Dot plots
A dot plot uses a number line. For every item of data, you put one dot above its value. Equal values stack up.
It is good for small sets of numbers (up to about 30). You can quickly see the most common value (the mode), the smallest and largest values, and any gaps or odd values (outliers).
Bar graphs
A bar graph compares separate groups, called categories (fruits, sports, cities). Each bar has the same width. Its height (or length) shows the number. Bars have equal gaps, because the categories are separate.
Parts of a good bar graph: a title, labelled axes, a scale that goes up in equal steps and starts at 0. A double bar graph puts two bars side by side for each category to compare two groups (for example boys and girls).
Histograms and class intervals
When there are many different numbers (heights, marks, times), we group them into class intervals such as 130โ140, 140โ150. The number in each interval is its frequency.
A histogram draws one bar for each interval. The bars touch, because the number line has no gaps. Usual rule: 140 goes in 140โ150, not in 130โ140 (the lower limit is included, the upper limit is not).
| Bar graph | Histogram |
|---|---|
| Categories (words) | Numbers in intervals |
| Gaps between bars | No gaps |
| Order of bars can change | Order fixed by the number line |
Line graphs
A line graph shows how one quantity changes over time: temperature over a day, a child's height each year, a phone's battery level. Time goes on the x-axis. Plot a point for each reading and join the points in order.
A line going up means an increase, going down means a decrease, flat means no change. A steeper line means faster change. Points between readings are only estimates.
Choosing the right graph
- Few numbers, want to see each one โ dot plot.
- Compare categories โ bar graph (or pie chart for parts of a whole).
- Many numbers, want the spread โ histogram.
- Change over time โ line graph.
Spotting misleading graphs
A graph can tell a false story even when every number is correct. Check these things:
- Where does the axis start? A y-axis that starts at 50 instead of 0 makes small differences look big. A zig-zag break mark should warn you.
- Is the scale even? Steps like 0, 10, 20, 50, 100 squash or stretch parts of the graph.
- Picture size: if a picture is twice as tall and twice as wide, it looks 4 times bigger, not 2.
- Missing labels or title: you cannot tell what is measured.
- Cherry-picking: showing only a few months that suit the story.
- 3D effects that tilt bars so front bars look larger.
Key formulas and definitions
- Frequency = how many times a value (or interval) occurs
- Class interval 140โ150: includes 140, excludes 150
- Width of interval = upper limit โ lower limit
- Bar graph: gaps; histogram: no gaps
- Line graph: x-axis = time, joined points show change
- Misleading check: axis starts at 0? even scale? labels? fair picture size?
Worked examples
1. Dot plot data: books read by students: 0, 1, 1, 2, 2, 2, 3. Draw it in words and find the mode.
Number line 0โ3. One dot above 0, two above 1, three above 2, one above 3. The tallest stack is above 2, so the mode is 2 books.
2. Favourite sport votes: cricket 12, football 8, badminton 5. Which graph should you draw and what goes on each axis?
A bar graph (separate categories). x-axis: sports, with equal gaps. y-axis: number of votes, scale 0, 2, 4 โฆ 12.
3. Marks of 10 students: 12, 18, 25, 27, 31, 33, 36, 38, 41, 47. Make a frequency table with intervals 10โ20, 20โ30, 30โ40, 40โ50.
10โ20: 12, 18 โ 2. 20โ30: 25, 27 โ 2. 30โ40: 31, 33, 36, 38 โ 4. 40โ50: 41, 47 โ 2. Total 10 โ. Draw touching bars of heights 2, 2, 4, 2.
4. Temperature at 6 am, 9 am, 12 pm, 3 pm, 6 pm: 22, 27, 32, 33, 28 ยฐC. When did it rise fastest?
Changes: +5, +5, +1, โ5. The rise is 5 ยฐC between 6โ9 am and between 9 amโ12 pm, so those two parts of the line are equally steep and steepest upward.
5. A bar graph shows Brand A = 98 and Brand B = 100 with the y-axis starting at 96. The bar for B looks twice as tall as A. Is that fair?
Heights shown: 98 โ 96 = 2 and 100 โ 96 = 4, so B looks 2 times bigger. Real ratio: 100 รท 98 โ 1.02, only 2% more. The graph is misleading; the axis should start at 0.
6. Is 50 counted in the interval 40โ50 or 50โ60?
In 50โ60. By the usual rule the lower limit is included and the upper limit is not.
Common mistakes
- Leaving gaps between bars in a histogram. Its bars must touch.
- Using a line graph for categories (like fruits). Lines only make sense when the x-axis is in order, usually time.
- Putting a value equal to the upper limit in the lower interval. 150 goes in 150โ160.
- Trusting bar heights without checking that the y-axis starts at 0.