Normal distribution

The bell curve

Many natural measurements (heights, test scores, measurement errors) form a normal distribution: a symmetric, bell curve centred on the mean μ.

Features:

• Mean = median = mode, at the centre.
• Symmetric about the mean.
• Tails approach (but never reach) the axis.
• Shape set by μ (centre) and σ (spread).

The empirical rule (68–95–99.7)

Example: Adult heights: μ = 170 cm, σ = 8 cm.

• 68% between 162 and 178 cm
• 95% between 154 and 186 cm
• 99.7% between 146 and 194 cm

Because of symmetry: 34% between μ and μ + σ; 13.5% between μ + σ and μ + 2σ; 2.5% above μ + 2σ.

z-scores

A z-score measures how many standard deviations a value is from the mean:

z = (x − μ) / σ

Example: A 186 cm person: z = (186 − 170)/8 = 2.

z-scores let you compare values from different distributions.

Example: 82 on a test (μ = 70, σ = 6): z = 2. 90 on another (μ = 80, σ = 10): z = 1. The first is relatively better.

The standard normal distribution

Converting to z-scores gives the standard normal distribution (μ = 0, σ = 1). Tables or technology give the area under the curve (the proportion of data) to the left of any z.

Finding proportions

Example: What percent of adults are taller than 182 cm?

• z = (182 − 170)/8 = 1.5
• Area left of 1.5 = 0.9332
• Above: 1 − 0.9332 = 6.7%

Between two values: P(162 < x < 182) = P(−1 < z < 1.5) = 0.9332 − 0.1587 = 77.5%.

Working backwards

Example: What height is the 90th percentile?

• z for 0.90 ≈ 1.28
• x = μ + zσ = 170 + 1.28(8) ≈ 180.2 cm

Technology

Spreadsheets: =NORM.DIST(x, μ, σ, TRUE) gives the area to the left; =NORM.INV(p, μ, σ) gives the value. Calculators have normalcdf and invNorm.

Is my data normal?

Check that a histogram is roughly bell-shaped and symmetric, with no strong skew. Not all data is normal (incomes are usually right-skewed).

Worked examples

Example 1: μ = 50, σ = 5. What percent lies between 45 and 55? 68%.

Example 2: z-score of 38 when μ = 50, σ = 4? −3.

Example 3: Percent below z = −1? 15.9%.

Common mistakes

• Forgetting that the area is a proportion (multiply by 100 for percent).
• Using the left-area when you need the right-area.
• Assuming all data is normal.

Try it yourself

1. μ = 100, σ = 15. What range contains about 95% of values?
2. z-score of 130 in that distribution?
3. Percent of values above z = 2?
4. Value at z = −1.5 when μ = 60, σ = 4?

Answers: 1) 70 to 130 2) 2 3) about 2.3% (2.5% by the empirical rule) 4) 54

Quick recap

• The normal distribution is a symmetric bell curve.
• Empirical rule: 68–95–99.7.
• z-scores and the standard normal distribution give the area under the curve.