Intuition The ONE core ideaA pile of numbers has a middle (the mean) and a width (how far the numbers stray from that middle). This whole topic is just a careful recipe for turning "how wide is the spread?" into a single honest number — and every symbol below is one small brick in that recipe.
Before you can understand variance, you have to be able to read it. The parent note throws a lot of little symbols at you very fast: ∑ x ˉ x i n σ 2 μ x 2 each one from absolute zero , anchors it to a picture, and says why the topic can't live without it .
Everything starts with a list of numbers you actually measured — test marks, dart distances, heights, whatever.
Definition Data value and the subscriptx i data value . We write the whole collection as x 1 , x 2 , x 3 , … , x n
The letter x
The little number underneath, called a subscript , is just a name tag telling us which one . x 1 x 2 x i i i 1 n
Look at the figure. Each dot on the number line is one x i x what kind of thing , the subscript tells you which specific one .
Intuition Why the topic needsx i x i i 1 n
n n how many data values there are — the number of dots on the line. In 2 , 4 , 4 , 6 , 9 n = 5
Writing x 1 + x 2 + x 3 + x 4 + x 5 "add up a whole sequence."
∑ ∑ S for S um) is a machine that adds. Read it left-to-right like a sentence:
∑ i = 1 n x i = x 1 + x 2 + x 3 + ⋯ + x n
The i = 1 below = "start the counter at 1."
The n above = "stop when the counter reaches n
The x i after = "here is the thing you add each time."
Look at the figure. The counter i 1 , 2 , 3 , … x i n
Worked example Reading a sigma out loudi = 1 ∑ 5 x i 2 , 4 , 4 , 6 , 9 2 + 4 + 4 + 6 + 9 = 25 i = 1 ∑ 5 x i 2 4 + 16 + 16 + 36 + 81 = 153
∑ x i 2 ( ∑ x i ) 2 Wrong feeling: the square can slide outside the sum. It cannot.
∑ x i 2 = 4 + 16 + 16 + 36 + 81 = 153 ( ∑ x i ) 2 = 2 5 2 = 625 2 inside means "square each term first." Position of the exponent changes everything — watch it like a hawk.
Intuition Why the topic needs∑ ∑
To ask "how far do numbers stray from the middle?" we first need the middle .
x ˉ mean (everyday "average") is the total divided by the count:
x ˉ = n 1 ∑ i = 1 n x i x x-bar ") is the universal signal for "the average of the x
Look at the figure. Imagine the number line is a see-saw and each data value is a weight sitting where it lands. The mean is the exact balance point — the pivot where the plank tips neither left nor right. Values far to the right are counterweighted by values far to the left.
Definition Population meanμ x ˉ where the data came from :
x ˉ sample — a handful you grabbed to represent a bigger group.μ whole population — every single member, none missing.
They are computed identically; the different letters just remind you whether you measured everyone (μ some (x ˉ n n − 1
Intuition Why the topic needs the mean firstSpread is always measured from a centre . The mean is that centre. See Mean, Median, Mode — measures of central tendency for why the mean (not median) is the natural partner of variance — it's the balance point, and squaring distances is happiest around a balance point.
Now that we have a centre, we can ask each value: how far are you from it?
Look at the figure. Each coloured arrow runs from the mean to a data point . Arrows pointing right are positive deviations (orange), arrows pointing left are negative (blue). The length of the arrow is how far off that value is; the direction is the sign.
Intuition The magic cancellation — why we can't stop hereAdd up every arrow: the rightward ones exactly cancel the leftward ones, because the mean is the balance point. In symbols, ∑ d i = ∑ ( x i − x ˉ ) = 0 always . That is the whole reason the parent note squares the deviations — to destroy the signs so the distances stop cancelling. Hold onto this arrow picture; it is what "( x i − x ˉ ) 2
The shortcut formula in the parent note lives or dies on telling these apart.
Definition Mean of the squaresx 2 Square every value first, then average.
x 2 = n 1 ∑ x i 2 whole x 2 inside , per value.
Definition Square of the mean( x ˉ ) 2 Average first, then square the single answer.
( x ˉ ) 2 = ( n 1 ∑ x i ) 2 x outside , applied once at the end.
Worked example Watch them differData 2 , 4 , 4 , 6 , 9
x 2 = 5 153 = 30.6 ( x ˉ ) 2 = 5 2 = 25 5
They are not equal — and the parent note's shortcut says their difference is the variance : 30.6 − 25 = 5.6
Read the bar's position. Bar over x 2 x
σ 2 σ σ ∑ statistic , not the sum machine) is the standard deviation : a single number for the typical arrow-length, in the data's own units.
Its square σ 2 variance : the average of the squared arrow-lengths, in squared units.
σ 2 ≥ 0 σ = σ 2
σ σ ∑ σ
mean x-bar = total over n
deviation = value minus mean
square the deviation to kill signs
variance = average squared deviation
mean of squares vs square of mean
standard deviation = square root of variance
Coefficient of Variation, Normal Distribution
Read it top to bottom: raw data need a count and a sum to build the mean ; the mean lets us measure deviations ; squaring and summing them gives variance ; a square root gives standard deviation , which then powers Coefficient of Variation and the Normal Distribution .
Reveal each and check you can answer instantly before tackling the parent note.
What does the subscript in x i A name tag saying which data value — the i i n
What does i = 1 ∑ n x i Start a counter i x i i = n
Is ∑ x i 2 ( ∑ x i ) 2 No. ∑ x i 2 ( ∑ x i ) 2
What is n The count — how many data values you have.
Give the formula for the mean x ˉ x ˉ = n 1 ∑ i = 1 n x i
What picture is the mean? The balance point of a see-saw with a weight at each data value.
Difference between μ x ˉ μ x ˉ
Define the deviation d i d i = x i − x ˉ
What does ∑ ( x i − x ˉ ) Exactly 0 — the mean is the balance point, so rightward and leftward deviations cancel.
x 2 ( x ˉ ) 2 x 2 ( x ˉ ) 2
Variance σ 2 σ σ 2 is in squared units;
σ = σ 2 restores the original units.
