z-scores and the 68-95-99.7 idea

What you will be able to do

• find and explain the z-score of a value, and turn a z-score back into a value;
• compare values from different scales with z-scores;
• use the 68-95-99.7 idea on mound-shaped data, and know when not to use it.

Two subjects, two scores

Aisha scores 72 in Maths and 78 in English in a made-up school test. Her English mark is higher. Is it her better result? It depends on the other students. In Maths the mean was 60 and the SD was 8 marks. In English the mean was 74 and the SD was 5 marks.

The z-score

The z-score of a value is

It counts how many standard deviations the value lies from the mean. A positive z-score is above the mean, a negative one below it, and 0 is exactly at the mean.

• Maths: $z = \dfrac{72 - 60}{8} = 1.5$.
• English: $z = \dfrac{78 - 74}{5} = 0.8$.

So, compared with the other students, Aisha did better in Maths. Her Maths mark is 1.5 SDs above the mean, while her English mark is less than 1 SD above it. Marks and minutes, metres and seconds all cancel in the division, so a z-score has no units. That is why it can compare values from different scales.

Going back. If you know the z-score, then $x = \text{mean} + z \times \text{SD}$. In Maths, a z-score of $-1$ means a mark of $60 - 8 = 52$.

When smaller is better. In a race, a time below the mean is good, so a good runner has a negative z-score. A long jump is the other way round. Compare two sports by how many SDs each athlete beat the mean, with the sign turned the right way.

The 68-95-99.7 idea

Many data sets are mound-shaped: one peak in the middle, roughly symmetric, with values thinning out on both sides. Heights, many measurement errors and the masses of eggs often look like this. For such data a useful rule holds:

• about $68\%$ of the values lie within 1 SD of the mean ($z$ between $-1$ and $1$);
• about $95\%$ lie within 2 SDs;
• about $99.7\%$, almost all, lie within 3 SDs.

Let us test it. The program below makes up 400 heights at random with a fixed seed and counts them. Their mean is about 163.1 cm and their SD about 7.2 cm.

The counts are 277 heights within 1 SD, 384 within 2 SDs and all 400 within 3 SDs. That is $69.25\%$, $96\%$ and $100\%$, close to the rule. The exact areas behind 68, 95 and 99.7 come from the normal distribution, which you can meet in chapter 4 of this collection.

When not to use it. The rule is for mound-shaped data. For skewed data it can be badly wrong. One rule does hold for any shape: at least $75\%$ of the values lie within 2 SDs of the mean, and at least about $89\%$ within 3 SDs. It is called Chebyshev's rule. It promises less, but it never fails.

You should find that a larger SD spreads the bands apart and flattens the mound. With a mean of 163 cm and an SD of 7 cm, about $95\%$ of the heights lie between 149 cm and 177 cm.

Practise with new numbers

Try it in Python

The program makes the 400 made-up heights with a fixed seed, finds the z-score of each, and counts how many lie within 1, 2 and 3 standard deviations.

Common mistakes

• Forgetting the sign. A value below the mean has a negative z-score.
• Dividing by the variance. Divide by the standard deviation.
• Using 68-95-99.7 on skewed data. It is for mound-shaped data only.
• Comparing raw scores across scales. Compare z-scores instead.

Remember these

• $z = \dfrac{x - \text{mean}}{\text{SD}}$; $x = \text{mean} + z \times \text{SD}$.
• A z-score has no units, so it compares values from different scales.
• Mound-shaped data: about $68\%$, $95\%$ and $99.7\%$ within 1, 2 and 3 SDs.
• Any shape: at least $75\%$ within 2 SDs (Chebyshev).

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The mean is 50 and the SD is 4. Find the z-score of 56 and of 44.

2. In a test with mean 65 and SD 10, which mark has a z-score of 2.3?

3. Masses of apples are mound-shaped with mean 150 g and SD 12 g. Between which masses do about $95\%$ of the apples lie?

4. For the apples of question 3, about what percentage weigh more than 162 g?

5. Incomes in a town are strongly skewed. Can you say that about $95\%$ lie within 2 SDs of the mean?

References

• OpenStax, Introductory Statistics, Chapter 2 (Descriptive Statistics), section on measures of the spread of the data, and Chapter 6 (The Normal Distribution), section on the standard normal distribution.