Big picture
Data are observations collected to answer a question. Data handling means gathering, organising, representing and interpreting those observations. It also introduces probability, a numerical way of describing chance when possible outcomes can be identified.
A class might record how pupils travel to school, measure plant heights or compare borrowing across months. Different questions need different representations. A pie chart describes parts of a whole, a histogram groups numerical observations and a line graph shows change across ordered values. A good display helps the reader understand rather than exaggerating a preferred conclusion.
Learning goals
Organise raw observations with tally marks and frequency tables.
Form equal-width class intervals and interpret a histogram.
Select and interpret bar, double-bar, line and pie charts.
Calculate pie-chart angles, fractions and percentages.
Identify outcomes, sample spaces and events in simple experiments.
Calculate probabilities for equally likely outcomes and explain experimental variation.
Core ideas
1. Collect purposeful data
A raw data set records observations before organisation. Decide what each observation represents and use consistent units. A height list mixing centimetres with metres cannot be compared without conversion. A travel survey needs a clear rule, such as recording the main mode used on the surveyed day, so that one pupil is not accidentally counted twice.
A frequency is the number of times a value or category appears. Tally marks help keep count; groups of five are easier to check than long ungrouped runs. For accessibility in this lesson, write a frequency table with numbers instead of drawing tally strokes.
Worked example. The numbers of borrowed books for ten pupils are 1, 2, 0, 1, 3, 2, 1, 0, 2, 1. Count each value:
Books | Frequency |
|---|---|
0 | 2 |
1 | 4 |
2 | 3 |
3 | 1 |
Total | 10 |
Check: 2 + 4 + 3 + 1 = 10 pupils. The most frequent value, or mode, is 1 book, not the largest value 3.
2. Summaries and what they hide
The range is highest minus lowest. The mean is total of observations divided by their number. The median is the middle value after sorting; for an even count, average the two middle values. These familiar summaries support interpretation but do not replace the distribution.
Worked example using the book data. Total books = 0 × 2 + 1 × 4 + 2 × 3 + 3 × 1 = 13. Mean = 13 ÷ 10 = 1.3 books per pupil. Ordered data are 0, 0, 1, 1, 1, 1, 2, 2, 2, 3. The fifth and sixth values are both 1, so median = (1 + 1) ÷ 2 = 1. Range = 3 − 0 = 3 books.
The mean need not be an observed whole-number count. It describes the average across pupils. A claim that every pupil borrowed 1.3 books would confuse an average with individual observations.
3. Grouped data and interval conventions
For many different numerical values, use class intervals. With the convention 10–20 meaning 10 ≤ value < 20, the lower endpoint is included and the upper endpoint goes into the next interval. Thus 20 belongs to 20–30, not 10–20. State the convention to avoid overlapping classes.
Worked example. Plant heights in centimetres are 12, 18, 21, 26, 29, 32, 35, 38, 42, 47, 49, 53. Equal class width = 20 − 10 = 10 cm.
Height interval, cm | Frequency |
|---|---|
10–20 | 2 |
20–30 | 3 |
30–40 | 3 |
40–50 | 3 |
50–60 | 1 |
The frequencies total 12 plants. Grouping makes the pattern easier to see but hides exact individual heights. You can identify a most frequent interval, but cannot reconstruct every measurement from the grouped table alone.
4. Bar graphs and double-bar graphs
A bar graph represents categories using bars of equal width and a common numerical scale. Gaps separate the categories because they are distinct. Label both axes and give a title. If each grid step represents 5 pupils, a bar reaching the fourth step represents 20 pupils.
A double-bar graph compares two related data sets across the same categories, using paired bars and a key. Suppose a club lends 18 English and 12 Hindi books in one week, then 24 English and 20 Hindi books in the next. Totals are 18 + 12 = 30 and 24 + 20 = 44; the increase is 14 books. Compare corresponding categories, not merely whichever two bars are nearest.
The numerical axis should normally begin at zero for a straightforward bar-length comparison. A truncated axis can exaggerate differences; if a break is unavoidable, it must be clearly marked and interpreted cautiously.
5. Histograms and line graphs
A histogram shows grouped numerical data. For the equal-width classes used here, adjacent rectangles touch and their heights represent frequencies. Draw the plant-height intervals on the horizontal axis and frequency on the vertical axis. Rectangles over 20–30, 30–40 and 40–50 all reach height 3.
Unlike a bar graph, the horizontal scale represents contiguous numerical intervals. The bars touch even if no observation equals a boundary. This introductory method assumes equal class widths; unequal-width histograms require a different treatment beyond this lesson.
