Measures of Spread

Range, IQR, variance and standard deviation for raw and grouped data

SPEC 2.3 • STATISTICAL MEASURES

Why Measure Spread?

Two datasets can have the same mean but be very different in how spread out the data is.

Dataset A

49, 50, 50, 51

Mean = 50

Dataset B

20, 40, 60, 80

Mean = 50

Key Concept

Measures of spread (or dispersion) describe how spread out data is from the centre. They complement measures of central tendency to give a complete picture of a dataset.

Range

Formula

Range = Maximum - Minimum

The range is the simplest measure of spread - the difference between the largest and smallest values.

Advantages

Easy to calculate and understand.

Limitations

Only uses two values (extremes), so is heavily affected by outliers. Ignores all other data points.

IQR and Interpercentile Range

Interquartile Range

IQR = Q_{3} - Q_{1}

The range of the middle 50% of data

Interpercentile Range

The difference between any two percentiles. Common examples:

10th to 90th percentile range:

P_{90} - P_{10}

(middle 80%)

5th to 95th percentile range:

P_{95} - P_{5}

(middle 90%)

Why Use IQR?

IQR is not affected by outliers, making it more robust than range.

01Worked Example - IQR (Continuous)

The table shows journey times for 80 commuters. Estimate the IQR using linear interpolation.

Time (t min)0-1010-2020-3030-4040-50

Cumulative Freq520507280

Q₁ position:

\frac{80}{4} = 20

th value → at boundary of 10-20 class

Q₁ = 20 minutes

Q₃ position:

\frac{3 \times 80}{4} = 60

th value → in 30-40 class

Linear interpolation:

Q_{3} = 30 + \frac{60 - 50}{72 - 50} \times 10 = 30 + \frac{10}{22} \times 10 = 34.5 min

IQR = 34.5 − 20 = 14.5 minutes

What is Sₓₓ?

Definition

Sₓₓ is the sum of squared deviations from the mean.

S_{xx} = \sum (x - \overset{x}{ˉ})^{2}

Sₓₓ vs Variance - Key Differences

SₓₓVariance (σ²)Sum of squared deviationsMean of squared deviations

S_{xx} = \sum (x - \overset{x}{ˉ})^{2}

σ^{2} = \frac{S _{xx}}{n}

Gets larger with more dataIndependent of sample sizeUsed in intermediate calculationsUsed for comparing datasets

When to Use Each

Sₓₓ: Intermediate step; often given in exam questions for you to use.

Variance/SD: Final answer for comparing spread between datasets.

Variance

Variance is the average squared deviation from the mean.

Variance (σ²)

σ^{2} = \frac{\sum x ^{2}}{n} - \overset{x}{ˉ}^{2}

"Mean of squares minus square of mean"

Easiest Method

"Mean of squares minus square of mean" is the quickest calculation method. Find

\frac{\sum x ^{2}}{n}

first, then subtract

\overset{x}{ˉ}^{2}

.

Alternative using Sₓₓ:

σ^{2} = \frac{S _{xx}}{n}

Units

Variance has squared units. If data is in cm, variance is in cm².

Standard Deviation

Standard deviation is the square root of the variance.

Standard Deviation (σ)

σ = \frac{\sum x ^{2}}{n} - \overset{x}{ˉ}^{2}

"Square root of (mean of squares minus square of mean)"

Easiest Method

Calculate variance using "mean of squares minus square of mean", then take the square root.

Alternative using Sₓₓ:

σ = \frac{S _{xx}}{n}

Why Use Standard Deviation?

Has the same units as the original data

More interpretable than variance

Roughly represents the "average distance from the mean"

Worked Example02

Calculate the variance and standard deviation of: 3, 5, 7, 8, 12

n = 5, so

\overset{x}{ˉ} = \frac{35}{5} = 7

σ^{2} = \frac{291}{5} - 7^{2} = 58.2 - 49 = 9.2

σ = 9.2 = 3.03

(3 s.f.)

