Statistics with Grouped Data
Senior 2 · Mathematics
Statistics with Grouped Data
Lesson 5 of 5
25 min
A table gives exact numbers; a graph shows the shape at once: where the data bunch up, whether they are spread out, and whether a few values sit far away. Newspapers, district reports and national examinations all expect you to draw and read three graphs of grouped data: the histogram, the frequency polygon and the pie chart.
A histogram is a bar graph for grouped numerical data. The horizontal axis is a number line of the variable, marked with the class boundaries; the vertical axis shows frequency. For classes of equal width, each bar's height is the class frequency.
Because the classes follow one another with no gaps (the boundary 14.5 ends one class and starts the next), the bars touch. That is the key difference from an ordinary bar chart of categories.
Figure 1. Histogram of the walking times of 40 students. Diagram: Ganzaa.
To draw one: choose a scale; mark the boundaries 4.5, 14.5, …, 54.5 on the horizontal axis; draw each bar from boundary to boundary with height equal to the frequency; label both axes and give a title.
A frequency polygon joins the points (midpoint, frequency) with straight lines. To close it, add an empty class at each end, with frequency 0: here the classes before 5–14 and after 45–54 have midpoints −0.5 and 59.5.
Figure 2. Frequency polygon of the walking times, closed with zero-frequency points at −0.5 and 59.5. Diagram: Ganzaa.
If you draw the polygon on top of the histogram, it passes through the middle of the top of each bar. Frequency polygons are especially useful for comparing two sets of data on one graph, for example the walking times of two schools.
A pie chart shows each class as a slice of a circle, with the angle proportional to the frequency:
angle = (frequency ÷ total) × 360°
Worked example
For the class 25–34: 13 ÷ 40 × 360° = 117°. All the angles: 45°, 108°, 117°, 63°, 27°, which add up to 360°.
Figure 3. Pie chart of the walking times. Diagram: Ganzaa.
A pie chart shows each class as a share of the whole, but it hides the order of the classes along the number line, so for grouped numerical data the histogram is usually the better choice.
Tip
Histogram: boundaries, touching bars. Polygon: midpoints, closed at both ends. Pie chart: angles that add up to 360°. Before drawing a pie chart, check your angles add up to 360°; before drawing a polygon, check you have added the two zero points.
Marks out of 50 of 30 students: 0–9: 2; 10–19: 5; 20–29: 9; 30–39: 10; 40–49: 4.
Which numbers mark the edges of the histogram bars?
Solution: the class boundaries −0.5, 9.5, 19.5, 29.5, 39.5, 49.5.
List the points plotted for the frequency polygon, including the closing points.
Solution: (−5.5, 0), (4.5, 2), (14.5, 5), (24.5, 9), (34.5, 10), (44.5, 4), (54.5, 0).
Find the pie-chart angle of each class.
Solution: 2 ÷ 30 × 360° = 24°; then 60°, 108°, 120° and 48°. Check: 24 + 60 + 108 + 120 + 48 = 360.
In a pie chart of 60 students, a class has an angle of 90°. How many students are in it?
Solution: 90 ÷ 360 × 60 = 15 students.
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
