B. Probability Distributions · Continuous Distributions and the Normal Distribution

The Normal Distribution and Its Properties

B2.6About 65 minutes

Learning goals

I can describe the key properties of a normal distribution.
I can sketch a normal curve for a given mean and standard deviation, with a correct scale.
I can explain how changing μ or σ changes the curve.
I can recognize situations that a normal distribution models well, and ones it does not.

Words to know

normal distribution
A continuous probability distribution with a symmetric, bell-shaped density curve, completely determined by its mean μ and standard deviation σ.

X ~ N(μ, σ²)
Notation meaning "X is normally distributed with mean μ and variance σ²". For example, N(32, 10.2²) has mean 32 and standard deviation 10.2.

inflection point
A point where a curve changes from bending downward to bending upward. On a normal curve they are at μ − σ and μ + σ.

asymptote
A line that a curve approaches more and more closely without reaching it. The x-axis is a horizontal asymptote of the normal curve.

standard normal distribution
The normal distribution with μ = 0 and σ = 1, written N(0, 1). It is used with z-scores later in this unit.

The most famous curve in statistics

Measure the heights of hundreds of maple seedlings of the same age, the masses of newborn babies at a hospital, or the volume of syrup a machine puts in each bottle. Draw a density histogram. Again and again you get the same shape: a single hump in the middle, falling away evenly on both sides. This bell shape is the normal distribution.

It shows up whenever a measurement is the result of many small, independent influences adding together: for a seedling, a little more sun here, a little less water there, slightly different genes. The ups and downs mostly cancel, so most values are near the middle and extreme values are rare.

Key features of a normal curve. It is symmetric about μ, where the mean, median and mode are equal. Half the area (0.5) is on each side. The curve changes the way it bends at μ − σ and μ + σ, and the tails get closer and closer to the axis without touching it.

Properties of every normal distribution

The density curve is bell-shaped with a single central peak.
It is symmetric about the mean μ. So the mean, median and mode are equal, and half the area is on each side of μ.
The total area under the curve is 1.
The inflection points are exactly one standard deviation from the mean, at μ − σ and μ + σ. This gives you a way to see σ on a graph.
The tails extend forever in both directions and approach the x-axis (an asymptote) without touching it. In practice, almost all of the area (about 99.7%) is within 3 standard deviations of the mean.
A normal distribution is completely determined by μ and σ. We write X ~ N(μ, σ²).

For reference, the equation of the curve is

f(x) = (1 / (σ√(2π))) e^(−(x − μ)² / (2σ²)).

You will not need to integrate this in this course; you will find areas with the empirical rule, z-tables and technology.

What μ and σ do to the curve

Changing μ slides the curve left or right. The shape stays the same.
Changing σ stretches or squeezes it. A larger σ gives a wider, flatter curve; a smaller σ gives a narrower, taller one. Because the area must stay 1, the peak height is 1/(σ√(2π)): doubling σ halves the peak height.

Left: two normal curves with the same standard deviation and different means; one is a slid copy of the other. Right: two curves with the same mean; the one with the larger σ is wider and only half as tall.

Worked example

Example 1: Sketching a normal curve with a scale

The heights of 16-month-old maple seedlings at a nursery are normally distributed with mean 32 cm and standard deviation 10.2 cm. Write this in N(μ, σ²) notation and list the values to mark on the horizontal axis of a sketch.

X ~ N(32, 10.2²).
Centre: μ = 32.
One σ each way: 32 − 10.2 = 21.8 and 32 + 10.2 = 42.2. These are the inflection points.
Two σ: 32 ± 20.4 gives 11.6 and 52.4.
Three σ: 32 ± 30.6 gives 1.4 and 62.6. Draw the curve almost touching the axis at these ends.

Mark 1.4, 11.6, 21.8, 32, 42.2, 52.4, 62.6 cm. The peak is above 32 and the curve bends at 21.8 and 42.2.

Worked example

Example 2: Normal or not?

For each situation, decide whether a normal model is reasonable. Explain.

Masses of 500 g bags of flour filled by a machine.
Waiting times at a walk-in clinic.
Summary of 200 test marks: mean 71, median 71, Q1 = 64, Q3 = 78, min 45, max 97.
Summary of 200 house prices in a town: mean $780 000, median $650 000.

1. Fill amounts vary slightly around a target because of many small causes. Normal is reasonable.
2. Most waits are short but a few are very long, so the distribution has a long right tail. Times also cannot go below 0. Skewed, not normal.
3. Mean = median, and Q1 and Q3 are the same distance (7) from the median. The data look symmetric. Normal is reasonable (a histogram would confirm it).
4. The mean is much larger than the median, a sign of a few very expensive houses pulling the mean up. Skewed right, not normal.

Normal: 1 and 3. Not normal: 2 and 4.

