Observation x	Deviation x − 4	Squared deviation
2	−2	4
4	0	0
4	0	0
6	2	4
Total	0	8

Step 3: Divide by n

There are four observations.

σ² = 8/4 = 2

Variance = 2

Verification

The squared deviations are nonnegative, so the variance must be nonnegative.

The result is positive because the observations are not all equal.

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6. Standard deviation

Standard deviation is the nonnegative square root of variance.

We write it as σ, read as “sigma.”

σ = √(σ²)

For the previous example:

Variance = 2

Therefore:

Standard deviation = √2 ≈ 1.414

Why take a square root?

Squaring deviations changes their units.

If observations are measured in centimeters:

• Deviations are in centimeters.
• Squared deviations are in square centimeters.
• Variance is in square centimeters.
• Standard deviation returns to centimeters.

Interpretation

Standard deviation measures spread around the mean using all observations.

It is the root-mean-square deviation, not the ordinary average of the absolute distances from the mean.

A smaller standard deviation indicates observations that are more tightly clustered around their mean.

A larger standard deviation indicates greater spread.

It does not mean that every observation is exactly one standard deviation from the mean.

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7. Basic properties of dispersion

Property 1: Nonnegative values

Range ≥ 0

Variance ≥ 0

Standard deviation ≥ 0

A negative variance is a calculation error.

Property 2: Zero dispersion

Variance is zero exactly when all observations are equal.

For:

7, 7, 7, 7

• Mean = 7.
• Every deviation = 0.
• Variance = 0.
• Standard deviation = 0.
• Range = 0.

Property 3: Data order does not matter

Rearranging observations does not change the mean, range, variance, or standard deviation.

Property 4: Standard deviation and variance rank spread consistently

If one dataset has a larger variance than another, it also has a larger standard deviation.

This follows because the square-root function increases for nonnegative inputs.

Property 5: Comparisons need context

Compare dispersion meaningfully when datasets measure the same quantity using the same units.

A larger standard deviation does not automatically mean worse performance. Whether greater variability is desirable depends on the context.

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8. Worked example 3: Same mean, different spread

Compare:

Dataset A: 4, 5, 6

Dataset B: 0, 5, 10

Both have mean 5.

Dataset A

Squared deviations:

1, 0, 1

Variance = 2/3

Standard deviation = √(2/3) ≈ 0.816

Range = 2

Dataset B

Squared deviations:

25, 0, 25

Variance = 50/3

Standard deviation = √(50/3) ≈ 4.082

Range = 10

Conclusion

Dataset B is more dispersed according to all three measures.

The same mean does not imply the same variability.

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9. A second formula for variance

Variance can also be calculated using:

Variance = mean of the squared observations − square of the mean

In symbols:

σ² = (x₁² + x₂² + … + xₙ²)/n − x̄²

Understand the difference

These are not the same:

• Mean of squares: Add x₁², x₂², and so on, then divide by n.
• Square of the mean: Find x̄ first, then square it.

Why the formula works

Expanding a squared deviation gives:

(x − x̄)² = x² − 2xx̄ + x̄²

When averaged over the dataset:

• The average of x² is the mean of squares.
• The average of x is x̄.

Therefore:

Variance = mean of squares − 2x̄² + x̄²

So:

Variance = mean of squares − x̄²

Worked example 4

Find the variance and standard deviation of:

1, 2, 3, 4

Mean:

x̄ = 10/4 = 2.5

Mean of squares:

(1² + 2² + 3² + 4²)/4 = 30/4 = 7.5

Variance:

σ² = 7.5 − 2.5² = 7.5 − 6.25 = 1.25

Standard deviation:

σ = √1.25 = √5/2 ≈ 1.118

Calculation caution

Keep exact fractions or sufficient decimal precision during intermediate calculations.

Premature rounding can distort the final variance.

For small datasets, the deviation table is often easiest to check.

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10. Frequency tables

When a value occurs several times, its squared deviation must be counted that many times.

