Which set has the smaller range?

The 3, 4, 5 set. Its extreme values are closer together, although both sets have the same average.

Which distances count: those below the centre, above it, or both?

Both. Both measure distance from the chosen centre, so neither is discarded.

What number divides the total distance here?

Four. The two rows together contain four observations.

Why does taking absolute values after adding fail here?

Opposite differences cancel. Taking each distance first preserves both contributions.

What unit belongs to this standard deviation?

Centimetres. The square root changes square centimetres back into centimetres.

How many times must the squared distance for 3 count?

Twice. Two observations have that value, and each contributes to variance.

Which changes when 10 is added back: the mean or the variance?

The mean. Adding equally to every reading leaves its distance from the mean unchanged.

What replaces 5^2 in the square total?

3^2. The replacement contributes its square because this total contains squared readings.

Teach Statistics in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather paper and a pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

Average travel time alone does not show how much journeys vary.
The spread of daily spending helps families plan a buffer.
Variation in manufactured lengths shows how consistently pieces are made.

What to teach

How to teach it

Seeing spread

Two sets share mean 4 but have different gaps between their extremes.

1. Measures of dispersion and range

Spread shows variation; range measures the gap between extremes.

Show: Look at the diagram. Find the shared mean, 4, then compare the arrows joining each set's extremes.
Ask: With matching means, will these sets be equally consistent?
Say: Set B has range 2; Set A has range 4. Matching means do not guarantee matching spreads.
Used in: The largest clothing measurement minus the smallest gives the range. This helps makers compare size variation.

If your child says…“The same average means equally consistent values.”

Ask: Are the extreme dots equally far apart in both sets?
What’s true: An average does not tell us how far apart the values are.

Mean deviation

Distances from the same observations to their mean and median.

2. Mean deviation for ungrouped data

Average distance from a chosen centre.

Show: Look at the diagram's ordered values: 1, 2, 6. Compare distances from mean 3 and median 2, the middle value.
Ask: Will averaging the distances give the same result from each centre?
Say: The distances average 2 about the mean and $\frac{5}{3}$ about the median.
Used in: Mean deviation describes water-use variation, helping plan supplies.

If your child says…“The differences cancel, so there is no spread.”

Ask: Do all the dots sit at the centre?
What’s true: Distances cannot cancel because each distance is non-negative.

$\bar{x}=\frac{1(1)+3(3)}{4}=2.5$

$\mathrm{M.D.}(\bar{x})=\frac{1|1-2.5|+3|3-2.5|}{4}$

3. Mean deviation for grouped data

Count repetitions; represent intervals by midpoints.

Show: Write these lines for intervals $[0,2)$ and $[2,4)$ , with frequencies 1 and 3. Say: “Use midpoints 1 and 3. X-bar means mean; M.D. means mean deviation.”
Ask: What estimated mean deviation will the final division give?
Say: The estimate is 0.75.
Used in: Grouped delivery times help businesses estimate variation without listing every delivery.

If your child says…“There are two rows, so I divide by two.”

Ask: How many observations do the two frequencies count altogether?
What’s true: The divisor counts observations, including repetitions.

$\mathrm{M.D.}(3)=\frac{|-2|+|2|}{2}=2$

$\frac{|(-2)+2|}{2}=0$

$\sigma^2=\frac{(1-3)^2+(5-3)^2}{2}=4$

4. Limitations of mean deviation

Absolute distances measure spread but make further algebra less convenient.

Show: Write this line for 1 and 5, whose mean is 3. Say: “M.D. means mean deviation. Adding the signed differences first gives zero. Squaring each difference and averaging gives variance 4.”
Ask: What will averaging squared readings and then subtracting the squared mean give?
Say: It also gives variance 4. Squaring allows this shortcut, while absolute values cannot simply move outside a sum.
Used in: Squared deviations simplify calculations when analysing variation in repeated measurements.

If your child says…“Distance bars work like ordinary brackets.”

Ask: Why does adding the signed differences first give a different answer?
What’s true: Taking absolute values before adding can differ from taking an absolute value after adding.

Variance and standard deviation

$\bar{x}=\frac{1+1+1+1+6}{5}=2\text{ cm}$

$(-1)+(-1)+(-1)+(-1)+4=0$

$\sigma^2=\frac{1+1+1+1+16}{5}=4\text{ cm}^2$

5. Variance and standard deviation

Variance: average squared distance from the mean; standard deviation: non-negative square root.

