1. Chapter at a Glance
Numbers tell stories. A list of marks, temperatures or daily sales is hard to understand at a glance, but put it on a dot plot or a line graph and patterns jump out: where values cluster, what is typical, what is unusual and how things change over time. This chapter teaches you to organise data, draw and read dot plots and line graphs, and summarise data with the mean, median and mode.
- Raw data, frequency tables and range.
- Dot plots — showing every value as a dot.
- Mean, median and mode, and when to use each.
- Line graphs for change over time; reading trends.
- Double line graphs for comparing two sets.
- Spotting misleading graphs.
Note: all data in this chapter is practice data made up for learning, not real measurements.
2. Organising Data
Raw data is data as it was collected, in no particular order. To make sense of it we sort it and count how often each value occurs (its frequency).
Practice data — number of books read last month by 15 students:
| Books | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|
| Frequency | 2 | 5 | 3 | 3 | 1 | 1 |
Range = highest − lowest = 7 − 2 = 5. The range tells us how spread out the data is.
3. Dot Plots
A dot plot places a dot above a number line for each data value. Stacks of dots show frequencies, and we can see clusters, gaps and outliers immediately.
Reading the plot: most students read 3–5 books; 3 is the most common value; 6 and 7 are less common.
4. Mean, Median and Mode
- Mean = sum of all values ÷ number of values.
- Median = the middle value when data is arranged in order. For an even count, the mean of the two middle values.
- Mode = the value that occurs most often.
For the books data (sorted: 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 7):
- Sum = 59, so mean = 59 ÷ 15 ≈ 3.93.
- 15 values → median is the 8th value = 4.
- Mode = 3 (5 times).
Find the median of 12, 7, 15, 9, 20, 11.
Solution: Sorted: 7, 9, 11, 12, 15, 20. Middle two are 11 and 12. Median = (11 + 12) ÷ 2 = 11.5.
Pocket money of 5 friends (₹): 50, 60, 60, 70, 500. Which average describes them best?
Solution: Mean = 740 ÷ 5 = ₹148, but four of the five get much less. Median = ₹60 better represents the group. An extreme value (an outlier) pulls the mean but barely affects the median.
| Measure | Best used when |
|---|---|
| Mean | Values are fairly even, no extreme outliers |
| Median | Data has outliers (e.g. incomes, house prices) |
| Mode | Finding the most popular choice (shoe size, favourite colour) |
5. Line Graphs
A line graph shows how a quantity changes over time. Points are plotted for each time and joined by straight lines. Time goes on the horizontal axis.
Practice data — height of a plant (cm) measured weekly:
| Week | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Height (cm) | 2 | 4 | 7 | 11 | 13 | 14 |
- Rising line → increase; falling line → decrease; flat line → no change.
- Steeper segment → faster change.
- Points between plotted times give only estimates (reading "in between" is called interpolation).
6. Double Line Graphs
Two line graphs on the same axes let us compare two sets of data over the same time, e.g. the practice-test scores of two students across five tests. Use a key (legend) to show which line is which. Where the lines cross, the two values are equal.
7. Misleading Graphs
Graphs can give a wrong impression if drawn carelessly or on purpose:
- Axis not starting at zero — small changes look huge.
- Uneven scale — equal steps on the axis do not represent equal amounts.
- Missing labels or units — the reader cannot tell what is measured.
Before believing a graph, read the axis labels, the units and where each axis starts.
8. Solved Problems (NCERT Style)
The mean of 5 numbers is 18. Four of them are 15, 20, 12 and 25. Find the fifth.
Solution: Total = 5 × 18 = 90. Known sum = 72. Fifth = 18.
Using the plant data, find the total growth from week 1 to week 4 and the average weekly growth in that period.
Solution: 13 − 4 = 9 cm over 3 weeks → 3 cm per week.
The mean weight of 10 students is 40 kg. A new student of 51 kg joins. Find the new mean.
Solution: New total = 400 + 51 = 451; new mean = 451 ÷ 11 = 41 kg.
