Standard Deviation and Variance

Measures of Spread

Measures of center: Mean, median, mode (where data clusters)

Measures of spread: Range, variance, standard deviation (how spread out data is)

Why important: Two datasets can have same mean but very different spreads

Example: Same Mean, Different Spread

Data set A: 10, 10, 10, 10 (no variation) Data set B: 0, 5, 15, 20 (lots of variation)

Both have mean = 10, but B is much more spread out

Range

Range: Difference between maximum and minimum

Formula: Range = max - min

Pros: Easy to calculate

Cons: Only uses two values, sensitive to outliers

Example: Calculate Range

Data: 5, 8, 12, 15, 20

Range: 20 - 5 = 15

Variance

Variance: Average of squared deviations from mean

Measures: How far data points are from mean, on average

Symbol: σ² (population), s² (sample)

Formula (population):

σ² = Σ(x - μ)² / N

Where:

• x = each data value
• μ = population mean
• N = number of values

Formula (sample):

s² = Σ(x - x̄)² / (n - 1)

Where:

• x̄ = sample mean
• n = sample size
• (n - 1) is called "degrees of freedom"

Example 1: Calculate Variance

Data: 2, 4, 6, 8, 10

Step 1: Find mean

μ = (2 + 4 + 6 + 8 + 10)/5 = 30/5 = 6

Step 2: Find deviations from mean

2 - 6 = -4
4 - 6 = -2
6 - 6 = 0
8 - 6 = 2
10 - 6 = 4

Step 3: Square deviations

(-4)² = 16
(-2)² = 4
0² = 0
2² = 4
4² = 16

Step 4: Find average

σ² = (16 + 4 + 0 + 4 + 16)/5 = 40/5 = 8

Variance: 8

Example 2: Using Table

Data: 1, 3, 3, 5, 8

Mean μ = 4

Variance:

σ² = (9 + 1 + 1 + 1 + 16)/5 = 28/5 = 5.6

Standard Deviation

Standard deviation: Square root of variance

Symbol: σ (population), s (sample)

Same units as original data (variance has squared units)

Formula:

σ = √(σ²) or s = √(s²)

Interpretation: Average distance from mean

Example 1: From Variance

Variance = 25

Standard deviation:

σ = √25 = 5

Example 2: Complete Calculation

Data: 10, 12, 14, 16, 18

Mean: μ = 14

Deviations: -4, -2, 0, 2, 4

Squared: 16, 4, 0, 4, 16

Variance:

σ² = (16 + 4 + 0 + 4 + 16)/5 = 40/5 = 8

Standard deviation:

σ = √8 ≈ 2.83

Interpretation: On average, values are about 2.83 units from mean

Population vs Sample

Population: All members of a group

• Use σ, σ²
• Divide by N

Sample: Subset of population

• Use s, s²
• Divide by (n - 1) for better estimate

Example: Sample Standard Deviation

Sample data: 5, 7, 9, 11

Mean: x̄ = 8

Squared deviations: 9, 1, 1, 9

Sample variance:

s² = (9 + 1 + 1 + 9)/(4 - 1) = 20/3 ≈ 6.67

Sample standard deviation:

s = √(20/3) ≈ 2.58

Properties of Standard Deviation

σ ≥ 0 (always non-negative)

σ = 0 only when all values are identical

Larger σ means more spread out data

Units same as data (unlike variance)

Adding constant to all data: σ unchanged

Multiplying all data by constant k: σ multiplied by |k|

Example: Effect of Transformation

Original data: 2, 4, 6, 8 (σ = 2.24)

Add 10 to each: 12, 14, 16, 18

• Mean changes to 15
• σ stays 2.24

Multiply by 3: 6, 12, 18, 24

• Mean becomes 15
• σ becomes 3(2.24) = 6.72

Interpreting Standard Deviation

Small σ: Data clustered near mean

Large σ: Data widely dispersed

Compare datasets: Larger σ means more variability

Example: Compare Two Classes

Class A scores: Mean = 75, σ = 5

• Most scores 70-80 (close to mean)

Class B scores: Mean = 75, σ = 15

• Scores range 60-90 (very spread out)

Same average, but Class B has more variation

Using Calculator/Technology

Most calculators have built-in functions:

• σₓ or σ (population)
• sₓ or s (sample)

Enter data in list, use statistics function

Empirical Rule (68-95-99.7)

For approximately normal distributions:

68% of data within 1σ of mean

95% of data within 2σ of mean

99.7% of data within 3σ of mean

Example: Apply Empirical Rule

Heights: mean = 170 cm, σ = 10 cm

Within 1σ: 160-180 cm (68%)

Within 2σ: 150-190 cm (95%)

Within 3σ: 140-200 cm (99.7%)

Outliers and Standard Deviation

Outlier: Value significantly different from others

Outliers increase standard deviation

Common rule: Value is outlier if more than 2σ from mean

Example: Identify Outlier

Data: 10, 12, 14, 16, 50

Mean: 20.4 Standard deviation: ≈ 16.4

Check 50:

50 - 20.4 = 29.6
29.6 > 2(16.4) = 32.8?  No

Borderline, but significantly affects σ

Coefficient of Variation

Compares variability across different scales

CV = (σ/μ) × 100%

Useful when comparing datasets with different units or means

Example: Compare Variation

Heights: Mean = 170 cm, σ = 10 cm

CV = (10/170) × 100% ≈ 5.9%

Weights: Mean = 70 kg, σ = 8 kg

CV = (8/70) × 100% ≈ 11.4%

Weights have more relative variation

Real-World Applications

Quality control: Product consistency (low σ desired)

Finance: Investment risk (σ measures volatility)

Testing: Score consistency across students

Weather: Temperature variability

Sports: Player/team consistency

Example: Manufacturing

Target bolt length: 5.0 cm σ = 0.05 cm (tight tolerance)

95% of bolts within:

5.0 ± 2(0.05) = 4.9 to 5.1 cm

Acceptable quality control

Practice