In this guide
Ask a class to draw a bar graph of their favourite fruits and some bars will be wide and some narrow, some will start above the line, and the numbers on the side will jump from 2 to 5 to 10. The children have seen graphs; they have not yet understood that a graph is a fair picture of numbers. Data handling in CTET tests both halves of this: reading and calculating from graphs, and knowing how children should learn to collect and represent data.
Paper I expects tally marks, pictographs and simple bar graphs, at the NCERT Class 1–5 level. Paper II adds frequency tables, double bar graphs, pie charts, and the mean, median, mode and range of ungrouped data. This guide covers both.
The data-handling cycle
Data work in school should follow the same cycle a statistician uses:
- Pose a question children care about: How do we come to school? Which month has the most birthdays?
- Collect the data by asking, counting or measuring.
- Organise it with tally marks and a frequency table.
- Represent it as a pictograph, bar graph or pie chart.
- Interpret it: What does the graph tell us? What surprised us? What new question does it raise?
Most textbook exercises start at step 4 with ready-made data. The pedagogy questions reward the teacher who starts at step 1 with the children's own data.
Organising data
Tally marks are made in groups of five: four upright strokes and a fifth across them. Seventeen is three full groups and two strokes. Grouping in fives makes counting quick and fits naturally with skip counting.
A frequency table lists each value or category with how often it occurs. The frequencies add up to the total number of observations, which is a quick check.
Representing data
| Form | Best for | Must have |
|---|---|---|
| Pictograph | Young children; small counts | A key (one picture = how many), pictures of the same size |
| Bar graph | Comparing categories | Bars of equal width, equal gaps, a scale starting at 0, labelled axes |
| Double bar graph | Comparing two sets, such as boys and girls, or two years | A legend showing which bar is which |
| Pie chart | Parts of a whole | Angles that add to 360° |
Pictograph. If one picture stands for 10 children, 4½ pictures show 45 children. Without the key, the pictograph means nothing.
Pie chart. Central angle = (value ÷ total) × 360°. If 60 of 240 students choose cricket, the cricket sector is 60/240 × 360 = 90°.
Worked example (reading a bar graph). A bar graph shows how 40 Class 5 children come to school: walk 12, cycle 8, bus 15, auto 5.
- How many more come by bus than by cycle? 15 − 8 = 7.
- What fraction walk? 12/40 = 3/10.
- In a pie chart of the same data, the bus sector would be 15/40 × 360 = 135°.
Mean, median, mode and range
| Measure | How to find it | Notes |
|---|---|---|
| Mean | Sum of values ÷ number of values | Affected by very large or small values |
| Median | Middle value after arranging in order; for an even count, the mean of the two middle values | Not affected much by extreme values |
| Mode | The value that occurs most often | There can be more than one mode |
| Range | Highest value − lowest value | A measure of spread, not of average |
Worked example 1. Data: 3, 5, 5, 6, 11.
- Mean = 30 ÷ 5 = 6.
- Median = the third value = 5.
- Mode = 5.
- Range = 11 − 3 = 8.
Worked example 2 (even count). Scores: 12, 15, 18, 20, 22, 27. Median = (18 + 20) ÷ 2 = 19. Sum = 114, so mean = 114 ÷ 6 = 19.
Worked example 3 (missing value). The mean of five numbers is 20. Four of them are 18, 22, 15 and 21. The five numbers add to 5 × 20 = 100, and the four add to 76, so the fifth is 24.
Worked example 4 (effect of a change). If 5 is added to every value, the mean, median and mode all rise by 5, and the range does not change.
Why the median matters. Monthly incomes in a small group: ₹10,000, ₹12,000, ₹12,000, ₹14,000 and ₹2,00,000. The mean is ₹2,48,000 ÷ 5 = ₹49,600, which describes nobody. The median, ₹12,000, is a better picture of a typical income.
Paper II may also touch on chance: when a fair coin is tossed, heads and tails are equally likely, so the probability of heads is 1/2; when a die is rolled, the probability of a 4 is 1/6.
Children's errors with data
| Error | Example | Teaching response |
|---|---|---|
| No key, or pictures of different sizes in a pictograph | Big pictures look like more | Make pictographs with identical stickers and agree on a key first |
| Bars of different widths, or a scale that does not start at 0 | A small difference looks huge | Compare two graphs of the same data; ask which one is fair |
| Uneven scale | Axis marked 0, 2, 5, 10 | Mark scales with equal steps on squared paper |
| Median found without ordering | Median of 9, 2, 7, 4, 3 given as 7 (the middle as written) instead of 4 | Always arrange first: 2, 3, 4, 7, 9 |
| Mode taken as the largest value | Mode of 2, 3, 3, 9 said to be 9 | Stress "most often", using tallies |
| Mean must be one of the values | Mean of 1, 2, 4 "can't be 7/3" | Share cubes equally among towers to see mean as levelling out |
| Confusing a value with its frequency | Reading the bar height as the category | Label axes in words: "number of children" |
Teaching data handling
- Use the class itself: birthdays by month, ways of coming to school, favourite games, heights.
- Human and block graphs: children stand in lines by category, or stack one block each, before drawing anything.
- Talk about the graph: What is the most common? The least? What would change if new children joined?
- Compare representations: the same data as a tally, a pictograph and a bar graph. Which is easiest to read, and why?
- Mean as fair share: towers of cubes of heights 2, 5 and 5 are levelled to 4 each; the mean of 2, 5 and 5 is 12 ÷ 3 = 4.
- Link to other subjects: rainfall and temperature in EVS or science.
See maths teaching aids for classroom materials.
Practice set
- In a pictograph, one picture stands for 5 children. How many children do 7 pictures show?
- Find the median of 8, 2, 6, 4, 10.
- Find the mode of 2, 3, 3, 4, 4, 4, 5.
- Find the mean of the first five natural numbers.
- Find the median of 7, 3, 9, 5, 11, 1.
- In a class of 36, 9 children like kabaddi. What is the angle of the kabaddi sector in a pie chart?
- The mean of 4 numbers is 15. When a fifth number is added, the mean becomes 16. What is the fifth number?
- A Class 4 teacher wants children to learn bar graphs. Which is best: (a) copy a bar graph from the textbook (b) collect data on how classmates come to school, tally it and draw a graph (c) memorise the parts of a graph (d) watch the teacher draw a graph?
Answers
- 35.
- Ordered: 2, 4, 6, 8, 10, so 6.
- 4.
- (1 + 2 + 3 + 4 + 5) ÷ 5 = 3.
- Ordered: 1, 3, 5, 7, 9, 11. Median = (5 + 7) ÷ 2 = 6.
- 9/36 × 360 = 90°.
- 5 × 16 − 4 × 15 = 80 − 60 = 20.
- (b). It follows the full data-handling cycle with data that matters to the children.
What to do next
- Write the "must have" rules for pictographs, bar graphs and pie charts from memory.
- Solve ten mean, median and mode questions, including even counts and missing values.
- Plan one data activity for a Class 3 class and one for Class 7, starting from a question.
- Continue with patterns.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Central Board of Secondary Education website .
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