Sₓₓ Formulas

Two equivalent ways to calculate Sₓₓ:

Definitional Formula

S_{xx} = \sum (x - \overset{x}{ˉ})^{2}

"Sum of (each value minus the mean) squared"

Computational Formula

S_{xx} = \sum x^{2} - \frac{( \sum x ) ^{2}}{n} = \sum x^{2} - n \overset{x}{ˉ}^{2}

"Sum of squares minus (sum squared divided by n)"

Exam Tip

The computational formula is faster when given Σx² and Σx. Both give the same answer!

Worked Example03

For 8 observations: Σx = 120 and Σx² = 2040. Calculate Sₓₓ, variance and standard deviation.

Sₓₓ:

S_{xx} = \sum x^{2} - \frac{( \sum x ) ^{2}}{n} = 2040 - \frac{12 0 ^{2}}{8} = 2040 - 1800 = 240

Variance:

σ^{2} = \frac{S _{xx}}{n} = \frac{240}{8} = 30

SD:

σ = 30 = 5.48

(3 s.f.)

Variance with Frequency Data

When data has frequencies, we weight each value by its frequency:

Variance with Frequency

σ^{2} = \frac{\sum f x ^{2}}{\sum f} - \overset{x}{ˉ}^{2}

"Mean of squares minus square of mean" (using frequencies)

Easiest Method

Still use "mean of squares minus square of mean" — just weight everything by frequency f.

Setting Up Your Table

Create columns for: x (value or midpoint), f, fx, fx²

For grouped data, x = class midpoint

04Worked Example - Frequency

Find the standard deviation for the number of pets owned:

Pets (x)01234

Frequency (f)512841

\overset{x}{ˉ} = \frac{44}{30} = 1.467

σ^{2} = \frac{96}{30} - 1.46 7^{2} = 3.2 - 2.15 = 1.05

σ = 1.05 = 1.02

pets (3 s.f.)

05Worked Example - Grouped

Estimate the standard deviation of telephone call lengths:

Length (l min)0 < l ≤ 55 < l ≤ 1010 < l ≤ 1515 < l ≤ 20

Frequency41552

σ^{2} = \frac{2262.5}{26} - (\frac{220}{26})^{2} = 87.02 - 71.60 = 15.42

σ = 15.42 = 3.93

minutes (3 s.f.)

Sₓₓ, Variance and SD Summary

The Relationship

MeasureFormulaWhat It Means

Sₓₓ

\sum x^{2} - \frac{( \sum x ) ^{2}}{n}

Total squared deviation

Variance σ²

\frac{\sum x ^{2}}{n} - \overset{x}{ˉ}^{2}

Average squared deviation

SD σ

Variance

Spread in original units

Quick Conversion

Sₓₓ → Variance: divide by n

Variance → SD: take square root

Sₓₓ → SD:

σ = \frac{S _{xx}}{n}

Common Exam Pattern

Given Sₓₓ = 450 and n = 10:

Variance = 450 ÷ 10 = 45

SD = √45 = 6.71 (3 s.f.)

Using Your Calculator

Your Casio calculator can find statistical values quickly using STAT mode.

Raw Data

1. MODE → STAT → 1-VAR2. Enter each value, press = after each3. AC → SHIFT + 1 (STAT)4. Select Var for statistics

Frequency Data

1. SHIFT → MODE (Setup)2. Scroll to STAT → Frequency: On3. MODE → STAT → 1-VAR4. Enter x in col 1, freq in col 2

Available Statistics

Sample sizenSumΣxSum of squaresΣx²Meanx̄Population SDσxSample SDsx

Exam Tip

Use calculator to check manual calculations. For grouped data, use midpoints as x values with their frequencies.

