DSE Maths | Measures of Dispersion: Tutor Prep Guide | Standard Deviation, Variance and Box Plots

2026-11-05

Tutor lesson-prep guides · Secondary Maths

This lesson fixes one habit: students can run the formula but cannot say what a large standard deviation means, so every comparison question becomes a guess.

Objectives — 1) Explain that standard deviation measures how far values sit from the mean. 2) Compute variance and standard deviation in five steps and round to three significant figures. 3) Draw a box plot from the five-number summary and use it to compare two distributions.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Mean	What is the mean of 2, 4, 6 and 8?	5
Squares and roots	Is the square of a negative number positive or negative?	Positive
Frequency tables	How do you total values from a frequency table?	Multiply each value by its frequency, then add

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
5 minutes	Opening check	Two sets of five values with the same mean but different spread: which class is more consistent?	Vote and give a reason
8 minutes	From mean to deviation	Measure spread by the gap from the mean and expose the cancelling problem	Compute the five gaps and add them
12 minutes	Five-step routine	Work one full example writing every step on the board	Repeat it and check the rounding
10 minutes	Box plots	Extract the five-number summary from the same data and draw the plot	Draw one and label the median
10 minutes	Comparing two sets	Compare with standard deviation and interquartile range	Write two comparison sentences

03 Teaching script (read it out)

Scene 1: The mean is not the whole story

Write two sets. A: 48, 49, 50, 51, 52. B: 20, 35, 50, 65, 80. Ask which class performed more consistently.
Most answer A. Ask what evidence they used and whether it can be written as a formula. This motivates a measure of distance from the mean.

Students should say the values are closer together, and accept that a number is needed to compare.

Scene 2: Deviations cancel

Have students compute the gaps. For A they are minus two, minus one, zero, one, two, which sum to zero. Ask whether zero means no spread, then introduce squaring before adding.

Students should name the cancelling problem and record it in their notes.

Scene 3: The five steps

Model: find the mean, find each gap, square the gaps, average the squares to get the variance, then take the root. Work B fully and round to three significant figures.

Students should recite all five steps and state that the fourth step divides by the number of values.

Scene 4: Box plot and comparison

From B take the minimum, lower quartile, median, upper quartile and maximum, then draw the box and whiskers. Ask what a longer box means.
Then model two comparison sentences, one about centre and one about spread, each containing a number.

They should say the middle half is more spread out, and write both sentences with figures.

04 Question types and worked steps

Type	Example	Steps	Answer
Direct computation	Find the standard deviation of 4, 6, 8, 10 and 12.	1) Mean 8 2) Gaps from minus four to four 3) Sum of squares 40 4) Variance 8 5) Standard deviation about 2.83	The standard deviation is about 2.83 to three significant figures.
Comparison	Two classes share a mean. Class A has standard deviation 4.2, class B 9.1. Which is more consistent?	1) Compare the values 2) The smaller one is more consistent 3) Say the marks are more concentrated	Class A is more consistent because its standard deviation is smaller, so marks cluster closer to the mean.
Effect of a change	Every value increases by 5. What happens to the standard deviation?	1) The mean rises by 5 2) Every gap is unchanged 3) The standard deviation is unchanged	It is unchanged because the distance of each value from the mean has not changed.
Reading a box plot	Find the interquartile range from a box plot and compare two groups.	1) Read the quartiles 2) Subtract 3) Compare box lengths	The interquartile range is the upper quartile minus the lower quartile, and the longer box holds the more spread middle half.

Sanity check (teach this): A standard deviation can never be negative; data that are all equal give zero; adding a constant shifts the mean but not the standard deviation; multiplying by a constant multiplies the standard deviation by its absolute value. Round only at the end.

05 Independent practice (3 questions)

Question	Answer	What to watch
Find the variance of 2, 4 and 6	8 divided by 3, about 2.67	Did they square before averaging?
What does a standard deviation of zero mean	Every value is identical	Did they say all values are equal?
Every value is tripled. What happens to the standard deviation?	It triples	Did they use the absolute value?

06 Common mistakes and what to say

Mistake	Student writes	Say this
Adding raw gaps	They add up to zero, so the standard deviation is zero.	The sum is zero because positives and negatives cancel, not because there is no spread. Square first.
Dividing by n minus one	I divided by n minus one.	For this syllabus the variance divides by n unless the question says otherwise.
Comparing means only	Both classes have the same mean so they are the same.	Same mean, but 4.2 against 9.1. Which class carries more risk? Talk about spread.
Rounding too early	I rounded after the first step.	Early rounding shifts the final answer. Keep full precision until the last line.

07 Homework (paper, section and question numbers)

S6 Normal Distribution guide, section 6, question type table row 1 — Standard deviation is the entry ticket for normal distribution questions
S6 Probability guide, section 7, independent practice question 2 — Comparison questions often sit inside probability contexts
Subject index: secondary maths prep guides — Redo the five-step routine before the next lesson

08 Exit check (last 5 minutes)

Question	Answer	Decision
Say what standard deviation measures	How far values sit from the mean	Saying it is the range means reteach
Complete one full computation	All five steps, three significant figures	Missing the square or the root means redo
Write one comparison with numbers	Class A at 4.2 is more concentrated than class B at 9.1	About the same is not acceptable

09 Tutor FAQ

Which year is this for

Mainly S5. In S4 the first act and the five-number summary can run alone.

Should students use the calculator statistics mode

Yes, demonstrate it once, but also require one hand computation so the steps are understood.

Standard deviation or interquartile range

With extreme values the interquartile range is more robust; otherwise compare standard deviations, and follow the question when it specifies.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.