Sec 4 E-Math Statistical Diagrams notes

A table stores raw facts; diagrams turn them visible. Bar graphs compare separate categories — heights equal frequencies, and the bars stay separated because the categories are discrete. Pictograms repeat a symbol per unit (e.g. one car icon per 10 cars); they are friendly to read but a partial symbol is hard to judge. Line graphs connect points to expose trends over time. Dot diagrams stack one dot per data value, ideal for small discrete sets — the tallest stack is the mode.

Choosing between them: time trend → line graph; comparing categories → bar graph; small discrete data → dot diagram; fractions of a total → pie chart (next section). The question "state why this diagram was chosen" wants the purpose, not a description.

A pie chart shares one whole between categories: the full circle is 360°, and each category’s angle = (frequency ÷ total) × 360°. Because the total circle represents the grand total, pie charts answer "what fraction?" questions at a glance — but they hide absolute numbers unless the total is stated.

1. 1Total the frequencies

   Class of 40 by transport: walk 12, bus 10, car 8, cycle 6, other 4 — total 40.

2. 2Fraction × 360°

   Walk: (12/40) × 360° = 108°; bus 90°; car 72°; cycle 54°; other 36°.

3. 3Check

   108 + 90 + 72 + 54 + 36 = 360° ✓ — the angles must always total 360°.

Two pie charts compare proportions within each group, not group sizes: a small school’s pie can look identical to a large school’s. Comparing totals needs bar graphs or the raw numbers.

A histogram displays grouped continuous data in equal class intervals with touching bars (continuity). The horizontal axis uses class boundaries (e.g. 10 ≤ w < 20) and the vertical axis the frequency. A stem-and-leaf diagram splits each value into a stem (leading digit(s)) and leaf (last digit) — it is a picture and a sorted list at once, so the median and mode can be read exactly.

Histogram questions love totals and fractions: from the parcels table, the modal class is 10 < m ≤ 20 (12 parcels), and the fraction heavier than 20 kg is (9 + 5)/33 = 14/33. Quote fractions or percentages — never read a single parcel’s exact mass from a histogram, because grouping destroyed that information.

A cumulative frequency table adds up as it goes: each row gives how many values are less than or equal to its upper boundary. Plotted (upper boundary against running total) and joined with a smooth curve, it estimates medians (read at n/2), quartiles (n/4 and 3n/4) and any percentile. A box-and-whisker plot compresses the same story into five numbers: minimum, lower quartile Q1, median, upper quartile Q3, maximum.

1. 1Read a class count

   Between 160 and 170: 25 − 12 = 13 students.

2. 2Median position

   n/2 = 17.5 → the 18th value; from the table it sits in the 160 < h ≤ 170 class.

3. 3Quartiles

   Q1 at n/4 = 8.75 → the 9th value; Q3 at 3n/4 = 26.25 → the 27th value.

Every diagram has a job description. Trends over time → line graph. Comparing categories → bar graph. Fractions of one whole → pie chart. Every individual value → stem-and-leaf or dot diagram. Spread and medians → box plot. Exam marks come from naming the purpose ("a line graph shows the trend over time"), not from naming the diagram alone.

• Truncated axis: starting the y-axis at 90 instead of 0 makes a small change look enormous.
• Unequal pictogram symbols: a bigger icon for a bigger bar inflates the impression twice over.
• Pie charts with different totals: comparing angles across pies ignores the group sizes.
• 3-D effects: perspective distorts which slice looks largest.

Inference questions ask what a diagram permits: from a box plot you may compare medians and spreads; you may not claim causation, nor read individual values that grouping has destroyed. Keep conclusions inside the data.