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📈 Introduction to Measure of Dispersion

In this lesson, we continue exploring statistics by focusing on Measures of Dispersion—a vital concept that tells us how spread out data values are around the center (mean). Even if two datasets have the same mean, their variability can be drastically different.

We’ll cover:

• What dispersion means
• Why it’s important
• Detailed explanation and formulas of Variance and Standard Deviation
• Comparison using multiple examples

🎯 Why Measure Dispersion?

Measures of central tendency (like mean, median, and mode) show where the center of a dataset lies. But they don’t tell us how much variation exists around that center.

Let’s illustrate this with two datasets:

Dataset A:

2, 2, 3, 3

Dataset B:

1, 1, 5, 5

Both datasets have 4 values.

Step 1: Calculate the Mean

Dataset A Mean (μ):

(2 + 2 + 3 + 3) / 4 = 10 / 4 = 2.5

Dataset B Mean (μ):

(1 + 1 + 5 + 5) / 4 = 12 / 4 = 3

👉 Though the means differ slightly, we’ll soon see how spread (dispersion) differs much more significantly.

📐 1. Variance (σ² or S²)

Variance measures the average squared deviation from the mean.

📌 Population Variance Formula (σ²):

σ2=1N∑i=1N(xi−μ)2\sigma^2 = \frac{1}{N} \sum_{i=1}^{N}(x_i – \mu)^2

Where:

• xix_i = individual value
• μ\mu = population mean
• NN = number of values in population

📌 Sample Variance Formula (S²):

S2=1n−1∑i=1n(xi−xˉ)2S^2 = \frac{1}{n - 1} \sum_{i=1}^{n}(x_i - \bar{x})^2

Where:

• xˉ\bar{x} = sample mean
• nn = sample size
• We use (n – 1) instead of nn to correct bias in the estimate (this is called Bessel’s correction).

🔍 Example 1: Population Variance for Dataset A

2, 2, 3, 3

Mean: μ=2+2+3+34=2.5\mu = \frac{2 + 2 + 3 + 3}{4} = 2.5

Now calculate squared deviations:

• (2−2.5)2=0.25(2 – 2.5)^2 = 0.25
• (2−2.5)2=0.25(2 – 2.5)^2 = 0.25
• (3−2.5)2=0.25(3 – 2.5)^2 = 0.25
• (3−2.5)2=0.25(3 – 2.5)^2 = 0.25

Sum: 0.25+0.25+0.25+0.25=10.25 + 0.25 + 0.25 + 0.25 = 1

Variance: σ2=14×1=0.25\sigma^2 = \frac{1}{4} \times 1 = 0.25

🔍 Example 2: Population Variance for Dataset B

1, 1, 5, 5

Mean: μ=1+1+5+54=3\mu = \frac{1 + 1 + 5 + 5}{4} = 3

Squared deviations:

• (1−3)2=4(1 – 3)^2 = 4
• (1−3)2=4(1 – 3)^2 = 4
• (5−3)2=4(5 – 3)^2 = 4
• (5−3)2=4(5 – 3)^2 = 4

Sum: 4+4+4+4=164 + 4 + 4 + 4 = 16

Variance: σ2=164=4\sigma^2 = \frac{16}{4} = 4

📊 Comparison:

➡️ Dataset B has a much larger spread, even though its mean isn’t too different from A.

🌟 2. Standard Deviation (σ or S)

Standard Deviation is the square root of variance, making it easier to interpret since it’s in the same unit as the original data.

📌 Formulas:

• Population Standard Deviation:

σ=σ2\sigma = \sqrt{\sigma^2}

• Sample Standard Deviation:

S=S2S = \sqrt{S^2}

✅ Example (Using Dataset B)

We already calculated: σ2=4\sigma^2 = 4

So, σ=4=2\sigma = \sqrt{4} = 2

This means data points deviate on average by 2 units from the mean.

🔬 Sample Variance Example (step-by-step)

Let’s take sample data:
4, 8, 6, 5, 3

Step 1: Mean xˉ=(4+8+6+5+3)/5=26/5=5.2\bar{x} = (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.2

Step 2: Squared deviations

• (4 – 5.2)² = 1.44
• (8 – 5.2)² = 7.84
• (6 – 5.2)² = 0.64
• (5 – 5.2)² = 0.04
• (3 – 5.2)² = 4.84

Step 3: Sum = 14.8

Step 4: Divide by (n – 1) = 4 S2=14.8/4=3.7S^2 = 14.8 / 4 = 3.7

Step 5: Standard Deviation = S=3.7≈1.92S = \sqrt{3.7} \approx 1.92

🧠 Why Divide by (n – 1) in Sample Variance?

This adjustment is called Bessel’s correction. It makes the sample variance an unbiased estimator of the population variance by compensating for the fact that a sample underestimates population variability.

💼 Real-Life Applications

• Finance: Risk of an asset (volatility) is measured using standard deviation.
• Education: Variability in student scores across tests.
• Manufacturing: Product consistency monitored via standard deviation in dimensions.
• Weather Forecasting: Comparing temperature fluctuation across days.
• Machine Learning: Feature scaling often involves standard deviation.

📌 Key Takeaways

✅ Variance gives a mathematical understanding of spread, but its unit is squared.

✅ Standard deviation is more interpretable and widely used.

✅ Two datasets can have the same mean but very different variances, indicating different levels of consistency or spread.

✅ Always use (n – 1) in the denominator when dealing with sample data.

📚 Summary Table