Statistics and Dispersion

Measures of central tendency

• Mean: $\bar{x} = \dfrac{\sum x}{n}$. For grouped data, use class marks: $\bar{x} = \dfrac{\sum fx}{\sum f}$.
• Median: the middle value of the ordered data (average of the middle two if $n$ is even).
• Mode: the most frequent value.

Measures of dispersion

Measure	How to find it
Range	largest − smallest
Inter-quartile range (IQR)	$Q_3 - Q_1$
Standard deviation $\sigma$	$\sqrt{\dfrac{\sum (x - \bar{x})^2}{n}}$ (use the calculator)

The IQR is not affected by extreme values; the range and standard deviation are.

Box-and-whisker diagram

It shows five numbers: minimum, $Q_1$, median, $Q_3$, maximum. The length of the box is the IQR.

Effect of changing the data

If every datum $x$ becomes $ax + b$:

• mean, median and mode become $a(\text{old}) + b$;
• range, IQR and standard deviation become $|a| \times (\text{old})$. Adding $b$ does not change the spread.

Standard score

$$z = \frac{x - \bar{x}}{\sigma}$$

It tells you how many standard deviations a value is above or below the mean, so you can compare results from different tests.

Normal distribution

For data that follow a normal distribution, about

• 68% lie within $1\sigma$ of the mean,
• 95% lie within $2\sigma$,
• 99.7% lie within $3\sigma$.

Common mistake: adding a constant to every datum changes the mean but not the standard deviation.