9. Master Formula Table
| Measure | Formula / rule |
|---|---|
| Range | Highest − Lowest |
| Mean | Sum of values ÷ number of values |
| Mean from frequency table | Σ(value × frequency) ÷ Σ frequency |
| Median (n odd) | ((n + 1)/2)th value |
| Median (n even) | Mean of (n/2)th and (n/2 + 1)th values |
| Mode | Most frequent value |
| Rate of change on a line graph | Change in value ÷ change in time |
10. Master Summary Table
| Tool | Shows | Exam focus |
|---|---|---|
| Frequency table | How often each value occurs | Build from raw data |
| Dot plot | Every value, clusters, outliers | Read mode and spread |
| Mean | Fair-share average | Missing value problems |
| Median | Middle value | Data with outliers |
| Line graph | Change over time | Trends, steepest rise |
| Double line graph | Comparison over time | Where lines cross |
11. Mind Map
- Frequency table
- Range
- Dot plots
- Clusters, outliers
- Mean, median, mode
- Outlier effect
- Trends over time
- Misleading graphs
12. Tips and Tricks
- Always sort the data before finding the median.
- For "missing value" questions, work with totals: total = mean × count.
- On a line graph, the steepest segment answers "when was the change fastest?"
- Taking the middle of unsorted data as the median.
- Forgetting that data can have more than one mode — or none.
- Joining points on a line graph for data that is not continuous over time.
13. Important One-liners
- Range measures spread; mean, median and mode measure the centre.
- An outlier changes the mean much more than the median.
- In a dot plot, the tallest stack is the mode.
- Line graphs are used for data that changes over time.
- The steeper the line, the faster the change.
- Total = mean × number of values.
14. Exam-pattern Questions with Detailed Answers
Find the range of 14, 9, 22, 17, 6, 19.
Show answer
22 − 6 = 16.
Answer: 16
Find the mode of 4, 7, 4, 9, 7, 4, 5.
Show answer
4 occurs 3 times — more than any other value.
Answer: 4
Which graph is best for showing a city's monthly rainfall over a year?
- Dot plot
- Line graph
- Pictograph of fruits
- Frequency table only
Show answer
Change over time is best shown by a line graph.
Answer: (B)
Find the median of 31, 25, 40, 28, 36, 22, 45, 30.
Show answer
Sorted: 22, 25, 28, 30, 31, 36, 40, 45. Middle two: 30 and 31. Median = 30.5.
Answer: 30.5
Find the mean from this frequency table: marks 5, 6, 7, 8 with frequencies 2, 4, 3, 1.
Show answer
Σ(value × f) = 10 + 24 + 21 + 8 = 63; Σf = 10. Mean = 6.3.
Answer: 6.3
The mean of 6 numbers is 20. If one number, 30, is removed, what is the new mean?
Show answer
Total = 120; new total = 90 for 5 numbers; mean = 18.
Answer: 18
Practice data: a shop's daily customers Mon–Fri are 40, 45, 38, 60, 52. Between which two consecutive days did the number rise most?
Show answer
Changes: +5, −7, +22, −8. Largest rise is Wed → Thu.
Answer: Wednesday to Thursday (+22)
A bar graph of two values, 98 and 102, has its vertical axis starting at 96. Why is it misleading?
Show answer
The bars look 2 and 6 units tall, so the second seems 3 times the first, though it is only about 4% more. The axis should start at 0 (or show a clear break).
Answer: The cut axis exaggerates a small difference
The mean of 8 numbers is 15. When two more numbers are added, the mean becomes 16. If one of the new numbers is 18, find the other.
Show answer
Old total 120; new total 160. The two new numbers sum to 40, so the other is 22.
Answer: 22
Write a set of 5 whole numbers with mean 10, median 8 and mode 6.
Show answer
Need sum 50, middle 8, and 6 appearing most. Try 6, 6, 8, x, y with x + y = 30 and both above 8 and different: 12 and 18. Set: 6, 6, 8, 12, 18. Check: sum 50 ✓, median 8 ✓, mode 6 ✓.
Answer: e.g. 6, 6, 8, 12, 18
A student's mean score in 4 tests is 72. What must they score in the 5th test to raise the mean to 75?
Show answer
Needed total = 375; current total = 288; 5th score = 87.
Answer: 87
Data: 3, 5, 5, 6, 8, 9, 40. Find the mean and median, and say which better represents the data and why.
Show answer
Sum = 76; mean ≈ 10.86. Median = 6. The value 40 is an outlier that drags the mean above six of the seven values. The median (6) is more representative.
Answer: Mean ≈ 10.86, median 6; median is better
15. Quick Revision Summary
- Organise raw data in a frequency table; range = highest − lowest.
- Dot plots show each value; the tallest stack is the mode.
- Mean = sum ÷ count; median = middle of sorted data; mode = most frequent.
- Outliers affect the mean more than the median.
- Line graphs show change over time; steeper means faster.
- Check axes and scales to avoid being misled.