11-15-2025, 04:40 PM
Chapter 13 — Variance & Standard Deviation
Mean, median, and mode tell us the “centre” of the data.
But we also need to know: How spread out is the data?
Variance and standard deviation measure:
• consistency
• variability
• how much values differ from the mean
These ideas are ESSENTIAL for real statistics, science, finance, physics, and data analysis.
---
13.1 What Is Variability?
Example:
Two students score:
Student A: 7, 7, 7, 7
Student B: 3, 7, 11, 7
Both have the SAME mean (7)…
…but Student B’s scores are much more spread out.
Variance + Standard deviation measure this spread.
---
13.2 Step-by-Step: Variance (σ²)
Variance tells us:
how far each value is from the mean (on average).
To calculate variance:
1. Find the mean
2. Subtract the mean from each value
3. Square each difference
4. Find the average of these squared values
Formula (population variance):
Variance = average of (value − mean)²
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13.3 Example — Variance Calculation
Data: 4, 6, 8
Step 1 — Mean
(4 + 6 + 8) / 3 = 6
Step 2 — Differences from mean
4 − 6 = −2
6 − 6 = 0
8 − 6 = 2
Step 3 — Square differences
(−2)² = 4
0² = 0
2² = 4
Step 4 — Average of squared differences
Variance = (4 + 0 + 4) / 3 = 8/3 ≈ 2.67
Variance = 2.67
---
13.4 Standard Deviation (σ)
Standard deviation is simply:
the square root of the variance
Why do we take the square root?
Because variance is in “squared units.”
Standard deviation gives us a number back in the SAME units as the data.
Example continued:
Variance ≈ 2.67
Standard deviation = √2.67 ≈ 1.63
Meaning:
Most values lie within about 1.63 of the mean.
---
13.5 What Standard Deviation Tells You
Small standard deviation → data is tightly packed
(balanced, consistent, predictable)
Large standard deviation → data is spread out
(unstable, unpredictable, more variation)
Examples:
• Test scores with low SD → students performed similarly
• Stock with high SD → risky investment
• Machine with low SD → reliable performance
---
13.6 Variance & SD With Frequency Tables
Example:
Value | Freq
2 | 3
5 | 1
7 | 2
Step 1 — Mean
Mean = (2×3 + 5×1 + 7×2) / 6
= (6 + 5 + 14) / 6
= 25 / 6
≈ 4.17
Step 2 — Differences from mean
2 − 4.17 = −2.17
5 − 4.17 = 0.83
7 − 4.17 = 2.83
Step 3 — Square and multiply by frequency
(−2.17)² × 3
(0.83)² × 1
(2.83)² × 2
Step 4 — Total ÷ total frequency gives variance
SD = √variance
(You don’t need exact decimals for understanding.)
---
13.7 Why Squared Differences?
Students always ask this.
Reasons:
• stops negatives cancelling positives
• amplifies large differences (outliers)
• creates a smooth mathematical measure
• is essential for advanced statistics
Variance is the foundation of:
• normal distribution
• hypothesis testing
• Z-scores
• confidence intervals
• regression
• machine learning
This is why it matters so much.
---
13.8 A Real-World Example
Two sales teams each have a mean of £500 in weekly sales.
Team A: 480, 510, 495, 505, 510
Team B: 300, 450, 500, 650, 700
Team A → low SD → reliable and consistent
Team B → high SD → unpredictable and unstable
Businesses use SD to evaluate:
• risk
• reliability
• staff performance
• machine quality
• investment behaviour
---
13.9 Exam-Style Questions
1. Find the variance and standard deviation of:
3, 7, 7, 9
2. A frequency table shows:
Value | Freq
2 | 2
6 | 1
10 | 1
Find the mean and variance.
3. Explain in words what a high SD means.
4. Two students have the same mean score.
Why might one still be considered “more consistent”?
5. A machine produces parts with SD = 0.2 cm.
Another produces SD = 0.9 cm.
Which is higher quality, and why?
---
13.10 Chapter Summary
• Variance measures spread using squared differences
• Standard deviation is the square root of the variance
• Small SD → consistent
• Large SD → variable
• Mean alone cannot describe a dataset
• These measures are essential for higher statistics
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
Mean, median, and mode tell us the “centre” of the data.
