MathematicsChapter 12: Basic Statistics

Range, Variance, Standard Deviation

Range = max - min (crude spread).

Learning objectives

Explain range, variance, standard deviation using the correct definitions, conditions, and notation.
Apply SD = sqrt(variance), grouped mean = sum fx / sum f correctly in routine and exam-style questions.
Show each important step, simplify the result, and check that it satisfies the original conditions.

Definition

Range = max - min (crude spread). Variance is the average of squared deviations from the mean (spread around the mean). Standard deviation is the positive square root of variance, in the same units as the data. Grouped mean uses x-bar = sum f xsum f with x the class mark.

Mathematical form

mean = 23010

mean = 23

Range = max - min (crude spread). Variance is th

bar = sum f xsum f with x the class mar

Where / symbols

RANGE, VARIANCE, STANDARD DEVIATION: Range = max - min (crude spread). Variance is the average of squared deviations from the mean (spread around the mean). Standard deviation is the positive square root of variance, in the same units as the data. Grouped mean uses x-bar = (sum f x) / (sum f) with x the class mark.
RANGE AND SD: Range uses only two values, so one outlier inflates it. SD uses every value. Variance is SD squared, not the other way around.
GROUPED MEAN: Class mark is the midpoint of the class. Multiply by frequency, add, divide by n = sum f. This estimates the mean when raw lists are gone.

Relationship

SD = sqrt(variance)
grouped mean = sum fx / sum f

Overview

Location vs spread:

Range and Sd

Range uses only two values, so one outlier inflates it. SD uses every value. Variance is SD squared, not the other way around.

Examples Range = max-min · SD = sqrt(variance) · Variance is squared units

Grouped Mean

Class mark is the midpoint of the class. Multiply by frequency, add, divide by n = sum f. This estimates the mean when raw lists are gone.

Examples sum fx / sum f · Not sum x / n ignoring f · Marks, not limits, for the mean formula

WHAT — RANGE, VARIANCE, STANDARD DEVIATION

WHY — Why this topic matters

Range, Variance, Standard Deviation matters because statistics turns raw data into useful information about position, spread, and relationship. Probability describes chance, while graphs and summary measures support fair comparisons.

HOW — How to solve and explain it

Start by identifying what is given and what must be found. Work through RANGE AND SD, then GROUPED MEAN. Worked: class marks 10, 20, 30 with f = 2, 3, 5. Find the mean. sum f = 2+3+5 = 10 sum f x = 210 + 320 + 5*30 sum f x = 20 + 60 + 150 = 230 mean = 23010 mean = 23 Finish by checking signs, restrictions, units, and whether the answer matches the question.

Worked example

class marks 10, 20, 30 with f = 2, 3, 5. Find the mean.
sum f = 2+3+5 = 10
sum f x = 2*10 + 3*20 + 5*30
sum f x = 20 + 60 + 150 = 230
mean = 23010
mean = 23

In simple words

Do not take SD as variance squared. SD is the square root. Variance is the square.

Key points

SD = sqrt(variance)
grouped mean = sum fx / sum f
SD same units as data
x = class mark

Exam preparation

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