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AI HL · 13.2

Probability Distributions

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13.2.7

The normal distribution: properties and when it is a good model

13.2.7

The normal distribution: properties and when it is a good model

Target: I can describe a normal curve: symmetric, bell-shaped, with its peak at the mean, and write \(X \sim N(\mu, \sigma^2)\). (+3 more)

By the end of this section

Key points

The normal distribution

  • A normal distribution is continuous. Probability is the area under the curve; the total area is 1.
  • The curve is a symmetric bell shape with its peak at the mean \(\mu\). The mean, median and mode are all equal.
  • We write \(X \sim N(\mu, \sigma^2)\): \(\mu\) is the mean and \(\sigma\) is the standard deviation, so \(\sigma^2\) is the variance.
  • A larger standard deviation gives a wider, flatter curve. The area under it is still 1.

The 68–95–99.7 rule

  • About 68% of values lie within 1 standard deviation of the mean.
  • About 95% lie within 2 standard deviations, and about 99.7% within 3.
  • The curve is symmetric, so the percentage in each tail is half of what is left over: 16% below \(\mu - \sigma\) and 2.5% above \(\mu + 2\sigma\).
  • The rule gives approximate answers. Exact areas come from the calculator in the next lesson.

From the binomial to the normal

  • On a Galton board a ball goes left or right at each pin. The number of right bounces is binomial, \(B(n, 0.5)\).
  • Drop many balls and the columns form a bell shape. The more rows, the closer the pile is to a normal curve.
  • With p = 0.5 the binomial is symmetric. With p near 0 or 1 it is skewed, and the normal shape fits badly.

Is a normal model suitable?

  • On a box plot, look for symmetry: median in the middle of the box and whiskers about equal. A long tail on one side means skew, and a normal model is poor.
  • Look at the limits. If the mean is within 2 standard deviations of a value that cannot be passed (such as 0 for a time), a normal model gives probability to impossible values.
  • Discrete counts with few values, such as a number of siblings, are poorly modelled by a continuous normal curve.
  • Give two reasons when asked, each tied to the data: for example, skewed box plot, and values limited at 0.

Explore: the normal distribution

A probability is an area under the curve. Drag the top of the curve to move the mean, the teal point on its side to change σ, and the orange points under the axis to choose the area. Or type exact values. Say how big the area is, then press Show answer.

Drag with the mouse or a finger, or select a point (Tab) and use the arrow keys. The axis fits itself to the curve when you let go.

Drag a point to change the shaded area.

  1. Drag the top of the curve, then the teal point. Which one moves the curve, and which one changes its shape? Why does a wider curve have to be lower?
  2. Press "Mark μ ± σ, 2σ, 3σ". Drag the orange points to μ − σ and μ + σ, then 2σ and 3σ. Say each area before you show it. Does it change when you change μ or σ?
  3. X ~ N(250, 16). What do you type for σ? Find P(X < 246).
  4. In "Find a value", choose Above and an area of 0.05. Is the value above or below the mean? Why does the calculator need the area 0.95?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
The Galton board

A Galton board has 12 rows of pins. A ball bounces left or right at each pin, each with probability 0.5, independently. Let B be the number of times a ball bounces right, so \(B \sim B(12, 0.5)\). Many balls are dropped and they pile up in columns by their value of B. Find the mean of B and say what shape the pile has.

Working
  1. Mean

    \(np = 12 \times 0.5 = 6\)

    A ball bounces right 6 times on average, so the middle column is the tallest.

  2. Spread

    \(\sigma = \sqrt{12 \times 0.5 \times 0.5} = 1.73\)

    Most balls land within a column or two of the middle.

  3. Shape

    Symmetric, one peak in the middle, tails that fall away on both sides.

    With p = 0.5 the binomial is symmetric. As the number of rows grows, the columns follow a smooth bell shape: the normal curve.

Video walkthrough coming
The 68–95–99.7 rule

The mass of a loaf from a bakery is normally distributed with mean 500 g and standard deviation 8 g. Use the 68–95–99.7 rule to find the percentage of loaves that weigh more than 516 g.

464472480488496504512520528536Mass (g)
Working
  1. In standard deviations

    \(516 = 500 + 2 \times 8\)

    516 g is 2 standard deviations above the mean.

  2. Within 2 sd

    About 95% of loaves are between 484 g and 516 g.

    The rule: about 68% within 1 sd, 95% within 2 sd, 99.7% within 3 sd.

  3. Both tails

    \(100 - 95 = 5\%\) outside

    The curve is symmetric, so the 5% splits equally above and below.

  4. One tail

    \(5 \div 2 = 2.5\%\)

    About 2.5% of loaves are heavier than 516 g.

Video walkthrough coming
A skewed box plot

The box plot shows the time, in minutes, that 60 patients waited at a clinic. A student suggests a normal distribution as a model. Give two reasons it is not suitable.

Wait0510152025303540Waiting time (minutes)
Working
  1. Read the box plot

    Min 2, Q1 4, median 7, Q3 15, max 36

    Compare the two halves of the box and the two whiskers.

  2. Reason 1: not symmetric

    Median − Q1 = 3 but Q3 − median = 8

    A normal distribution is symmetric about the mean. This one is skewed to the right.

  3. Reason 2: the whiskers

    Lower whisker 2, upper whisker 21

    The long upper tail has no mirror on the left. Waiting times also cannot go below 0, so the curve is cut off on the left.

Video walkthrough coming
A mean close to a limit

The number of minutes a train is late is modelled by a normal distribution with mean 3 and standard deviation 2.5. Explain why a normal distribution is not a good model.

−7.5−5−2.52.557.51012.515Minutes late
Working
  1. Distance to the limit

    \(3 \div 2.5 = 1.2\) standard deviations

    A late time of 0 is the limit. It is only 1.2 standard deviations below the mean.

  2. Compare with the rule

    From the calculator: normal cdf with lower = −10^99, upper = 0, μ = 3, σ = 2.5: 0.115

    The model gives about 11.5% to negative times. A negative time cannot happen.

  3. Conclude

    The mean is less than 2 standard deviations from 0, so the model is poor.

    A good normal model has the mean well away from any limit, at least 2 standard deviations.

Video walkthrough coming