- Mean equals the total of observations divided by their number.
- The mean balances positive and negative deviations.
- Adding a value above the current mean increases the mean.
- Adding k to every value increases the mean by k.
- Median is the central value after sorting and is less sensitive to outliers.
- Frequency tables require weighted totals for mean.
- Spreadsheets use formulas such as =SUM and =AVERAGE.
- Read axes, units, scales, legends and sources before interpreting line graphs.
Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.
Tales by Dots and Lines — NCERT Class 8 Mathematics Chapter 12
Direct answer: Tales by Dots and Lines (Ganita Prakash Part II, Chapter 5) explains the mean as a balance point, how mean and median respond to changing data, missing values, frequency tables, spreadsheet formulas, line graphs, infographics and responsible data interpretation.
Why can two groups have the same average but very different experiences? Why might one extremely large value change the mean but hardly affect the median? Data stories become more trustworthy when we understand both their numbers and their pictures.
Textbook scope: Sections 5.1 The Balancing Act and 5.2 Visualising and Interpreting Data, printed pages 103–133. The worked problems below distinguish identifiable textbook prompts from original additional practice; they do not claim to reproduce every official exercise.
5.1 The Balancing Act: mean as the balance point
Arithmetic mean = sum of observations divided by their count. For 2, 4, 9, mean=(2+4+9)/3=5. Distances below 5 are 3 and 1, totaling 4; distance above is 4. This balancing property holds for any finite numerical data set. Unlike the median, the mean need not be an observed value.
What happens when data are added or removed?
If n values have mean m and a new value x is added, new mean=(nm+x)/(n+1). It increases if x>m, decreases if x<m, and remains unchanged if x=m. Removing an observation x gives new mean=(nm−x)/(n−1), for n>1. Removing a value above the old mean decreases the mean; removing one below increases it.
Unchanging mean and relative transformations
Adding k to every observation adds k to the mean. Multiplying every observation by k multiplies the mean by k. To add three values while preserving mean 10, choose 11,12,7 because their sum is 30. A common mistake is thinking adding a single value greater than the mean leaves it unchanged.
Median and changing data
Sort the values first. For odd n, the median is the middle observation; for even n, it is the average of the two middle observations. Example: 2,3,9,10,50 has median 9 but mean 14.8. Adding or removing a value can change which positions are central; unlike the mean, the median is resistant to an extreme outlier.
Find missing values and correct an error
If n values have mean m, their total is nm. A missing value equals nm minus the sum of the known values. If an entry was wrongly recorded as w instead of c, corrected total=old total−w+c, so corrected mean=(old total−w+c)/n.
Mean and median with frequencies
For a frequency table with values xᵢ and frequencies fᵢ, mean=Σ(fᵢxᵢ)/Σfᵢ. For median, use cumulative frequencies to locate the middle position(s), not the middle distinct value. Example: value 2 occurs twice, 4 occurs three times, 9 occurs once. Total=6; sum=2×2+4×3+9=25; mean=25/6; ordered values 2,2,4,4,4,9 have median 4.
Spreadsheets: cells, ranges and formulas
A spreadsheet cell is identified by column letter and row number, e.g. B3. A range B3:G3 includes six cells. Use =SUM(B3:G3) to total them and =AVERAGE(B3:G3) to calculate their mean. Spreadsheet formulas begin with =. AVERAGE generally ignores blank cells but includes numerical zeros, so inspect missing entries before interpreting results.
5.2 Visualising and Interpreting Data
A line graph plots quantities against an ordered horizontal variable, often time, and connects the points. Always check the axis labels, units, scale, legend and source. Steeper segments indicate a larger rate of change only when horizontal intervals and axis scaling are comparable. A line graph of monthly temperature describes observed changes; it does not alone prove why they occurred.
Interpreting infographics and activity strips
An infographic combines numbers, icons and concise text. A strip chart can show time spent sleeping, studying and playing. Check whether category totals match the full day (24 hours) and whether percentages sum to 100% when categories are exhaustive and nonoverlapping. Different visual scales or selective start points can exaggerate differences.
