4.3 Measures of central tendency and dispersionIB Maths: Applications and Interpretation HL: Revision notes

The mean is $\bar{x}=\frac{\sum x}{n}$. The median is the middle value of the ordered data (the mean of the two middle values if $n$ is even). The mode is the most frequent value. Example: $2,0,3,1,4,1,2,5,1,1$ ordered is $0,1,1,1,1,2,2,3,4,5$: mean $=\frac{20}{10}=2$, median $=\frac{1+2}{2}=1.5$, mode $=1$. For a frequency table, $\bar{x}=\frac{\sum fx}{\sum f}$, which your GDC can do from lists.

For grouped data the individual values are unknown, so use the mid-interval value of each class (the mean of its end points) to estimate the mean: $\bar{x}\approx\frac{\sum f\,m}{\sum f}.$ The modal class is the class with the greatest frequency, for equal class widths only. Example: classes $0\le t<10,\dots,40\le t<50$ with frequencies $6,14,18,9,3$. Mid-interval values $5,15,25,35,45$ give $\sum fm=1140$ and $\bar{x}\approx\frac{1140}{50}=22.8$. Modal class: $20\le t<30$.

Range $=\text{max}-\text{min}$. Interquartile range $\text{IQR}=Q_3-Q_1$ measures the spread of the middle half and is not affected by outliers. The standard deviation $\sigma$ measures typical distance from the mean; the variance is its square, $\sigma^2$. Use your GDC (one-variable statistics) to find $\sigma$. At SL the data set is treated as the population unless stated otherwise, so use the population standard deviation $\sigma_x$; a sample version $s_{n-1}$ gives a slightly larger value. Hand calculation, $\sigma=\sqrt{\frac{\sum(x-\bar{x})^2}{n}}$, can help understanding but is not required. Example: for $0,1,1,1,1,2,2,3,4,5$, $\sum(x-2)^2=22$, so variance $=2.2$ and $\sigma=1.48$.

Quartiles divide ordered data into four equal parts: $Q_1$ (lower quartile), the median $Q_2$ and $Q_3$ (upper quartile). For discrete data, find them with your GDC. There are different methods for quartiles (for example whether the median is included in each half when $n$ is odd), so a value from your GDC may differ slightly from a hand calculation. Use the GDC value. Example: $10.8,11.0,11.0,11.2,11.2,11.4\mid11.4,11.6,11.6,11.8,12.0,12.0$ gives $Q_1=11.1$, $Q_3=11.7$, $\text{IQR}=0.6$.

If every data item has a constant $k$ added or subtracted, the mean, median and quartiles change by $k$, but the standard deviation, variance, IQR and range are unchanged. If every item is multiplied by $k$, the mean, median, quartiles, standard deviation, IQR and range are all multiplied by $k$, and the variance by $k^2$. Example: temperatures with mean $30$ and $\sigma=4$, converted by $F=1.8C+32$: new mean $=1.8(30)+32=86$; new $\sigma=1.8\times4=7.2$ (the $+32$ does not change spread); new variance $=7.2^2=51.84$.