1. Chapter at a Glance
A cricketer's batting average, the average rainfall of a month, the "typical" family size in a village — all these are ways of summarising a large collection of numbers by one representative value. In this chapter you will "connect the dots" in data: organise observations, draw dot plots and double bar graphs, and find the mean, median, mode and range. You will also learn which summary is most suitable in which situation.
- Organising raw data; frequency tables and dot plots.
- The arithmetic mean (average) and how to calculate it.
- The median (middle value) for odd and even numbers of observations.
- The mode (most frequent value) and the range (spread).
- Double bar graphs for comparing two sets of data.
- Choosing the best average and spotting misleading summaries.
2. Organising Data
Raw data are the numbers as collected, for example the heights (cm) of 9 students: 142, 150, 138, 145, 150, 147, 139, 150, 144. To understand them we arrange them in order and count how often each value appears.
Arranged: 138, 139, 142, 144, 145, 147, 150, 150, 150.
Dot plots
A dot plot places a dot above a number line for each observation. Repeated values stack up, so clusters, gaps and the most common value are visible at a glance.
3. The Arithmetic Mean
The mean (average) shares the total equally among all observations:
For the heights: sum = 138 + 139 + 142 + 144 + 145 + 147 + 150 + 150 + 150 = 1305, and there are 9 observations. Mean = 1305 ÷ 9 = 145 cm. (Always add twice — a small slip changes the mean.)
- The mean lies between the smallest and the largest value.
- If every value increases by k, the mean increases by k.
- The sum of the differences of values from the mean is always 0 — the mean is the "balance point".
A batter scored 45, 12, 78, 0 and 65 runs in five innings. Find the batting average (mean).
Solution: Sum = 200; mean = 200 ÷ 5 = 40 runs.
4. The Median
The median is the middle value when the data are arranged in order.
- If the number of observations n is odd, the median is the ((n + 1)/2)th value.
- If n is even, the median is the mean of the two middle values (the (n/2)th and (n/2 + 1)th).
Find the median of 12, 7, 19, 4, 15, 9.
Solution: Ordered: 4, 7, 9, 12, 15, 19 (n = 6, even). Middle values: 9 and 12. Median = (9 + 12) ÷ 2 = 10.5.
5. Mode and Range
- The mode is the value that occurs most often. Data may have one mode, more than one mode, or no mode.
- The range = largest value − smallest value. It shows how spread out the data are.
For the heights: mode = 150 cm, range = 150 − 138 = 12 cm.
- Mean: uses every value; best when there are no extreme values.
- Median: not affected by a few very large or very small values (good for incomes, house prices).
- Mode: best for "most popular" questions (shoe size, favourite colour) — works even for non-numerical data.
Monthly incomes (₹ thousand) of 6 households: 18, 20, 22, 24, 26, 300. Compare mean and median.
Solution: Mean = 410 ÷ 6 ≈ 68.3 thousand; median = (22 + 24) ÷ 2 = 23 thousand. The single rich household pulls the mean far up; the median describes a typical household much better.
6. Double Bar Graphs
A double bar graph shows two sets of data side by side for each category, using two colours and a key. It is ideal for comparisons — for example, marks of a class in two tests, or boys and girls in different sports.
| Subject | Test 1 average | Test 2 average | Change |
|---|---|---|---|
| Maths | 62 | 71 | +9 |
| Science | 68 | 66 | −2 |
| English | 74 | 79 | +5 |
(Practice data.) Reading the pairs shows that Maths improved most and Science fell slightly.
7. Solved Problems (NCERT Style)
The mean of 5 numbers is 18. Four of them are 12, 20, 15 and 25. Find the fifth.
Solution: Total = 5 × 18 = 90. Sum of four = 72. Fifth = 18.
The mean age of 10 players is 16 years. A coach aged 38 joins them. Find the new mean.
Solution: Old total = 160. New total = 198 for 11 people. New mean = 198 ÷ 11 = 18 years.
Find the mean, median and mode of 3, 5, 5, 6, 8, 9, 13.
Solution: Mean = 49 ÷ 7 = 7. Median = 4th value = 6. Mode = 5. Range = 13 − 3 = 10.
