4.3 Measures of central tendency and dispersionIB Maths: Analysis and Approaches HL: Revision notes

For a list of $n$ values the mean is $\bar{x} = \frac{\sum x}{n},$ and for a frequency table it is $\bar{x} = \frac{\sum fx}{\sum f}$. At SL the data set is treated as the whole population, so the formula booklet writes the mean as $\mu$.

The median is the middle value once the data are in order: it is the $\left(\frac{n+1}{2}\right)$th value. With an even number of values, take the mean of the two middle values. The mode is the most frequent value; a data set can have more than one mode or none.

If you know the mean of a group, you know its total: total $= n\bar{x}$. This is the quickest way to handle a value being added or removed.

When data are grouped into classes such as $25 \le t < 30$, the exact values are lost. To estimate the mean, replace every value in a class by its mid-interval value (here 27.5) and use $\frac{\sum fx}{\sum f}$.

The modal class is the class with the highest frequency. On this course the modal class is only used with equal class widths.

The answer is an estimate because it assumes the values in each class are centred on the mid-interval value.

Dispersion measures how spread out the data are.

• Range $=$ largest $-$ smallest: simple, but depends only on the two most extreme values.
• Interquartile range $\mathrm{IQR} = Q_3 - Q_1$: the spread of the middle 50% of the data, so it is not affected by extreme values.
• Standard deviation $\sigma$: a typical distance of the values from the mean. The variance is $\sigma^2$.

On this course you find the standard deviation and variance with technology only: enter the data (with frequencies, or mid-interval values for grouped data) into your GDC and read off $\sigma$. The standard deviation has the same units as the data; the variance has squared units.

The lower quartile $Q_1$ and upper quartile $Q_3$ split the ordered data into quarters. For discrete data you find them with technology. Be aware that different methods exist: some calculators exclude the median and take the median of each half, others interpolate. For 38, 39, 41, 42, 44, 45, 46, 47, 48, 51, 120, one method gives $Q_1 = 41$ and $Q_3 = 48$, another gives $41.5$ and $47.5$. IB markschemes accept values from standard methods.

If every value is transformed by $y = ax + b$:

• mean: $\bar{y} = a\bar{x} + b$ (the median and mode transform the same way);
• standard deviation: $\sigma_y = |a|\,\sigma_x$, because adding $b$ shifts the data but does not change the spread;
• variance: $\sigma_y^2 = a^2\sigma_x^2$.

Example: temperatures with mean $18.4\,^{\circ}\mathrm{C}$ and $\sigma = 3.5\,^{\circ}\mathrm{C}$ converted by $F = 1.8C + 32$ have mean $65.12\,^{\circ}\mathrm{F}$ and $\sigma = 6.3\,^{\circ}\mathrm{F}$.