Normal Distribution

The normal distribution (bell curve) has these key properties:

• Symmetric: The left side is a mirror image of the right side
• Unimodal: One peak at the centre
• Mean = Median = Mode: All three measures of centre are equal
• Asymptotic: Tails approach but never touch the horizontal axis
• Total area under the curve = 1 (represents 100% of the data)

The normal distribution is completely defined by two parameters:

• Mean $\mu$: Locates the centre of the distribution
• Standard deviation $\sigma$: Controls the width and spread

The normal distribution is bell-shaped, symmetric about the mean μ, with spread determined by standard deviation σ. Mean = median = mode. Total area under the curve = 1. Most data lies within 3σ of the mean.

Pause, copy the five properties of the normal distribution: bell-shaped, symmetric about μ, mean = median = mode, total area under curve = 1, and most data lies within 3σ of the mean into your book.

The normal distribution is bell-shaped, symmetric about the mean μ, with spread determined by σ; mean = median = mode, and the total area under the curve equals 1. The five core properties describe the shape, but the empirical rule gives the critical percentages: 68% of data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.

For any normal distribution with mean $\mu$ and standard deviation $\sigma$:

• 68% of data falls within $\mu \pm \sigma$ (within 1 standard deviation)
• 95% of data falls within $\mu \pm 2\sigma$ (within 2 standard deviations)
• 99.7% of data falls within $\mu \pm 3\sigma$ (within 3 standard deviations)

Example: IQ scores: $\mu = 100$, $\sigma = 15$.

• 68% of people have IQ between $100 - 15 = 85$ and $100 + 15 = 115$
• 95% of people have IQ between $100 - 30 = 70$ and $100 + 30 = 130$
• 99.7% of people have IQ between $100 - 45 = 55$ and $100 + 45 = 145$

Only 0.3% of people have IQ below 55 or above 145.

Empirical rule: 68% of data lies within 1σ, 95% within 2σ, 99.7% within 3σ of the mean. By symmetry, half these percentages lie on each side. Memorise as 68-95-99.7.

Pause, copy the empirical rule: 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ, and the symmetric split: half of each percentage lies on each side of the mean (e.g., 34% between μ and μ+1σ) into your book.

The empirical rule gives 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ, by symmetry, half each percentage lies on each side. For one-sided intervals or non-standard bands, use symmetry: the percentage from μ to μ+1σ is 68/2 = 34%, and from μ+1σ to μ+2σ is (95−68)/2 = 13.5%. Build any interval by adding or subtracting these half-band values.

Finding percentages in a range:

Example: Year 12 student heights, $\mu = 170$ cm, $\sigma = 8$ cm. What percentage are taller than 178 cm?

• $178 = 170 + 8 = \mu + \sigma$
• 68% are within $\mu \pm \sigma$, so 32% are outside
• By symmetry, 16% are above $\mu + \sigma = 178$ cm

Identifying unusual values:

• Values more than 2 SD from the mean are unusual (only 5% of data)
• Values more than 3 SD from the mean are very unusual (only 0.3% of data)

To find the percentage of data in a given interval: calculate how many standard deviations each boundary is from the mean, apply 68/95/99.7 (whole-band or half-band using symmetry), then subtract or add as needed.

Pause, copy the three-step interval process: (1) count how many standard deviations each boundary is from μ; (2) apply the matching 68/95/99.7 percentage; (3) use symmetry to split or construct one-sided/non-standard intervals, into your book.

Applying the empirical rule, using symmetry and subtraction to find percentages in any interval, gives numerical answers. When comparing two normal distributions, you need a qualitative description: the distribution with the larger mean has a higher centre (typically larger values), and the distribution with the larger standard deviation is more spread out (less consistent data).

Two normal distributions with the same mean but different standard deviations look very different:

• Same mean: Both centred at the same point on the horizontal axis
• Larger $\sigma$: Wider, flatter bell, data more spread out
• Smaller $\sigma$: Narrower, taller bell, data more concentrated around the mean

Example: Factory A: $\mu = 100$ mm, $\sigma = 2$ mm. Factory B: $\mu = 100$ mm, $\sigma = 5$ mm. Customer requires parts within $100 \pm 4$ mm.

• Factory A: $100 \pm 4 = \mu \pm 2\sigma$. By the 95% rule, 95% of parts meet spec.
• Factory B: $100 \pm 4 = \mu \pm 0.8\sigma$. This is less than 1 SD, roughly 58% meet spec.

To compare two normal distributions: state which has the larger mean (higher centre) and which has the larger standard deviation (more spread). A smaller σ means data is more consistent; a larger σ means more variability.

Pause, copy the comparison framework: larger mean → higher centre (typically larger values); larger standard deviation → more spread (less consistent); always describe the comparison using the actual variable names and units in context into your book.