But we also need to know: How spread out is the data?
Variance and standard deviation measure:
• consistency
• variability
• how much values differ from the mean
These ideas are ESSENTIAL for real statistics, science, finance, physics, and data analysis.
---
13.1 What Is Variability?
Example:
Two students score:
Student A: 7, 7, 7, 7
Student B: 3, 7, 11, 7
Both have the SAME mean (7)…
…but Student B’s scores are much more spread out.
Variance + Standard deviation measure this spread.
---
13.2 Step-by-Step: Variance (σ²)
Variance tells us:
how far each value is from the mean (on average).
To calculate variance:
1. Find the mean
2. Subtract the mean from each value
3. Square each difference
4. Find the average of these squared values
Formula (population variance):
Variance = average of (value − mean)²
---
13.3 Example — Variance Calculation
Data: 4, 6, 8
Step 1 — Mean
(4 + 6 + 8) / 3 = 6
Step 2 — Differences from mean
4 − 6 = −2
6 − 6 = 0
8 − 6 = 2
Step 3 — Square differences
(−2)² = 4
0² = 0
2² = 4
Step 4 — Average of squared differences
Variance = (4 + 0 + 4) / 3 = 8/3 ≈ 2.67
Variance = 2.67
---
13.4 Standard Deviation (σ)
Standard deviation is simply:
the square root of the variance
Why do we take the square root?
Because variance is in “squared units.”
Standard deviation gives us a number back in the SAME units as the data.
Example continued:
Variance ≈ 2.67
Standard deviation = √2.67 ≈ 1.63
Meaning:
Most values lie within about 1.63 of the mean.
---
13.5 What Standard Deviation Tells You
Small standard deviation → data is tightly packed
(balanced, consistent, predictable)
Large standard deviation → data is spread out
(unstable, unpredictable, more variation)
Examples:
• Test scores with low SD → students performed similarly
• Stock with high SD → risky investment
• Machine with low SD → reliable performance
---
13.6 Variance & SD With Frequency Tables
Example:
Value | Freq
2 | 3
5 | 1
7 | 2
Step 1 — Mean
Mean = (2×3 + 5×1 + 7×2) / 6
= (6 + 5 + 14) / 6
= 25 / 6
≈ 4.17
Step 2 — Differences from mean
2 − 4.17 = −2.17
5 − 4.17 = 0.83
7 − 4.17 = 2.83
Step 3 — Square and multiply by frequency
(−2.17)² × 3
(0.83)² × 1
(2.83)² × 2
Step 4 — Total ÷ total frequency gives variance
SD = √variance
(You don’t need exact decimals for understanding.)
---
13.7 Why Squared Differences?
Students always ask this.
Reasons:
• stops negatives cancelling positives
• amplifies large differences (outliers)
• creates a smooth mathematical measure
• is essential for advanced statistics
Variance is the foundation of:
• normal distribution
• hypothesis testing
• Z-scores
• confidence intervals
• regression
• machine learning
This is why it matters so much.
---
13.8 A Real-World Example
Two sales teams each have a mean of £500 in weekly sales.
Team A: 480, 510, 495, 505, 510
Team B: 300, 450, 500, 650, 700
Team A → low SD → reliable and consistent
Team B → high SD → unpredictable and unstable
Businesses use SD to evaluate:
• risk
• reliability
• staff performance
• machine quality
• investment behaviour
---
13.9 Exam-Style Questions
1. Find the variance and standard deviation of:
3, 7, 7, 9
2. A frequency table shows:
Value | Freq
2 | 2
6 | 1
10 | 1
Find the mean and variance.
3. Explain in words what a high SD means.
4. Two students have the same mean score.
Why might one still be considered “more consistent”?
5. A machine produces parts with SD = 0.2 cm.
Another produces SD = 0.9 cm.
Which is higher quality, and why?
---
13.10 Chapter Summary
• Variance measures spread using squared differences
• Standard deviation is the square root of the variance
• Small SD → consistent
• Large SD → variable
• Mean alone cannot describe a dataset
• These measures are essential for higher statistics
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive