Intuition The one core ideaA whole pile of numbers is hard to hold in your head, so we squeeze it down to one honest summary number — a centre. Everything in this topic is just three different, careful ways of asking "where is the middle of this pile?" (fair-share, halfway-point, most-crowded-spot).
Before you can read a single formula in the parent note, you need to earn every symbol in it. Below, each symbol gets: plain words → the picture → why the topic needs it. They are ordered so each one leans on the one before it.
Definition An observation and the countn observation is one single measured value in your data — one person's height, one test score. The letter n how many observations you have (the count of the pile).
Picture a row of pebbles on a table. Each pebble is one observation. Count them: that count is n
Why the topic needs it: every centre-measure divides or positions by how many things there are . Without n ÷ n n /2
x i x i i i i name tag for position: x 1 x 2 x i raw single observation — we will use a different letter for class midpoints later (Section 6), so this symbol never gets overloaded.
Picture the pebbles again, now numbered left to right: pebble #1, pebble #2, pebble #3. The value written under pebble #3 is x 3
Intuition Why bother with the subscript?Because we want to say "do this to every value" without writing them all out. Instead of "add x 1 x 2 x 3 x i i
Why the topic needs it: the mean, median and mode formulas all talk about "each value" or "each class". The subscript is the language for that.
∑ i = 1 ∑ n x i x i i = 1 i = n x 1 + x 2 + x 3 + ⋯ + x n
Intuition Reading the little numbers on∑ below the sigma (i = 1 where to start — begin at the first slot. The number above (n where to stop — the last slot. So i = 1 ∑ n i 1 n x i ∑ x i i = 1 n
Picture sweeping your hand across the whole row of pebbles — starting at pebble 1, ending at pebble n i = 1 ∑ n x i
∑ Wrong feeling: "the big symbol must mean something fancy like power or product."
Fix: it is only addition , done to a whole list at once, from the bottom index up to the top index. Nothing more.
Why the topic needs it: the very first formula, x ˉ = n ∑ i = 1 n x i ∑
x ˉ x ˉ mean — the fair-share value. The bar on top is just a hat that says "this is the average of the x
Intuition The picture: a see-sawImagine each observation is a weight placed on a ruler at its value. The mean x ˉ balances — the pushes to the left cancel the pushes to the right. That is why we can write ∑ ( x i − x ˉ ) = 0
Why the topic needs it: x ˉ all three symbols are defined — see Section 9).
Definition A class and its frequencyf i 0 – 10 10 – 20 class . The frequency f i how many observations fell into that class .
Picture sorting the pebbles into labelled jars. Jar "10 – 20 f = 8
Why the topic needs it: once data is in jars we no longer see individual pebbles, so every grouped formula (mean, median, mode) is written in terms of f i
Definition Midpoint, boundary, widthFor a class like 20 – 30
the lower boundary l = 20 starts ),
the class width h = 30 − 20 = 10 wide the range is),
the midpoint m i = 2 20 + 30 = 25 middle of the range).
Note we use a fresh symbol m i for the midpoint — not x i ∑ f i x i x i m i
Intuition Why a midpoint?Inside a jar we've lost the exact values. The fairest single stand-in for "somewhere between 20 and 30" is the middle, 25. So we pretend every one of those items sits at the midpoint m i
Why the topic needs it: grouped mean uses m i l h into a class.
Common mistake Boundary vs printed limitIf classes are printed as 20 – 29 , 30 – 39 gap . Convert to continuous boundaries (19.5 – 29.5 , … l 20 – 30 Histograms & Frequency Distributions for why continuity matters.
Definition The median symbolx ~ median x ~ the value that splits the ordered data into two equal halves — 50% below, 50% above. It sits at position n /2
Intuition Why "ordered" comes firstA middle only makes sense once things are lined up smallest-to-largest. On a jumbled pile there is no left-half and right-half. So step one is always sort , then read off the centre.
n Wrong feeling: "just grab one of the two centre numbers."
Fix: when n must average the two central values — picking only one gives a biased answer.