A line graph joins plotted values in their meaningful order, often time. For library visits of 10, 15 and 12 over three consecutive days, plot day versus visits and join successive points. Visits rise by 5, then fall by 3. Connecting unordered categories such as favourite fruits would suggest a continuity that the categories do not have.
6. Pie charts as parts of a whole
A pie chart divides a circle into sectors. The complete circle is 360°, representing the whole data set. Sector angle = category frequency ÷ total × 360°. Conversely, frequency = sector angle ÷ 360° × total.
Worked example. Forty pupils choose one preferred club each.
Club | Pupils | Fraction | Angle |
|---|---|---|---|
Reading | 12 | 12/40 | 108° |
Games | 16 | 16/40 | 144° |
Art | 8 | 8/40 | 72° |
Music | 4 | 4/40 | 36° |
For reading, angle = 12 ÷ 40 × 360° = 108°. Its percentage is 12 ÷ 40 × 100 = 30%. Check total angle: 108 + 144 + 72 + 36 = 360°. Games occupies 40% because 16 ÷ 40 × 100 = 40%. A 72° sector represents 72 ÷ 360 × 40 = 8 pupils.
Use a pie chart only when categories form a non-overlapping whole. If pupils can choose several clubs, their response counts cannot automatically be treated as shares of the pupil total.
7. Chance experiments, outcomes and events
A random experiment has possible outcomes, but its particular result is uncertain beforehand. Tossing a coin is an example. An outcome is one possible result; a sample space lists all outcomes; an event is a collection of outcomes meeting a condition.
For a standard die, sample space = {1, 2, 3, 4, 5, 6}. The event “even number” contains {2, 4, 6}. A fair die makes the six outcomes equally likely. Fairness is an assumption, not a consequence of simply listing outcomes.
A spinner divided into differently sized sectors does not make each labelled sector equally likely. Count sectors only when they are equal in area and the spinner mechanism is assumed fair. Always identify what the elementary outcomes actually are.
8. Calculate theoretical probability
For equally likely outcomes, P(event) = favourable outcomes ÷ total outcomes. Probability lies between 0 and 1. An impossible event has probability 0; a certain event has probability 1.
Worked example. On a fair die, prime outcomes are 2, 3 and 5. Probability = 3 ÷ 6 = 1/2. The number 1 is not prime. An outcome greater than 6 is impossible, so its probability is 0.
Worked example. A bag holds 5 red, 3 blue and 2 green identical-sized counters, mixed thoroughly. There are 5 + 3 + 2 = 10 counters. P(blue) = 3/10. P(not blue) = (5 + 2) ÷ 10 = 7/10, also 1 − 3/10. Colours are not equally likely categories: choosing among three colour names does not give probability 1/3 for each.
9. Experimental frequency and two-stage outcomes
An experimental probability is observed frequency divided by trial count. If a coin shows heads 18 times in 40 tosses, the experimental value is 18 ÷ 40 = 9/20 = 0.45. A fair coin's theoretical probability is 1/2, but a short trial need not give exactly half heads. Previous tosses do not make the next fair toss “owe” a particular result.
For two fair coins, list ordered outcomes HH, HT, TH and TT. Exactly one head occurs in HT or TH, so probability = 2 ÷ 4 = 1/2. At least one head occurs in HH, HT or TH, so probability = 3/4. The events “exactly one” and “at least one” are different.
Listing outcomes prevents the mistaken claim that zero, one and two heads are equally likely. One head corresponds to two elementary outcomes, whereas zero and two each correspond to only one.
A reliable approach to questions
Identify what was observed and whether categories overlap.
Organise counts and verify the grand total.
Choose a graph appropriate to categories, intervals, time or shares.
Read labels, scales, units and interval boundaries before calculating.
For chance, list equally likely outcomes before selecting favourable ones.
Check angle totals, probability bounds and whether the conclusion exceeds the data.
Connections that matter
Fractions and percentages convert counts into shares and probabilities.
Introduction to graphs develops scales, coordinates and time-based displays.
Science uses repeated observations while distinguishing measurements from conclusions.
Common mistakes
Putting an interval boundary in two classes: apply the stated endpoint convention.
Drawing gaps in an equal-width histogram: adjacent numerical classes touch.
Using 100° as a pie-chart whole: a full circle is 360°.
Treating colour names or head-count categories as equally likely: list the actual elementary outcomes.
Revision checklist
I can organise observations and check frequencies.
I can interpret simple statistical summaries.
I can distinguish bar graphs, histograms, line graphs and pie charts.
I can calculate sector angles and percentages.
I can list outcomes and compute simple probabilities.
I can explain why experimental results may differ from theoretical probability.