Show: Write this line for recorded lengths 1, 1, 1, 1, 6 cm. Say: “The mean length is 2 cm. Four differences are −1 cm, and the fifth is 4 cm. They add to zero. Sigma-squared means variance.”
Ask: What standard deviation will describe this spread in centimetres?
Say: The standard deviation is 2 cm, the square root of variance 4 square centimetres. Squaring prevents cancellation, and taking the root restores centimetres.
Used in: Packet-weight standard deviations express variation in familiar grams.

If your child says…“The standard deviation also uses square centimetres.”

Ask: What happens to square centimetres when we take the square root?
What’s true: Taking the square root restores the measurements' original unit.

$\bar{x}=\frac{1(0)+2(3)}{3}=2$

$\sigma^2=\frac{1(0-2)^2+2(3-2)^2}{3}=2$

6. Standard deviation of a frequency distribution

Each occurrence contributes its own squared distance from the mean.

Show: Write this line for 0 occurring once and 3 occurring twice. Say: “The three readings total 6, so their mean is 2. Count both threes when averaging values and squared distances.”
Ask: What standard deviation follows from this frequency calculation?
Say: The standard deviation is $\sqrt{2}$ because we root the variance.
Used in: Frequency tables summarise variation in production measurements, helping factories monitor consistency.

If your child says…“The squared distance for 3 only counts once.”

Ask: How many readings of 3 appear in the expanded list?
What’s true: Each occurrence contributes, so repeated values require repeated squared distances.

$\sigma_x=\sqrt{2^2\times1}$

$\sigma_x=\sqrt{4}$

7. Step-deviation method

Simplify readings, then restore centre and scale.

Show: Subtract 10 from readings 10, 14; halve to get 0, 2. Their mean and variance are 1. Write these lines. Say: “Sigma-x names original standard deviation.”
Ask: After doubling the simplified readings and adding 10, what will their standard deviation become?
Say: Standard deviation is 2: doubling doubles distances.
Used in: Step-deviation simplifies calculations with equally spaced class midpoints.

If your child says…“I must add the assumed centre back to the variance too.”

Ask: If a reading and its mean both rise equally, does their difference change?
What’s true: Adding the same number to every reading leaves spread unchanged. Multiplication changes standard deviation by the factor's absolute value, and variance by the factor squared.

$\sigma^2=\frac{10}{2}-\left(\frac{4}{2}\right)^2$

$\sigma^2=5-4$

8. Correcting observations using mean and variance

Repair both totals before recalculating.

Show: Two readings have mean 3, variance 4. Replace recorded 5 with 3. Write these lines. Say: “Total six becomes four; square total twenty-six becomes ten. Sigma-squared means variance.”
Ask: What corrected variance comes next?
Say: Variance is 1. Replace squared contributions in the square total; the count stays unchanged.
Used in: Corrected sales summaries improve stock planning.

If your child says…“I'll subtract 5 and add 3 in both totals.”

Ask: Did the recorded 5 contribute 5 or its square to the square total?
What’s true: A square total needs squared contributions removed and replaced.

A real problem

Control charts help distinguish routine variation from unusual problems.

Telephone engineers needed more reliable transmission.

Shewhart proposed his control chart on May 16, 1924.

Control limits sit at multiples of the plotted quantity’s standard deviation from the centre line.

These limits help separate routine variation from variation caused by unusual problems.

American Society for Quality Wikipedia NIST

An unbranded landline telephone on a plain desk.

Concept Lab · Common misconception

The tempting mistake in Statistics

Open standalone misconception page →

Does equal mean imply equal spread?

Wrong

Two sets with mean 5 must have the same spread.

Why it feels right. The mean is the number I use to describe a group.

Right

Spread measures how far values lie from the centre.

Worked example

Both groups balance at 5. Only Group B spreads to 1 and 9.

The sets {5, 5, 5} and {1, 5, 9} both have mean 5.
Their ranges are 0 and 8.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

Which set has the smaller range?

The 3, 4, 5 set. Its extreme values are closer together, although both sets have the same average.
Which distances count: those below the centre, above it, or both?

Both. Both measure distance from the chosen centre, so neither is discarded.
What number divides the total distance here?

Four. The two rows together contain four observations.
Why does taking absolute values after adding fail here?

Opposite differences cancel. Taking each distance first preserves both contributions.
What unit belongs to this standard deviation?

Centimetres. The square root changes square centimetres back into centimetres.
How many times must the squared distance for 3 count?

Twice. Two observations have that value, and each contributes to variance.
Which changes when 10 is added back: the mean or the variance?

The mean. Adding equally to every reading leaves its distance from the mean unchanged.
What replaces $5^2$ in the square total?

$3^2$ . The replacement contributes its square because this total contains squared readings.

15 more questions with answers

Teach Statistics in eight steps.

Why it matters