Cumulative frequency — ogive, median from graph

WHAT is cumulative frequency?

There are two flavours:

Less-than CF: total of all frequencies up to and including the upper boundary. Increases as you go right.
More-than CF: total of all frequencies from the lower boundary onwards. Decreases as you go right.

HOW to build the CF table (from first principles)

Suppose classes have frequencies $f_1, f_2, \dots, f_k$ . The less-than CF is defined by accumulation:

$CF_1 = f_1, \qquad CF_i = CF_{i-1} + f_i$

So by unrolling this recurrence:

$CF_i = f_1 + f_2 + \cdots + f_i = \sum_{j=1}^{i} f_j$

The last value is always $CF_k = N$ , the total number of observations. That's your sanity check.

Crucial rule — WHY use the upper boundary? For a less-than ogive, plot CF against the upper class boundary, not the midpoint. WHY? Because $CF_i$ counts everything below the upper boundary of class $i$ — so the point $(\text{upper boundary}, CF_i)$ is the honest statement "this many values are below this x."

The curve starts at the lower boundary of the first class with CF $=0$ and rises to $N$ .

Figure — Cumulative frequency — ogive, median from graph

HOW to find the median from the graph

Steps:

Compute $N/2$ (use $N/2$ , not $(N+1)/2$ , for grouped/continuous data — see mistake below).
On the y-axis, mark $N/2$ . Draw a horizontal line to the curve.
Drop a vertical line to the x-axis. Read the value → that's the median.

Twin-ogive method: Draw both the less-than and more-than ogives on the same axes. Where they intersect, drop a perpendicular to the x-axis — that x-value is the median.

The formula behind the graph (derivation)

The graph is a linear interpolation inside the median class. Let's derive it.

Let the median class be the class where CF first reaches or passes $N/2$ . Inside that class, assume frequency is spread uniformly. Let:

$L$ = lower boundary of median class
$CF_b$ = cumulative frequency of the class before the median class
$f$ = frequency of the median class
$h$ = class width

We need to go from CF $= CF_b$ (at $x=L$ ) up to $N/2$ . The extra count needed is $\left(\tfrac{N}{2} - CF_b\right)$ . Since $f$ observations are spread evenly across width $h$ , each unit of x adds $f/h$ to the count. So the horizontal distance to travel is:

$\Delta x = \frac{\frac{N}{2} - CF_b}{f/h} = \left(\frac{\frac{N}{2} - CF_b}{f}\right) h$

Worked Examples

Worked example Example 1 — Build CF and read the median

Marks of 40 students:

Marks	Frequency $f$
0–10	4
10–20	6
20–30	10
30–40	12
40–50	8

Less-than CF table:

Less than	CF
10	4
20	10
30	20
40	32
50	40

Why this step? Each CF = previous CF + current frequency (running total). Final CF $=40=N$ . ✓

Median: $N/2 = 20$ . The CF just before 20 is 10 (at "less than 20"), and CF reaches 20 at the top of the 20–30 class — so the median class is 20–30 (this is the class where the running total first reaches $N/2$ ). Why? The median class is where CF first hits or crosses $N/2=20$ ; CF goes from 10 (below 20) up to 20 within the 20–30 class, so that's the class containing the median position.

$L=20,\; CF_b=10,\; f=10,\; h=10$ $\text{Median} = 20 + \left(\frac{20-10}{10}\right)\times 10 = 20 + 1\times 10 = 30$ On the graph: go to $y=20$ , slide right to the curve, drop down → $x=30$ (the median sits exactly at the boundary here). ✓

Worked example Example 2 — Interpolation actually kicks in

Same table but $N/2$ lands mid-class. Frequencies: 0–10: 5, 10–20: 8, 20–30: 12, 30–40: 10, 40–50: 5. Total $N=40$ , $N/2=20$ .

CF: 5, 13, 25, 35, 40. First CF $\ge 20$ is 25 (at "less than 30"), so median class = 20–30. Why? CF jumps from 13 (below 20) to 25 (above 20) inside this class.

