Cumulative frequency — ogive, median from graph: Foundations

Before we can climb that curve, we need to earn every symbol the parent topic throws at you. We build them one at a time — no symbol appears here before it is defined in words and shown as a picture.

0. The raw material: an observation and a count

Look at Figure 1: each observation is a dot. We sweep them into buckets and count. That count is $f$ .

Figure — Cumulative frequency — ogive, median from graph

1. Classes and their boundaries

In Figure 2 each bucket is a strip on the number line. The left edge is the lower boundary; the right edge is the upper boundary; the distance across is $h$ .

Why does the topic obsess over the upper boundary? Because we are about to count everything below a line — and the natural place to put that line is the right edge of a bucket.

2. The running total: cumulative frequency

We know $f$ = "how many in this bucket." Now we ask a different question: how many in this bucket AND all buckets to its left? That running total is the star of the whole topic.

The symbol $\sum$ (a big Greek "S", for Sum) is just shorthand for "add these up":

$C F_{i} = f_{1} + f_{2} + \dots + f_{i} = \sum_{j = 1}^{i} f_{j}$

Read the right-hand side out loud: "start with $j = 1$ , add every $f_{j}$ up to $j = i$ ." The little $j$ is just a counter that walks through the buckets. Nothing mysterious — it's a running total, like adding up a shopping receipt line by line.

Figure 3 shows the receipt idea: each bar's height is $f$ ; the staircase behind them is the accumulating $C F$ .

3. $N$ and the halfway mark $N /2$

4. The median — what we are hunting for

On the ogive, the median is the x-value directly below the point where the curve reaches height $N /2$ . That is the entire payoff of building everything above.

Here is the last tool. When the halfway height $N /2$ lands inside a bucket, the ogive doesn't jump — it slopes. To read a value off a slope we assume the observations are spread evenly across the bucket, and slide along in proportion. That even-sharing rule is Linear interpolation.

$fraction into the bucket = \frac{\frac{N}{2} - C F _{b}}{f}$

where $C F_{b}$ is the running total just before this bucket and $f$ is this bucket's own count. Multiply that fraction by the width $h$ and add to the lower boundary $L$ :

$Median = L + (\frac{\frac{N}{2} - C F _{b}}{f}) \times h$

Every symbol here is now earned: $L$ = lower boundary of the median class, $C F_{b}$ = running total before it, $f$ = its count, $h$ = its width, $N /2$ = halfway height.

Prerequisite map

Each box is a symbol we built above; follow the arrows and you arrive at the median from the ogive. From the same curve you can later read Quartiles and percentiles from ogive, and you can compare the median against the Mean and mode of grouped data.

Equipment checklist

Hide the answers and test yourself — if any line stumps you, re-read its section.

What does the frequency

f

of a class mean?

How many observations fall into that class (its count).

What is a class's upper boundary, and why does the ogive use it?

The right edge of the bucket;

C F

counts everything below it, so the point sits honestly at that x-value.

What does

\sum_{j = 1}^{i} f_{j}

tell you to do?

Add up

f_{1}

through

f_{i}

— a running total.

What are the two anchor facts of

C F

C F_{1} = f_{1}

and

C F_{k} = N

What is

N

The total number of observations, the sum of all frequencies.

Why climb to height

N /2

(not

(N + 1) /2

) on an ogive?

The ogive is a smooth continuous curve, so we use the halfway height, not a discrete position.

In the median formula, what is

C F_{b}

The cumulative frequency of the class just before the median class.

What assumption does linear interpolation make inside the median class?

Observations are spread uniformly, so the curve is a straight ramp across the bucket.

How do you fix discontinuous classes like

10 - 19

Subtract 0.5 from lower and add 0.5 to upper →

9.5 - 19.5