Population vs Sample SD

Population SD (σ)

σ = \frac{S _{xx}}{n}

Divide by n

Sample SD (s)

s = \frac{S _{xx}}{n - 1}

Divide by n−1

When to Use Each

Population (σ): Data is the entire population

Sample (s): Data is a sample; gives "unbiased estimate"

Exam Tip

Use population SD (÷n) unless told it's a sample or asked for "unbiased estimate".

Worked Example06

A sample of 12 values has Sₓₓ = 132. Calculate both population and sample standard deviations.

Population SD:

σ = \frac{132}{12} = 11 = 3.32

(3 s.f.)

Sample SD:

s = \frac{132}{11} = 12 = 3.46

(3 s.f.)

Note: Sample SD is always slightly larger.

Comparing Data

When comparing two datasets, comment on both location and spread.

LOCATION

Compare mean or median:

"On average, A is higher/lower than B"

SPREAD

Compare SD or IQR:

"A is more/less consistent than B"

Interpretation

Larger SD/IQR = more variability, less consistency

Smaller SD/IQR = less variability, more consistency

Always interpret in context

Worked Example - Comparison07

Two factories produce bolts (target 50g):

Factory A: mean = 50g, SD = 1.2g

Factory B: mean = 50g, SD = 3.5g

Compare the factories.

Location: Both have same mean (50g) - on average, both meet target.

Spread: Factory A has smaller SD (1.2g vs 3.5g).

Conclusion: Factory A produces more consistent bolts. Factory B has greater variability. Factory A is more reliable.

Coding for Variance and SD

If data is coded using

y = \frac{x - a}{b}

:

Coding Rules

σ_{x} = b \times σ_{y} σ_{x}^{2} = b^{2} \times σ_{y}^{2}

Key Points

Adding/subtracting (−a): Does NOT change variance or SD

Multiplying/dividing (÷b): DOES change variance and SD

SD is multiplied by |b|, variance by b²

Why?

Shifting data (±a) moves all values by same amount - spread unchanged.

Scaling (×b) stretches/compresses data - spread changes.

Worked Example - Coding08

Heights coded using y = (x − 170)/5. Coded data: ȳ = 1.2, σ_y = 2.4. Find original mean and SD.

Coding:

y = \frac{x - 170}{5}

, so a = 170, b = 5

Original mean:

\overset{x}{ˉ} = b \overset{y}{ˉ} + a = 5 \times 1.2 + 170 = 176 cm

Original SD:

σ_{x} = b \times σ_{y} = 5 \times 2.4 = 12 cm

The "−170" doesn't affect SD, only "÷5" does.

Practice

Question 1: For the data 4, 6, 8, 10, 12, calculate the variance and standard deviation.

n = 5, Mean = 40/5 = 8

Variance = 360/5 − 8² = 72 − 64 = 8

SD = √8 = 2.83 (3 s.f.)

Practice

Question 2: Data is coded using y = (x − 50)/10. If σ_y = 1.5, find σₓ.

Coding: y = (x − 50)/10, so b = 10

σₓ = b × σ_y = 10 × 1.5 = 15

Practice

Question 3: IQR = 12, Q₁ = 25. Find Q₃. Range = 40, Max = 55. Find Min.

Q₃: IQR = Q₃ − Q₁ → 12 = Q₃ − 25 → Q₃ = 37

Min: Range = Max − Min → 40 = 55 − Min → Min = 15

Summary

Key Formulas

MeasureFormula

RangeMax − Min

IQRQ₃ − Q₁

Sₓₓ

\sum x^{2} - \frac{( \sum x ) ^{2}}{n}

Variance

\frac{\sum x ^{2}}{n} - \overset{x}{ˉ}^{2}

SD

Variance

Coding

σ_{x} = b \times σ_{y}

Key Exam Points

Variance: "mean of squares minus square of mean" is the easiest method

Sₓₓ = total; Variance = average (Sₓₓ/n)

When comparing: BOTH location AND spread

Coding: ±a doesn't change SD; ×b does