Cheat Sheet — Statistics

📊 Data & Frequency Basics

Class mark xᵢ = (lower + upper) ÷ 2
Class size h = upper − lower limit

Key terms

Term	Meaning
fᵢ	frequency of a class
cf	cumulative (running) total
n = Σfᵢ	total frequency

Continuous classes: if there are gaps (e.g. 10–19, 20–29), adjust to 9.5–19.5… before finding median/mode.

➗ Mean of Grouped Data

Direct: x̄ = Σfᵢxᵢ ÷ Σfᵢ
Assumed mean: x̄ = a + (Σfᵢdᵢ ÷ Σfᵢ), dᵢ = xᵢ − a
Step-deviation: x̄ = a + h·(Σfᵢuᵢ ÷ Σfᵢ), uᵢ = (xᵢ−a)/h

All three methods give the same mean. Choose step-deviation when class widths are equal.

Pick a as the class mark near the middle (often the one with the highest frequency) to keep dᵢ small.

🎯 Median of Grouped Data

Median = l + [(n/2 − cf)/f] × h

What each letter means

l	lower limit of median class
n/2	half the total frequency
cf	cf of class BEFORE median class
f	frequency of median class
h	class size

Median class = first class whose cf ≥ n/2.

🔝 Mode of Grouped Data

Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h

l	lower limit of modal class
f₁	frequency of modal class
f₀	frequency of class before
f₂	frequency of class after
h	class size

Modal class = the class with the highest frequency.

📈 Ogive (CF Curve)

Less than ogive → plot (upper limit, cf)
More than ogive → plot (lower limit, cf)

Median from a graph

Draw both ogives on the same axes
Their point of intersection gives the median
Or draw a line at y = n/2 to the less-than curve, then drop to the x-axis

Ogive is always smooth & free-hand — never join points with straight lines in the exam.

⭐ All Formulas at a Glance

Measure	Formula
Mean (direct)	Σfx / Σf
Mean (assumed)	a + Σfd/Σf
Mean (step-dev)	a + h·Σfu/Σf
Median	l + [(n/2−cf)/f]·h
Mode	l + [(f₁−f₀)/(2f₁−f₀−f₂)]·h
Empirical	3Md = Mo + 2x̄

🔗 Empirical Relationship

3 Median = Mode + 2 Mean
Mode = 3 Median − 2 Mean
Mean = (3 Median − Mode) ÷ 2

Use it to estimate the third measure when the other two are known.

Order on the number line: for a moderately skewed distribution, Mean, Median and Mode differ but stay close.

🧠 Exam Quick Recall

Median: build the cf column first, then find n/2.

Mode: spot the tallest frequency = modal class before plugging in.

Common mistakes: using (n+1)/2 instead of n/2 for grouped data; forgetting to make classes continuous; taking cf of the median class instead of the class before it.

Step-deviation check: u = (x − a)/h should be small whole numbers (…−2,−1,0,1,2…).