Averages and spreadIB MYP Maths Standard: Revision notes

The mean is the total divided by the number of values. The median is the middle value when the data are in order (with an even number of values, halfway between the two middle ones). The mode is the most common value. The range is the largest value minus the smallest, and it measures the spread. Example: $12, 15, 13, 12, 40, 14, 12$. Mean $=\frac{118}{7}=16.9$. Ordered: $12,12,12,13,14,15,40$ so the median is $13$, the mode is $12$ and the range is $40-12=28$.

In a frequency table, the total number of values is $\sum f$, and the total of all the values is $\sum fx$ (multiply each value by its frequency, then add). Mean $=\frac{\sum fx}{\sum f}$. The mode is the value with the highest frequency (not the frequency itself). For the median, use cumulative frequencies to find the middle position. Example: $0,1,2,3,4$ goals with frequencies $4,6,5,3,2$ (total $20$). $\sum fx=33$, so the mean is $1.65$. The mode is $1$. The 10th value is $1$ and the 11th is $2$, so the median is $1.5$.

In a grouped table the individual values are not known, so we can only estimate the mean. Use the midpoint of each class as the value for that class. Estimated mean $=\frac{\sum f\times\text{midpoint}}{\sum f}$. The modal class is the class with the highest frequency. The median class is the class containing the middle value. Example: classes $0$ to $<10$, $10$ to $<20$, $20$ to $<30$, $30$ to $<40$, $40$ to $<50$ with frequencies $4,10,15,8,3$. Midpoints $5,15,25,35,45$ give $\sum fx=960$ and an estimated mean of $\frac{960}{40}=24$.

• Mean: uses all the values, but is pulled by outliers (extreme values).
• Median: not affected by outliers, so it is best for skewed data, such as house prices or one very high value.
• Mode: the only average for categorical data (such as the most popular colour), and best when the most common value matters, such as the most common shoe size. Example: ages $12,12,12,13,14,15,40$: the mean $16.9$ is higher than the age of 6 of the 7 members, so the median $13$ is a better typical value.

To compare two data sets, make two comments: one about an average (mean or median) and one about the spread (range). Use the context, and compare the numbers. Example: Swift times have mean $20$ and range $4$. Rapid times have mean $18$ and range $24$. Rapid is quicker on average, but Swift is more consistent because its range is smaller. In real-life (criterion D) problems, link the comparison to the situation (for example, exam papers must arrive within $25$ minutes) and say how reliable the data are. Small samples may not be typical.