Standard Deviation and z-Scores: Quick Revision

A brief note on Standard Deviation and z-Scores. Read it before a test. Try each question at the end before you open its answer.

Measures of spread

• Deviations $x - \bar{x}$ always add up to zero.
• MAD: $\dfrac{\sum |x - \bar{x}|}{n}$.
• Population SD (whole group): $\sigma = \sqrt{\dfrac{\sum (x - \mu)^2}{n}}$.
• Sample SD (a sample): $s = \sqrt{\dfrac{\sum (x - \bar{x})^2}{n - 1}}$.
• Variance is the square of the SD, in squared units.

Shifting and stretching

• Add $a$: centres go up by $a$; spreads stay the same.
• Multiply by $b$: centres times $b$; SD, IQR and range times $|b|$; variance times $b^2$.
• Standardise: $z = \dfrac{x - \bar{x}}{\sigma}$ gives mean 0 and SD 1.

z-scores and the bands

• z-score: $z = \dfrac{x - \text{mean}}{\text{SD}}$, no units; back again, $x = \text{mean} + z \times \text{SD}$.
• Mound-shaped data: about $68\%$, $95\%$ and $99.7\%$ within 1, 2 and 3 SDs.
• Any shape: at least $75\%$ within 2 SDs and about $89\%$ within 3 SDs (Chebyshev).

A worked example

Five whole-class quiz scores are $6, 8, 9, 11$ and $16$.

1. Mean. $\dfrac{50}{5} = 10$.
2. Deviations. $-4, -2, -1, 1, 6$; squares add up to 58.
3. Population SD. $\sigma = \sqrt{\dfrac{58}{5}} \approx 3.41$.
4. z-score of 16. $\dfrac{16 - 10}{3.41} \approx 1.76$.

Practice with new numbers

A program to try

The program finds the mean, both standard deviations and the z-score of the largest value in the worked example.

Mistakes to avoid

• Mixing up $n$ and $n - 1$. Whole group: $n$. Sample: $n - 1$.
• Adding to the SD when every value is shifted.
• Dividing by the variance in a z-score.
• Using 68-95-99.7 on skewed data.

Check yourself

1. Find the population SD of $2, 4, 6$.

2. A data set has SD 5. Every value is multiplied by $-3$. Find the new SD.

3. The mean is 30 and the SD is 4. Find the z-score of 24.

4. Mound-shaped data have mean 70 and SD 5. Between which values do about $68\%$ lie?

Sources

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