice's central angle is its share of 360°: Walk = 8/20 × 360° = 144°. The four angles add to 360° and the frequencies to the total — the check most pie-chart questions turn on.

Worked example

A company has 30 Science, 70 Arts and 50 Commerce graduates. Find the pie-chart angle for Science.

Total $N = 30 + 70 + 50 = 150$ .
Apply the formula: $\theta_{\text{Science}} = \dfrac{30}{150} \times 360^\circ$ .
Simplify: $\theta_{\text{Science}} = \dfrac{1}{5} \times 360^\circ = 72^\circ$ .

Answer:

\theta_{\text{Science}} = 72^\circ

Practice this conceptself-check · 4 quick reps

Try it yourself

In a pie chart, four sectors have central angles in the ratio

1 : 2 : 3 : 4

. Find each angle.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
A category is $\tfrac{1}{4}$ of the total. Sector angle?
2.
A category is $50$ of a total $200$ . Angle?
3.
All sector angles of a pie chart must sum to?
4.
A category is $\tfrac{1}{3}$ of the total. Angle?

From the bank · past-year question

Example 2StatisticsMODERATE

The central angles

p

q

r

and

s

(in degrees) of four sectors in a Pie Chart satisfy the relation

9p=3q=2r=6s

. What is the value of

4p-q

[Q107 · Apr · 2023]

All angles MUST sum to $360^\circ$

If your computed angles don't add to 360, you have an arithmetic error. PYQs that give angle relations ("

9p = 3q = 2r = 6s

") use

p + q + r + s = 360

as the closing equation — without that, the system is underdetermined.

Drill 4 more on pie charts

Concept 3 of 3

Reading Frequency Tables — Mode, Cumulative, Median

Intuition

A frequency table compresses a lot of data into one grid. Three things you read directly off it: the MODAL class (highest frequency), the CUMULATIVE frequency up to any class (running total), and the MEDIAN (the value where cumulative frequency crosses

n/2

Definition

Modal class: the class with the highest frequency. Cumulative frequency at class $k$ : $\sum_{i \leq k} f_i$ . Median for grouped data: $M = L + \dfrac{n/2 - F}{f}\,h$ , where $L$ = lower bound of the median class, $F$ = cumulative frequency before it, $f$ = frequency of median class, $h$ = class width.

Median from a Grouped Frequency Distribution

M = L + \dfrac{\tfrac{n}{2} - F}{f}\,h

$L$ lower bound of the median class
$F$ cumulative frequency BEFORE the median class
$f$ frequency of the median class
$h$ class width

Worked example

Heights (cm) of 20 students:

150{-}155 \to 4,\ 155{-}160 \to 6,\ 160{-}165 \to 7,\ 165{-}170 \to 3

. Find the median height.

Total $n = 4 + 6 + 7 + 3 = 20$ , so $n/2 = 10$ .
Cumulative frequencies: 4, 10, 17, 20. The 10th observation falls at the END of the $155{-}160$ class — but $n/2 = 10$ is reached at the boundary, so by convention the median class is $160{-}165$ .
Identify: $L = 160,\ F = 10,\ f = 7,\ h = 5$ .
Apply: $M = 160 + \dfrac{10 - 10}{7} \times 5 = 160 + 0 = 160$ .

Answer:

M = 160

Practice this conceptself-check · 4 quick reps

Try it yourself

Weekly wages (₹) of 30 workers:

0{-}500 \to 3,\ 500{-}1000 \to 8,\ 1000{-}1500 \to 12,\ 1500{-}2000 \to 7

. Find the median wage.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Frequencies $3, 5, 4$ . Cumulative frequency after the 2nd class?
2.
The modal class is the one with?
3.
$n = 20$ . The median lies at which cumulative position?
4.
Frequencies $2, 4, 6$ . Last cumulative total?

From the bank · past-year question

Example 3StatisticsMODERATE

The frequency distribution of height of students of a class is given below: Height (cm): 160–162, 162–164, 164–166, 166–168 Number of Students: 12, 15, 24, 13

What is the median height of the class?

[Q102 · Apr · 2025]

Cumulative frequency is RUNNING total, not class total

The cumulative frequency at class

k

is the sum of frequencies from class 1 through class

k

— not the frequency of class

k

alone. Tripping on this turns every median-from-grouped-data question into nonsense.

Identify the median class FIRST, then plug into the formula

The median class is the class where the cumulative frequency first reaches or exceeds

n/2

. Don't pick the class with the highest frequency (that's the modal class) or the middle row of the table.

Drill 3 more on reading frequency tables — mode, cumulative, median

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 (3)

Histograms, Frequency Polygons & Ogives
Frequency Density (for unequal class widths)
$\text{Density} = \dfrac{\text{Frequency}}{\text{Class width}}$
Pie Charts
Sector Angle in a Pie Chart
$\theta_i = \dfrac{f_i}{N} \times 360^\circ \qquad \sum_i \theta_i = 360^\circ$
Reading Frequency Tables — Mode, Cumulative, Median
Median from a Grouped Frequency Distribution
$M = L + \dfrac{\tfrac{n}{2} - F}{f}\,h$

Watch out for (4)

Bar height $\neq$ frequency when class widths differ
→ Histograms, Frequency Polygons & Ogives
All angles MUST sum to $360^\circ$
→ Pie Charts
Cumulative frequency is RUNNING total, not class total
→ Reading Frequency Tables — Mode, Cumulative, Median
Identify the median class FIRST, then plug into the formula
→ Reading Frequency Tables — Mode, Cumulative, Median

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1StatisticsEASY

For given statistical data, the graphs for less than ogive and more than ogive are drawn. If the point at which the two curves intersect is

P

, then abscissa of point

P

gives the value of which one of the following measures of central tendency?

[Q114 · Apr · 2017]

Example 2StatisticsEASY

In an examination, 40% of candidates got second class. When the data are represented by a pie chart, what is the angle corresponding to second class?

[Q109 · Apr · 2017]

Example 3StatisticsMODERATE

The frequency distribution of height of students of a class is given below: Height (cm): 160–162, 162–164, 164–166, 166–168 Number of Students: 12, 15, 24, 13

The height which occurs most frequently in the class is

[Q103 · Apr · 2025]

Example 4StatisticsEASY

The frequency distribution of height of students of a class is given below: Height (cm): 160–162, 162–164, 164–166, 166–168 Number of Students: 12, 15, 24, 13

The most appropriate graphical representation of the given frequency distribution is

[Q104 · Apr · 2025]

Example 5StatisticsEASY

In a Maths test, 20% got first class. Central angle in Pie Chart?

[Q111 · Apr · 2018]

Drill every past-year question on this subtopic

14 questions from the bank — paginated, with cart and Word-export support.

Drill the 14 questions

Related notes

Measures of Central Tendency — grouped-data median and mode use these tables →