Mean, median, mode — calculation for raw and grouped data: Foundations

Before you can compute anything, you must be fluent in the little symbols and pictures the parent note throws at you. This page builds every one of them from nothing, in an order where each new idea leans only on the ones before it.

1. What "data" actually is

Figure — Mean, median, mode — calculation for raw and grouped data

We write each observation with a subscript to give it a name and a position.

Why the topic needs it: without addresses we couldn't say things like "average the 3rd and 4th values." The subscript lets us talk about any slot without knowing what number is in it yet.

2. Counting the pile: $n$ and $N$

3. The sum symbol $\sum$ — the star of the show

The single most important piece of new notation is the Greek capital sigma, $\sum$ . Everything else is arithmetic; this is the one symbol that scares people, so we build it slowly.

Read the figure like a conveyor belt: the counter $i$ ticks $1, 2, 3, \dots, n$ , and at each tick the value $x_{i}$ drops into a running total (the red bucket). When $i$ passes $n$ , the belt stops and the bucket holds the sum.

Why the topic needs it: every formula for the mean starts with $\sum$ . The grouped-data mean $\frac{\sum f x}{N}$ is literally "add up (frequency times midpoint) for each group, then divide."

4. The bar: $\overset{x}{ˉ}$ means "mean"

The mean is also the balance point of the data — the spot where the row of values would sit level on a see-saw.

Look at the red triangle (the pivot) under the see-saw: it sits exactly at $\overset{x}{ˉ}$ . Values far to the right (like a big outlier) push the pivot toward them — that is why the mean gets "pulled by extremes," a fact the parent note relies on but does not picture.

5. Sorting, position and the "middle" — median groundwork

6. Grouped data: intervals, class marks, width

When there are too many values to list, we sort them into buckets. This is where the capital $N$ and the letters $f, x, h, L$ enter.

In the figure, each bar is a class interval; its width along the axis is $h$ , its height is the frequency $f$ , and the red dot on top of the axis marks the class mark $x$ dead-centre. The lower edge of a bar is its lower boundary $L$ — the last new symbol.

Recall Why does the grouped median add a fraction of

$h$ to $L$ ? Because we assume the class's $f$ values are spread evenly across its width $h$ . To reach observation number $\frac{N}{2}$ we must step $(\frac{N}{2} - C F)$ values into a class holding $f$ values over width $h$ , i.e. a fraction $\frac{\frac{N}{2} - C F}{f}$ of the way across — so we walk that fraction of $h$ past the starting edge $L$ .

How the foundations feed the topic

Once these symbols feel natural, they unlock: Frequency Distributions (where classes and frequencies come from), Cumulative Frequency Graphs (the picture behind $C F$ and the grouped median), Weighted Mean (the $\sum f x / N$ idea generalised), Measures of Dispersion and Box Plots (spread around these centres), and Skewness (why mean, median and mode drift apart). Return to the parent: Mean, median, mode — calculation for raw and grouped data.

Equipment checklist

Cover the right side and test yourself — you are ready only if every line is instant.

x_{i}

means

the observation in slot number

i

(its address, not a fixed number)

n

means

how many observations there are, counted individually (raw data)

N

means

total number of observations found by summing frequencies (grouped data)

\sum_{i = 1}^{n} x_{i}

means

add up

x_{i}

for

i = 1, 2, \dots, n

\overset{x}{ˉ}

means

the mean, the total

\sum x_{i}

shared equally among

n

Ascending order means

smallest to largest, which fixes each value's position

Median position for odd

n

the

\frac{n + 1}{2}

-th value after sorting

Median position for even

n

average of the

\frac{n}{2}

-th and

\frac{n}{2} + 1

-th values

Class mark

x_{i}

midpoint of a class = (lower limit + upper limit) / 2

Class width

h

upper limit minus lower limit of a class

Frequency

f_{i}

how many observations landed in class

i

Cumulative frequency

C F

running total of frequencies up to a given class

Lower boundary

L

the left edge (lower limit) of a class