Critical data literacy and misleading conclusions
Ask who collected the data, when, how large the sample is, whether the values are averages, and whether categories are comparable. A claim about all children cannot be inferred from an average alone. Correlation between two plotted trends does not establish causation. A graph of electricity use cannot establish that there were no power cuts.
Formula and interpretation table
| Topic | Rule | Example |
|---|---|---|
| Arithmetic mean | Sum / number of values | (6+7+8)/3=7 |
| Mean after adding x | (nm+x)/(n+1) | n=4,m=12,x=18→13.2 |
| Mean after removing x | (nm−x)/(n−1) | n=5,m=10,x=14→9 |
| Missing observation | nm−sum of known | 4 values, mean 15, known sum 43→17 |
| Frequency mean | Σfx / Σf | 2 twice, 4 thrice→3.2 |
| Even-count median | Mean of two central values | 2,3,5,9→4 |
| Spreadsheet mean | =AVERAGE(B3:G3) | Average of numeric cells in range |
Textbook-style activities and learning checks
Draw dots above values 2,4,9 on a number line and demonstrate balancing around 5. Survey five daily temperatures and calculate both mean and median. In a spreadsheet, enter six observations, use SUM and AVERAGE, and compare to hand calculations. Draw a line graph of actual daily readings and explain the highest, lowest and largest consecutive change. Never invent observations for a real-world survey.
Fully worked questions: textbook-aligned examples and original practice
Each question is written in full, individually numbered, and followed by a calculation and final answer. Some prompts follow identifiable textbook examples; other questions are original supplementary practice, not official NCERT Figure it Out questions.
Question 1. Find the mean of the first 50 natural numbers.
Step-by-step solution: The first and last values are 1 and 50. Pair from the ends: each pair sums to 51, and there are 25 pairs. Sum=25×51=1275. Mean=1275/50.
Final answer: 25.5.
Question 2. Find the mean of the first 50 odd natural numbers.
Step-by-step solution: The sequence is 1,3,…,99. Pair first and last: sum 100; 25 pairs give total 2500. Divide by 50.
Final answer: 50.
Question 3. Find the mean of the first 50 positive multiples of 4.
Step-by-step solution: The sequence is 4,8,…,200. First plus last=204, so mean of the arithmetic progression=(4+200)/2.
Final answer: 102.
Question 4. Find the mean and median of 6, 7 and 8.
Step-by-step solution: Sum=6+7+8=21; mean=21/3=7. Sorted data have middle value 7.
Final answer: Mean 7; median 7.
Question 5. Explain why the mean of 2, 4 and 9 is a balance point.
Step-by-step solution: Mean=(2+4+9)/3=5. Below the mean, deviations are 5−2=3 and 5−4=1. Above the mean, 9−5=4. The distances balance.
Final answer: Mean 5; total deviation below and above is 4 each.
Question 6. A data set has four values with mean 12. Add 18. Find the new mean.
Step-by-step solution: Old sum=4×12=48. New sum=48+18=66; new count=5. New mean=66/5.
Final answer: 13.2.
Question 7. Five observations have mean 10. Remove an observation of 14. Find the new mean.
Step-by-step solution: Old total=5×10=50. Remaining total=50−14=36. Remaining count=4; mean=36/4.
Final answer: 9.
Question 8. A set has mean 10. Choose three new observations, two above 10 and one below, without changing the mean.
Step-by-step solution: Three new values must total 3×10=30. Choose 11 and 12 (both above 10); the remaining value is 30−23=7.
Final answer: 11, 12 and 7.
Question 9. Eleven values are 8, 3, 10, 13, 4, 6, 7, 7, 8, 8 and 5. Find the mean, then add 10 to every value.
Step-by-step solution: Total=79, n=11, mean=79/11≈7.1818. Adding 10 to each adds 110 to the sum; new mean=(79+110)/11=189/11.
Final answer: Original mean 79/11≈7.18; new mean 189/11≈17.18.
Question 10. The mean of 12, 14, x and 17 is 15. Find x.