$L=20,\; CF_b=13,\; f=12,\; h=10$ $\text{Median}=20+\left(\frac{20-13}{12}\right)\times10 = 20 + \frac{7}{12}\times10 = 20+5.83 = 25.83$ Why this step? We needed 7 more counts past 13 to reach 20; the 12 counts fill the width of 10 uniformly, so $7/12$ of the way in. The graph would read $\approx 25.8$ at $y=20$ . ✓

Common Mistakes (Steel-manned)

Active Recall

Recall Quick self-test (hide and answer)

What does the y-value of an ogive at $x=30$ literally mean?
Why do we use $N/2$ and not $(N+1)/2$ ?
In the twin-ogive method, what's special about the crossing point?
Derive the median formula from "count needed = $N/2 - CF_b$ ."

Answers: (1) number of observations less than 30. (2) Grouped data → smooth curve, halfway height. (3) less-than = more-than = $N/2$ → median. (4) $\Delta x = (N/2-CF_b)/(f/h)$ , add to $L$ .

Recall Feynman: explain to a 12-year-old

Imagine a queue of kids sorted shortest to tallest. "Cumulative frequency" is just counting "how many kids are shorter than this line I drew." If I keep drawing lines and counting, and plot it, I get a curve that always goes up. Now I want the middle kid. Half the kids is 20 (out of 40). So I go up the height axis to 20, walk right until I hit the curve, then look straight down — that height is the middle kid's height. Easy!

Flashcards

Cumulative frequency of a class is

the sum of that class's frequency and all frequencies before it (running total).

An ogive is a graph of

cumulative frequency (y) vs class boundary (x), joined by a smooth curve.

For a less-than ogive you plot CF against the

upper class boundary.

The final cumulative frequency always equals

N, the total number of observations.

To read the median from an ogive, go up the y-axis to

N/2, across to the curve, then down to the x-axis.

The median class is the class

inside which the cumulative frequency first reaches or crosses N/2 (CF goes from below N/2 to ≥ N/2).

Grouped-data median formula

Median = L + ((N/2 − CF_b)/f)·h.

In the twin-ogive method the median is the x-value at

the intersection of the less-than and more-than ogives.

Why N/2 (not (N+1)/2) for grouped data

because the ogive is treated as a continuous smooth curve, so we split the total height in half.

Meaning of L, CF_b, f, h in the median formula

lower boundary of median class, CF of class before it, frequency of median class, class width.

Connections

Median (measures of central tendency)
Frequency distribution table
Histogram and frequency polygon
Quartiles and percentiles from ogive
Mean and mode of grouped data
Linear interpolation

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, cumulative frequency ka matlab simple hai: har class tak ka "running total". Yaani "kitne log iss value se KAM hain?" Jaise "40 se kam marks kitno ne laaye?" Table mein har CF = pichhla CF + current frequency. Last CF hamesha N (total) hota hai — yeh tumhara check hai ki galti to nahi hui.

Ab ogive ek graph hai: x-axis pe class ka upper boundary, y-axis pe cumulative frequency, aur points ko smooth curve se jodo. Yaad rakho — midpoint nahi, upper boundary use karna hai, kyunki CF "upper boundary se neeche kitne" batati hai. Curve hamesha upar ki taraf badhta hai, kyunki total sirf badh sakta hai.

Median nikalna graph se bahut easy: y-axis pe $N/2$ tak jao, right slide karke curve ko touch karo, phir seedhe neeche x-axis pe utaro — wahi median hai. Ya phir less-than aur more-than dono ogive banao, jahaan wo cross karti hain, wahi median. Median class wahi hai jiske andar CF pehli baar $N/2$ tak pahunchti hai (CF $N/2$ se neeche se upar jaati hai) — off-by-one galti se bacho! Example 1 mein CF 10 se 20 tak 20–30 class ke andar badhti hai, isliye median class 20–30 hai, 30–40 nahi.

Formula bhi yahi kaam karta hai: $\text{Median} = L + \frac{N/2 - CF_b}{f}\times h$ . Isme graph ka interpolation hi chhupa hai — median class ke andar frequency ko uniformly maan ke, jitna count aur chahiye ( $N/2 - CF_b$ ), utna aage badho. Board exam mein $N/2$ use karo (grouped data), $(N+1)/2$ nahi — yeh sabse common galti hai!

Cumulative frequency — ogive, median from graph

WHAT is cumulative frequency?

HOW to build the CF table (from first principles)

HOW to draw an ogive

HOW to find the median from the graph

The formula behind the graph (derivation)

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Statistics & Probability — Intermediate

Connections