Formulas

n = total frequency

Mean = sum of (frequency × value) ÷ total frequency

Variance = sum of [frequency × squared deviation] ÷ total frequency

Alternatively:

Variance = sum of (frequency × value²) ÷ total frequency − mean²

Worked example 5

Find the mean, variance, and standard deviation for:

Measure	After y = kx + c
Mean	k × original mean + c
Range	|k| × original range
Variance	k² × original variance
Standard deviation	|k| × original standard deviation

Worked example 8

A dataset has:

• Mean = 4.
• Range = 8.
• Variance = 9.
• Standard deviation = 3.

Every value is transformed using:

y = −2x + 5

Then:

New mean = −2 × 4 + 5 = −3

New range = 2 × 8 = 16

New variance = (−2)² × 9 = 36

New standard deviation = 2 × 3 = 6

The negative multiplier reverses the order of the observations, but dispersion remains nonnegative.

Special case: k = 0

Every transformed observation becomes c.

Therefore, range, variance, and standard deviation all become zero.

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14. Interpreting consistency

Worked example 9

Two machines fill containers.

Their recorded fill volumes have:

• Machine A: mean 500 mL, standard deviation 2 mL.
• Machine B: mean 500 mL, standard deviation 7 mL.

Which machine has more consistent recorded volumes?

Machine A, because its smaller standard deviation indicates less variation around its mean.

What this does not prove

A smaller standard deviation alone does not establish accuracy relative to a target.

For example:

• Machine C could have mean 480 mL and standard deviation 1 mL.
• Its fills are consistent, but centered below a 500 mL target.

Consistency concerns spread. Accuracy relative to a target also requires considering location.

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15. Common mistakes and verification

Mistake 1: Averaging signed deviations

They sum to zero. Square them before averaging for variance.

Mistake 2: Forgetting the final division

The sum of squared deviations is not the variance until it is divided by n.

Mistake 3: Using n − 1 automatically

This lesson uses n. Follow the convention stated in the problem.

Mistake 4: Confusing variance with standard deviation

Standard deviation is the square root of variance.

Mistake 5: Squaring a negative deviation incorrectly

(−3)² = 9, not −9.

Mistake 6: Ignoring frequencies

A value occurring five times contributes five squared deviations.

Mistake 7: Multiplying variance by k instead of k²

Scaling every observation by 3 multiplies variance by 9.

Mistake 8: Thinking a shift changes dispersion

Adding 10 to every observation leaves variance and standard deviation unchanged.

Mistake 9: Averaging group variances without enough information

A combined variance depends on group sizes, group variances, and differences between group means.

Do not assume it is the simple average of the group variances.

Verification checklist

• Is the mean correct?
• Have I included every observation?
• Do the signed deviations sum to zero, allowing for rounding?
• Are all squared deviations nonnegative?
• Have I divided by the total number of observations?
• Is variance nonnegative?
• Does standard deviation squared reproduce variance?
• Have I used frequencies correctly?
• Are variance units squared and standard-deviation units original?
• Have I avoided premature rounding?

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16. Independent practice

Use the variance convention that divides by n.

Give exact answers where possible. Decimal approximations may be added to three decimal places.

Practice 1

Find the range, variance, and standard deviation of:

3, 3, 3, 3

Practice 2

Find the mean, variance, and standard deviation of:

2, 4, 6

Practice 3

Find the range, mean, variance, and standard deviation of:

−2, 0, 2, 4

Practice 4

Use the mean-of-squares formula to find the variance of:

1, 3, 5, 7

Practice 5

A dataset has eight observations. The sum of its squared deviations from the mean is 72.

Find its variance and standard deviation.

Practice 6

Find the mean, variance, and standard deviation from this frequency table.

Value	Frequency
1	2
3	4
5	2

Practice 7

Compare the variances of:

A: 8, 10, 12

B: 6, 10, 14

Which is more dispersed?

Practice 8

A dataset has mean 12, range 9, and standard deviation 4.

Every value is increased by 7.

Find the new mean, range, variance, and standard deviation.

Practice 9

A dataset has mean 5 and variance 4.

Every value is transformed using:

y = 3x − 2

Find the new mean, variance, and standard deviation.

Practice 10

A dataset has range 10 and standard deviation 6.

Every value is transformed using:

y = −x/2 + 8

Find the new range, variance, and standard deviation.

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17. Practice solutions

Solution 1

All observations equal 3.