Definition Cumulative frequencyc f cumulative frequency of a class is ==the running total: how many observations fall in that class or any earlier one ==.
Picture pouring jar 1 into a big tank, then jar 2, then jar 3 — reading the tank's level after each pour. Those running levels are the c f ogive .
Intuition Where this formula comes from — a straight-line rampLook at the figure below. On an ogive the cumulative frequency climbs from c f left boundary l c f + f right boundary l + h assume the f across the class, so between those two boundaries the climb is a straight ramp. The median is the height where the ramp crosses the level n /2 x ~ x ~ − l fraction of the width h 2 n − c f f
Why the topic needs it: the median is a position (n /2 c f where inside that class .
Definition The mode (ungrouped)The mode is the value that occurs most often . For raw data you simply tally how many times each value appears and pick the biggest tally.
Common mistake Modes can tie — and there can be more than oneA data set may have:
one mode (unimodal), e.g. 2 , 2 , 3 , 5 2 two modes (bimodal), e.g. 2 , 2 , 5 , 5 , 7 2 and 5 no mode — if every value appears equally often (e.g. 1 , 2 , 3 , 4
When ties happen, report all the tied values; do not silently pick one.
Definition The grouped-mode frequenciesf 0 , f 1 , f 2 modal class = the class with the highest individual frequency):
f 1 target class we build the answer inside),f 0 immediately before the modal class,f 2 immediately after the modal class.
Picture three bars side by side, the middle one (f 1 f 0 f 2 histogram .
Common mistake When the denominator2 f 1 − f 0 − f 2 f 0 = f 2 = f 1 0 breaks (division by zero). This signals there is no single clear peak — the top is flat. Likewise, if two different classes tie for the highest frequency , the distribution is bimodal and you should not force one modal class; either report both or use a finer grouping. The formula only applies when one class is a strict, isolated maximum.
Every grouped formula ends in a ratio — one quantity divided by another:
x ˉ = count total , median position fraction f n /2 − c f , mode lean fraction 2 f 1 − f 0 − f 2 f 1 − f 0 .
means here
A ratio answers "how much of one thing per one of another." Fair-share is total per person . The median position fraction is what portion of the way through the median class the middle person sits. The mode lean fraction is what portion of the class width the peak has leaned. Every centre-measure is secretly a "how-far-along" question — and a ratio is the tool that answers "how far along."
Read each arrow as "feeds into ". The four building blocks on the left combine into the three centre-measures on the right, which together form the topic.
Test yourself — cover the right side and answer before revealing.
What do the small numbers on i = 1 ∑ n Start adding at slot i = 1 i = n x i
What does the subscript i x i A position name-tag: x i i
Which symbol do we use for a class midpoint, and why not x i m i
What single operation does ∑ Add up every listed value into one total — pure addition, from the bottom index to the top.
What is x ˉ The mean; the pivot/balance point where pulls above and below cancel, so ∑ ( x i − x ˉ ) = 0
What is a frequency f i How many observations fall into that class (jar-count).
How do you get the total count n Add all frequencies: n = ∑ i = 1 k f i
For class 20 – 30 l h m i l = 20 h = 10 m i = 25
Ungrouped median when n Odd: the 2 n + 1 2 n 2 n + 1
Why must you sort before finding a median? A "middle" only exists once values are lined up smallest-to-largest.
What does cumulative frequency c f The running total of observations in this class and all earlier ones.
Why does the grouped-median formula work geometrically? The ogive climbs as a straight ramp across the median class; the median is where the ramp reaches height n /2
In the grouped median formula, which c f f c f before the median class; f of the median class itself.
Ungrouped mode with ties — what do you report? All tied values (unimodal / bimodal); if every value is equally frequent, there is no mode.
When does the grouped-mode formula break, and what does that mean? When 2 f 1 − f 0 − f 2 = 0
Which frequency is f 1 f 0 , f 2 f 1 f 0 f 2
State the empirical relation between the three centres. Mode ≈ 3 x ~ − 2 x ˉ
