The Normal Distribution

The normal distribution describes data such as heights, test marks and the weights of packaged food, which cluster around the mean in a bell shape. It was worth 4 marks in NESA's sample HSC paper and about 7 marks a year in recent HSC papers. Almost every question uses just two tools: the z-score formula and the 68–95–99.7 rule.

• A normally distributed dataset has a symmetrical bell-shaped curve, with mean, median and mode approximately equal.
• A z-score is the number of standard deviations a value is above or below the mean.
• The z-scores of a normal dataset have a mean of 0 and a standard deviation of 1.
• Use z-scores to compare results from different datasets.
• The empirical rule: about 68%, 95% and 99.7% of data have z-scores between −1 and 1, −2 and 2, and −3 and 3.
• Find probabilities with z-scores and the empirical rule, shown as shaded areas under the curve.
• Use z-scores to make judgements, such as whether a result is unusual.

When a large dataset is normally distributed, its histogram looks like a bell: highest in the middle and tailing off evenly on both sides.

• It is symmetrical about the mean, so half the data is above the mean and half below.
• The mean, median and mode are approximately equal, all at the centre.
• The standard deviation $\sigma$ controls the spread: a larger $\sigma$ gives a wider, flatter curve.
• The total area under the curve represents 100% of the data, so an area represents a percentage or a probability.

Data that is skewed (with a long tail on one side) or has two peaks is not normally distributed, and the rules on this page don't apply.

$x$ is the value, $\mu$ the mean and $\sigma$ the standard deviation.

• A positive z-score is above the mean; a negative z-score is below it; $z=0$ is exactly the mean.
• $z=2$ means "2 standard deviations above the mean".

Example. Bags of rice are filled with a mean of 500 g and a standard deviation of 4 g. A bag weighing 494 g has

so it is 1.5 standard deviations below the mean.

To go backwards from a z-score to a value, rearrange: $x=\mu +z\sigma$. A bag with $z=1.5$ weighs $500+1.5\times 4=506$ g.

Worked example: comparing results

Mia scored 71 in an English test (mean 64, standard deviation 5) and 78 in a Science test (mean 70, standard deviation 6.5). In which test did she do better compared with her class?

Her English z-score (1.4) is higher than her Science z-score (1.23, 2 d.p.), so she did better in English relative to her class, even though her Science mark is higher. The raw marks can't be compared directly because the tests have different means and spreads.

For normally distributed data:

Because the curve is symmetrical, split these in half for one side of the mean:

And the tails: 16% of data has $z>1$ (since $50-34=16$), 2.5% has $z>2$, and 0.15% has $z>3$. The same amounts lie below $-1$, $-2$ and $-3$.

To find a probability:

1. Draw a bell curve and mark the mean in the middle.
2. Mark $\mu \pm \sigma$, $\mu \pm 2\sigma$ and $\mu \pm 3\sigma$ underneath.
3. Convert the values in the question to z-scores.
4. Shade the region the question asks about.
5. Add or subtract the empirical rule percentages for the shaded pieces.

Worked example: bags of rice

The rice bags above have mean 500 g and standard deviation 4 g, and the weights are normally distributed.

Step 1: mark the scale.

(a) What percentage of bags weigh more than 504 g?

504 g has $z=1$. Shade the right tail beyond $z=1$: $50\%-34\%=16\%$.

(b) What percentage weigh between 488 g and 504 g?

These are $z=-3$ and $z=1$. Shade from $-3$ to 1, in two pieces either side of the mean:

(c) Bags under 492 g are rejected. Out of 600 bags, how many are expected to be rejected?

492 g has $z=-2$. The tail below $z=-2$ is 2.5%.

About 15 bags are expected to be rejected.

z-scores tell you how unusual a value is:

• Between −1 and 1: typical (about 68% of data).
• Beyond ±2: unusual (only about 5% of data is that far out).
• Beyond ±3: very rare (about 0.3% of data).

For example, a 513 g bag ($z=3.25$) would be very unusual. A quality manager might check whether the filling machine needs adjusting. When asked to justify a judgement, quote the z-score and the percentage it corresponds to.

• Swapping the order in $x-\mu$, which gives the wrong sign.
• Dividing by the variance or the mean instead of the standard deviation.
• Using 68% for one side of the mean. One side is 34%.
• Adding percentages for regions on the same side that overlap, instead of subtracting.
• Comparing raw scores from different tests instead of z-scores.
• Using the empirical rule on data that isn't normally distributed.

• Always sketch and shade the curve; it stops most errors and can earn a mark.
• Show the z-score calculation for every value, with the formula.
• In comparison questions, finish with a sentence: "Her result in English was better because its z-score is higher."
• Expected number questions: find the percentage first, then multiply by the total. Round to a whole number if counting people or items.

• Data analysis (Year 11): mean, median, mode and population standard deviation from a calculator.
• Relative frequency and probability: areas under the normal curve are probabilities, and expected numbers use $np$.
• Bivariate data analysis: the other Year 12 statistics focus area.