8. Master Formula Table
| Measure | Formula / rule |
|---|---|
| Mean | sum ÷ number of observations |
| Total from mean | mean × number of observations |
| Median (n odd) | ((n + 1)/2)th value in order |
| Median (n even) | mean of (n/2)th and (n/2 + 1)th values |
| Mode | most frequent value |
| Range | largest − smallest |
| Shift all values by k | mean, median, mode shift by k; range unchanged |
9. Master Summary Table
| Tool | Shows | Best for |
|---|---|---|
| Dot plot | Each value, clusters | Small data sets |
| Double bar graph | Two sets side by side | Comparisons |
| Mean | Equal share | Data without extremes |
| Median | Middle value | Data with extremes |
| Mode | Most common | Popularity, sizes |
| Range | Spread | Consistency |
10. Mind Map
- Order data
- Frequency
- Dot plots
- Mean
- Median
- Mode
- Range
- Extreme values
- Double bar graphs
- Choosing averages
11. Tips and Tricks
- Always arrange data in order before finding the median.
- To check a mean, see if the differences from it add up to zero.
- Use "assumed mean" for big numbers: for 98, 102, 105, 95, subtract 100 → −2, 2, 5, −5; mean of these = 0, so mean = 100.
- Taking the middle of unordered data as the median.
- Forgetting zero values (a score of 0 is still an observation).
- Saying the mean must be one of the data values — it need not be.
12. Important One-liners
- Mean = sum of observations ÷ number of observations.
- The median is the middle value of ordered data.
- The mode is the most frequent value.
- Range = largest − smallest value.
- Extreme values affect the mean much more than the median.
- Data can have more than one mode.
- A double bar graph compares two related data sets.
- The sum of deviations from the mean is zero.
13. Exam-pattern Questions with Detailed Answers
Find the mean of 4, 8, 6, 10, 12.
- 6
- 8
- 10
- 40
Show answer
Sum = 40; 40 ÷ 5 = 8.
Answer: (B) 8
Find the mode of 2, 3, 3, 5, 7, 3, 5.
Show answer
3 occurs three times, more than any other value.
Answer: 3
Find the range of 23, 41, 17, 35, 29.
Show answer
41 − 17 = 24.
Answer: 24
Find the median of 31, 27, 40, 35, 29, 33, 38, 26.
Show answer
Ordered: 26, 27, 29, 31, 33, 35, 38, 40. Middle values 31 and 33. Median = 32.
Answer: 32
The mean of 6 numbers is 15. If one number, 20, is removed, what is the new mean?
Show answer
Total = 90; after removal 70 for 5 numbers; mean = 14.
Answer: 14
Which average best describes the typical shoe size sold in a shop?
- Mean
- Median
- Mode
- Range
Show answer
The shop wants the most commonly sold size — the mode.
Answer: (C) Mode
The mean of x, x + 2, x + 4, x + 6 and x + 8 is 13. Find x.
Show answer
Sum = 5x + 20; mean = x + 4 = 13, so x = 9.
Answer: x = 9
Every value in a data set is increased by 5. What happens to the mean and to the range?
Show answer
The mean increases by 5; the range does not change (largest and smallest both rise by 5).
Answer: Mean +5; range unchanged
The mean of 8 observations is 25. Later it was found that 42 was wrongly recorded as 24. Find the correct mean.
Show answer
Wrong total = 200. Correct total = 200 − 24 + 42 = 218. Correct mean = 218 ÷ 8 = 27.25.
Answer: 27.25
Write a set of 5 whole numbers with mean 10, median 9 and mode 8.
Show answer
Sum must be 50. Middle (3rd) value 9; mode 8 → at least two 8s before it: 8, 8, 9, a, b with a + b = 25 and a, b ≥ 9, distinct from 8. Choose 10 and 15.
Set: 8, 8, 9, 10, 15 → mean 50/5 = 10 ✔, median 9 ✔, mode 8 ✔.
Answer: For example 8, 8, 9, 10, 15
A student's average in 4 tests is 72. What must she score in the 5th test to raise her average to 75?
Show answer
Current total = 288. Needed total = 5 × 75 = 375. Required score = 375 − 288 = 87.
Answer: 87
Two batters each have a mean of 40 runs in 5 innings. Batter A: 38, 42, 40, 39, 41. Batter B: 5, 90, 12, 85, 8. Who is more consistent? Use range.
Show answer
Range A = 42 − 38 = 4; range B = 90 − 5 = 85. A's scores vary much less.
Answer: Batter A is more consistent
14. Quick Revision Summary
- Organise data in order; dot plots show clusters and the mode.
- Mean = sum ÷ count; median = middle of ordered data; mode = most frequent; range = max − min.
- Median resists extreme values; mode suits "most popular".
- Double bar graphs compare two data sets with a key.
- Use totals (mean × count) to solve missing-value problems.