Range = 0

Variance = 0

Standard deviation = 0

Solution 2

Mean:

(2 + 4 + 6)/3 = 4

Squared deviations:

4, 0, 4

Variance = 8/3

Standard deviation = √(8/3) ≈ 1.633

Solution 3

Range:

4 − (−2) = 6

Mean:

(−2 + 0 + 2 + 4)/4 = 1

Deviations:

−3, −1, 1, 3

Squared deviations:

9, 1, 1, 9

Variance = 20/4 = 5

Standard deviation = √5 ≈ 2.236

Solution 4

Mean:

(1 + 3 + 5 + 7)/4 = 4

Mean of squares:

(1 + 9 + 25 + 49)/4 = 84/4 = 21

Variance = 21 − 4² = 5

Solution 5

Variance = 72/8 = 9

Standard deviation = √9 = 3

Solution 6

Total frequency:

2 + 4 + 2 = 8

Mean:

(1 × 2 + 3 × 4 + 5 × 2)/8 = 24/8 = 3

Weighted sum of squared deviations:

2 × (1 − 3)² + 4 × (3 − 3)² + 2 × (5 − 3)²

= 2 × 4 + 4 × 0 + 2 × 4 = 16

Variance = 16/8 = 2

Standard deviation = √2 ≈ 1.414

Solution 7

Both means are 10.

Dataset A:

Variance = (4 + 0 + 4)/3 = 8/3

Dataset B:

Variance = (16 + 0 + 16)/3 = 32/3

Dataset B is more dispersed.

Its variance is four times that of A, and its standard deviation is twice that of A.

Solution 8

Adding 7 changes only the mean.

New mean = 19

New range = 9

New variance = 4² = 16

New standard deviation = 4

Solution 9

New mean = 3 × 5 − 2 = 13

New variance = 3² × 4 = 36

New standard deviation = √36 = 6

Solution 10

The multiplier is −1/2, whose absolute value is 1/2.

New range = 10/2 = 5

New standard deviation = 6/2 = 3

New variance = 3² = 9

The added constant 8 does not affect dispersion.

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18. Lesson assessment — Variance & Dispersion Quiz

Suggested time: 12–15 minutes
Questions: 8 single-answer multiple-choice questions
Marks: 1 per question
Suggested course mastery target: 7/8

Instructions: Use variance calculated by dividing by n for every question.

Question 1

Find the range of:

−5, −1, 3, 7

A. 2
B. 7
C. 12
D. 14

Question 2

Find the variance of:

2, 4, 6

A. 8/3
B. 4
C. 8
D. √(8/3)

Question 3

A dataset has variance 25.

What is its standard deviation?

A. 25
B. 625
C. −5
D. 5

Question 4

Every observation in a dataset is increased by 10.

Which statement is correct?

A. The variance increases by 10.

B. The variance and standard deviation remain unchanged.

C. The standard deviation increases by 10.

D. The variance increases by 100.

Question 5

A dataset has variance 9. Every observation is multiplied by 2.

What is the new variance?

A. 18
B. 6
C. 36
D. 81

Question 6

A dataset has values 1 and 5, each occurring twice.

What is its variance?

A. 4
B. 2
C. 8
D. 16

Question 7

Two datasets measure the same quantity in the same units and have equal means.

Dataset A has standard deviation 2. Dataset B has standard deviation 5.

Which statement is correct?

A. Dataset A must contain fewer observations.

B. Every observation in B is greater than every observation in A.

C. The datasets must have different medians.

D. Dataset A has smaller variance and smaller standard deviation.

Question 8

A dataset has standard deviation 4.

Every value is transformed using:

y = −3x + 7

What is the new standard deviation?

A. −12
B. 12
C. 19
D. 144

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19. Instructor answer key

Tutor LMS placement: Put the assessment in a separate Quiz item if it will be graded. Keep the answer key and solutions separate from student-facing questions until submission.

Question	Answer	Skill assessed
1	C	Range
2	A	Variance from raw data
3	D	Standard deviation from variance
4	B	Adding a constant
5	C	Scaling variance
6	A	Repeated observations
7	D	Comparing dispersion
8	B	Scaling standard deviation