Representing dataIB MYP Maths Standard: Revision notes

A frequency table lists each value or category with its frequency (how many times it occurs). A tally chart is used to record data as you collect it, in groups of five. Continuous data are put in class intervals such as $0\leq h<10$, $10\leq h<20$ so that each value belongs to exactly one class. Check that the frequencies add up to the number of items in the data set.

A bar chart has bars of equal width with gaps for categorical or discrete data; the height shows frequency. A pictogram uses symbols with a key, for example one symbol $=2$ students, so $8$ students need $4$ symbols. In a pie chart each angle is $\frac{\text{frequency}}{\text{total}}\times360^\circ$. For $12$ out of $30$ students: $\frac{12}{30}\times360^\circ=144^\circ$. The angles must add up to $360^\circ$. Pie charts show proportions, not exact frequencies.

A line graph plots values over time and joins the points, to show a trend. Only use a line graph when the data change continuously, such as temperature across the day. A frequency polygon joins the midpoints of the tops of the class intervals. The midpoint of $20\leq x<30$ is $25$. Plot each (midpoint, frequency) point and join with straight lines.

A stem-and-leaf diagram splits each value into a stem (the tens) and a leaf (the units). The leaves in each row are written in order of size, and you must include a key, such as $4\mid1$ means $41$. Example: marks $41, 45, 41, 47, 45, 41$ give the row $4\mid1\ 1\ 1\ 5\ 5\ 7$. Repeated values are written out each time. The diagram keeps all the original data, so you can read off the median, mode and range.

A scatter graph plots pairs of values to show if there is a relationship (correlation). Positive correlation: as one increases, so does the other. Negative: as one increases, the other decreases. None: no pattern. A line of best fit passes through the mean point $(\bar{x},\bar{y})$ and close to all the points, with about equal numbers on each side. For temperatures $14, 18, 21, 25, 28, 30$ the mean is $\frac{136}{6}=22.7$. Interpolation (predicting inside the range of the data) is reliable. Extrapolation (predicting outside the range) may not be. An outlier is a point far from the pattern.

Choose the representation to fit the data and the purpose:

• categories (sports, colours): bar chart, pictogram or pie chart;
• change over time: line graph;
• two variables: scatter graph;
• showing every value in order: stem-and-leaf diagram. A graph is misleading if the vertical axis does not start at $0$ (bars look much taller than they should), the scale is uneven, axes are not labelled, a 3D effect distorts sizes, or the pictogram symbols have different sizes. Example: profits of $4.0$ and $4.5$ million drawn with an axis starting at $3.9$ million give bar heights of $0.1$ and $0.6$, so the second bar looks $6$ times as tall for only a $12.5\%$ increase.