Worked example

Example 3: Comparing two normal distributions

LED bulbs from Company A last X ~ N(25 000, 1000²) hours; bulbs from Company B last Y ~ N(25 000, 3000²) hours. Compare the curves. Which company's bulbs are more predictable? Which graph has the taller peak, and how many times as tall?

Same mean, so both curves are centred at 25 000 h.
B's standard deviation is 3 times A's, so B's curve is 3 times as wide.
The peak height is 1/(σ√(2π)), so A's peak is 3 times as tall as B's.
A smaller σ means lifetimes cluster closely around the mean.

Company A is more predictable. Its curve is narrower and 3 times as tall at the peak.

Just for fun

The normal curve once appeared on money. Germany's old 10-mark banknote showed the mathematician Carl Friedrich Gauss next to a drawing of the bell curve and its formula. The curve is also called the Gaussian distribution, after him.

Hands-on activity

Your own bell curve: the 5-second challenge

You need: a phone or watch with a stopwatch that shows hundredths of a second, paper and a pencil (or a spreadsheet).

Start the stopwatch, look away, and try to stop it at exactly 5.00 s. Record the time. Do this 40 times (share with family members if you like).
Make a frequency table with class width 0.2 s (for example 4.4 to 4.6, 4.6 to 4.8, ...).
Draw a histogram and a frequency polygon.
Find the mean, the median and the standard deviation of your 40 times.

What to look for: a rough bell shape centred near 5 s (you may run slightly early or late). The mean and median should be close. Mark x̄ − s and x̄ + s on your graph: the polygon should be changing from steep to flatter near those points. Keep your data for the next lesson.

Your turn

For X ~ N(60, 8²), state the mean, standard deviation and the x-values of the inflection points.

For X ~ N(60, 8²), what is P(X < 60)? Explain.

For X ~ N(60, 8²), list the values at μ ± 1σ, μ ± 2σ and μ ± 3σ.

A normal distribution has inflection points at 140 and 160. Find μ and σ.

Two normal curves have the same σ but μ = 20 and μ = 35. How are they related?

Which is more likely to be normally distributed: the number of hours of sunshine per day in Ottawa over a year, or the diameters of 1000 bolts made by one machine? Explain.

Why can't a normal distribution exactly model a variable such as mass, which cannot be negative?

More practice: warm-up

In a normal distribution, how do the mean, median and mode compare?

What is the total area under a normal curve?

What two numbers completely determine a normal distribution?

Does a normal curve ever touch the x-axis?

What happens to a normal curve when σ gets larger?

More practice: on your own

Birth masses at a hospital are modelled by N(3.4, 0.5²) kg. Give the masses at μ ± 1σ, μ ± 2σ and μ ± 3σ.

For the birth masses above, what fraction of babies are heavier than 3.4 kg according to the model?

Curve P is N(100, 15²) and curve Q is N(100, 5²). Which is taller, and by what factor?

A data set of 300 values has mean 48, median 48.5, and its histogram has one peak and is close to symmetric. Is a normal model reasonable? What else would you check?

Explain in your own words why many measurements in nature are approximately normal.

Challenge

Use the formula f(x) = (1/(σ√(2π))) e^(−(x − μ)²/(2σ²)) to find the peak height of N(0, 1) and of N(32, 10.2²), to 4 decimal places.

Show that f(μ + σ) = f(μ) × e^(−1/2) for any normal curve, and evaluate e^(−1/2).

A company says the time to assemble a desk is normal with μ = 50 min and σ = 30 min. Explain why this model is doubtful, using the property that almost all values lie within 3σ of μ.

Check your understanding

Quiz

Which statement is true for every normal distribution?

A normal curve has inflection points at 32 and 48. What is σ?

Curve A is N(70, 4²); curve B is N(70, 8²). Which statement is true?

Which variable is LEAST likely to be modelled well by a normal distribution?

For X ~ N(25, 3²), what is P(X > 25)?

What does the notation X ~ N(100, 15²) tell you?

For parents and teachers

Big idea (B2.6): the normal distribution is a symmetric bell curve, set completely by its mean μ and standard deviation σ. Mean = median = mode, the curve bends at μ ± σ, and it models many measurements that come from lots of small random effects.

How to help: Ask your teen to sketch N(32, 10.2²) with a labelled scale (1.4, 11.6, 21.8, 32, 42.2, 52.4, 62.6). Then play "normal or not?" with everyday examples: shoe sizes of adult women (roughly normal), the number of emails people get (skewed), sap from maple trees on one day (varies; ask why).

Common difficulties:

Reading N(μ, σ²) as N(μ, σ). In this lesson the second number is written as a square to remind you it is the variance.
Drawing the curve touching the axis, or drawing σ too wide (the inflection points should be about one-sixth of the visible width from the centre).
Assuming every data set is normal. Check the shape first.