Sec 1 Mathematics Statistical Data Handling notes

1. 11. Pose a question

   Decide what you want to find out, such as “How do the pupils in my class usually travel to school?”

2. 22. Collect the data

   Choose a method — a survey, a tally of observations, a measurement — and record the raw values.

3. 33. Organise the data

   Count the values into a frequency table so the mess of raw data becomes readable numbers.

4. 44. Represent the data

   Draw a pictogram, bar graph, pie chart, line graph or histogram that suits the data.

5. 55. Interpret and conclude

   Read the graph, answer the original question, and check whether the conclusion is fair.

The cycle runs in order: there is nothing to organise until data is collected, and nothing to interpret until it is represented. Exam questions often test a single link — for example, asking which stage comes before data collection (posing the question) or after representing the data (interpreting it).

A frequency table lists each category (or each class of values) beside the number of times it occurs — its frequency. Tallies are drawn as bundles of four with a fifth strike across them, so each completed bundle counts 5. A row showing three bundles and two extra strokes reads 3 × 5 + 2 = 17.

Two quick checks save marks: every tally row must agree with its frequency, and the frequencies must sum to the total surveyed. If a table of 20 pupils shows frequencies 15, 5 and 6, something is wrong — one row has been miscounted.

A pictogram repeats a symbol to show each frequency. The key states what one symbol is worth, for example one book symbol = 4 books. Rows of 3 symbols, 2½ symbols and 1 symbol then mean 12, 10 and 4 books. Half a symbol is allowed and means half the key value — but symbols must stay the same size throughout, or the picture lies.

A bar graph uses bars of equal width on labelled axes. The vertical scale should start at 0, increase in equal steps, and carry a label such as “Number of pupils”. Bars for categories are drawn with gaps between them because the categories are separate things, not a continuous scale.

A pie chart splits one whole into sectors. Because the full circle is 360°, each sector’s angle is its fraction of the total × 360°. The angles must always add up to 360° — that is the fastest way to check your construction.

Reading a pie chart reverses the idea: a sector of 126° out of 360° represents 126/360 = 7/20 of the total. For a pie chart of 40 members, the 126° sector covers 7/20 × 40 = 14 members. The pie chart shows shares of one whole — it cannot tell you how big each whole is.

A line graph plots points and joins them to show how something changes through time — temperature through a day, water level through a week, distance through a journey. Steeper means faster change; a flat stretch means no change at all.

A histogram looks like a bar graph but represents grouped continuous data — heights, masses, times — sorted into equal class intervals such as 120 ≤ h < 130. Because the classes share boundaries (one ends where the next begins), the bars are drawn with no gaps.

A line graph is wrong for categories: joining “vanilla” to “chocolate” to “strawberry” suggests a trend between things that are not on a scale. Likewise a pie chart is useless when the parts do not make a natural whole.

Graphs can also mislead. The classic tricks: a vertical axis that starts above zero, so bars differ by 10% but look like they differ by 500%; uneven steps on a scale; and pictogram symbols that grow instead of multiply. A careful reader always asks where does the scale start? before comparing bars.