Mathematics, five units · Grade 12

What is the normal distribution and how do you standardise a measurement?

The normal distribution is a symmetric bell curve centred on the mean, with its width set by the standard deviation. Standardising turns a measurement into a z-score: subtract the mean and divide by the standard deviation. The score says how many standard deviations the measurement is from the mean, which is what lets you read a probability from the table.

Learning objectives

Two numbers describe the whole curve

The mean fixes where the centre of the bell sits, and the standard deviation fixes how wide it is. Those two numbers describe the entire distribution — nothing further is needed.

The curve is symmetric about the mean, so half of it lies on each side. That fact saves calculation in a great many questions: the probability of being below the mean is exactly a half.

A rule of thumb worth remembering: about sixty-eight per cent of measurements fall within one standard deviation of the mean, and about ninety-five per cent within two. It does not replace the table, but it tells you immediately whether an answer is plausible.

Why standardise at all

Every normal distribution has its own mean and standard deviation, and no table could cover them all. Standardising moves any distribution onto one standard distribution, with a mean of zero and a standard deviation of one.

A z-score is not a unit of measurement but a relative distance. A score of 2 means two standard deviations above the mean, whether the measurement was a height in centimetres or a mark in an exam.

Reading the table

The table gives the probability of being below a given z-score. To get the probability of being above it, subtract from one; for the probability between two values, subtract the smaller probability from the larger.

A negative z-score usually has no row of its own, so use the symmetry: the probability below minus z equals the probability above z.

Worked examples

  1. Exam marks are normally distributed with mean 70 and standard deviation 10. What is the z-score of 85?

    1. Subtract the mean: 85 minus 70, that is 15
    2. Divide by the standard deviation: 15 over 10

    Answer: The z-score is 1.5

  2. In the same distribution, what is the probability of scoring below 70?

    1. 70 is exactly the mean, so its z-score is zero
    2. The curve is symmetric about the mean

    Answer: The probability is 0.5

  3. Heights are normally distributed with mean 170 and standard deviation 8. Which height has a z-score of minus 2?

    1. Minus 2 standard deviations is minus 16 centimetres
    2. Add to the mean

    Answer: The height is 154 centimetres

Common mistakes

Dividing by the variance instead of the standard deviation
The variance is the standard deviation squared. Dividing by it gives a z-score smaller by a factor of the standard deviation, and every probability read from the table will be wrong.
Subtracting the mean after dividing
The order is fixed: subtract the mean first, then divide. Reversing it produces a number with no meaning at all.
Giving up when a negative z-score is not in the table
The table relies on the symmetry of the curve. The probability below minus z equals the probability above z, which is one minus the table value for the positive z.

What to remember

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