Step-by-step solution: Required total=4×15=60. Known sum=12+14+17=43. Subtract 43 from 60.
Final answer: x=17.
Question 11. Six values have mean 15. One entry 28 should have been 18. Find the corrected mean.
Step-by-step solution: Recorded total=6×15=90. Correct total=90−28+18=80. Divide by 6.
Final answer: 40/3≈13.33.
Question 12. Find mean and median from a frequency table: 2 occurs twice, 4 occurs three times and 9 occurs once.
Step-by-step solution: Total frequency=2+3+1=6. Weighted total=2×2+4×3+9×1=25; mean=25/6. Sorted observations are 2,2,4,4,4,9; the middle two are both 4.
Final answer: Mean 25/6≈4.17; median 4.
Question 13. Find mean and median of 2,3,9,10,50 and explain the difference.
Step-by-step solution: Sum=74; mean=74/5=14.8. Ordered middle observation is 9. The large value 50 raises the mean much more than the median.
Final answer: Mean 14.8; median 9.
Question 14. A spreadsheet contains observations in B3:G3. Write formulas for total and mean.
Step-by-step solution: SUM adds the range; AVERAGE divides the total of numerical entries by their count. Include the leading equals sign.
Final answer: =SUM(B3:G3) and =AVERAGE(B3:G3).
Question 15. A shop records visitors Mon–Sun as 16,19,10,14,20,22,35 and buyers as 10,8,7,11,12,16,26. Describe a suitable two-line graph.
Step-by-step solution: Put days on horizontal axis and people on vertical axis (0 to 40, steps of 5). Plot and connect both series separately. Sunday has most visitors (35) and buyers (26); Wednesday has fewest visitors (10) and buyers (7).
Final answer: Plot both labelled series; Sunday peak, Wednesday lowest.
Question 16. Using the same shop data, on which day was the fraction of visitors purchasing largest?
Step-by-step solution: Calculate buyers/visitors for each day: 10/16=.625, 8/19≈.421, 7/10=.7, 11/14≈.786, 12/20=.6, 16/22≈.727, 26/35≈.743. Compare.
Final answer: Thursday, 11/14≈78.6%.
Question 17. Monthly maximum temperatures Jan–Jun are 24,27,33,37,41,38°C. Find the largest rise and the first fall.
Step-by-step solution: Consecutive changes are +3,+6,+4,+4,−3°C. Largest increase is +6 from February to March. First decrease is from May to June.
Final answer: Largest rise February–March, 6°C; first fall May–June.
Question 18. Find the mean and median of 24,27,33,37,41,38.
Step-by-step solution: Sum=200; mean=200/6=33⅓. Sort 24,27,33,37,38,41; middle two are 33 and 37, average 35.
Final answer: Mean 33⅓; median 35.
Question 19. A daily activity strip totals 8 h sleep, 7 h school, 2 h homework, 3 h play and 4 h other activities. Check total and find sleep percentage.
Step-by-step solution: Total=8+7+2+3+4=24 hours. Sleep share=8/24=1/3, or 33⅓%.
Final answer: 24 hours; sleep 33⅓%.
Question 20. A graph shows that the percentage of households using kerosene for lighting declined. Can it prove that there were no power cuts?
Step-by-step solution: A decline in the percentage using kerosene as a primary lighting source does not measure frequency of electricity interruptions. The graph lacks power-cut data.
Final answer: No; the conclusion is unsupported.
Question 21. A data set of five values has mean 8. Each value is multiplied by 3 and then reduced by 2. Find the new mean.
Step-by-step solution: A linear transformation y=3x−2 transforms the mean the same way: new mean=3×8−2.
Final answer: 22.
Question 22. The mean of 4, 6, 10 and x is 8. Find x, then find the median.
Step-by-step solution: Required total=4×8=32. Known sum=20, so x=12. Sorted values 4,6,10,12; median=(6+10)/2.
Final answer: x=12; median 8.
Question 23. A class has 20 students with mean score 60 and another 30 with mean 70. Find the combined mean.
Step-by-step solution: Combined total marks=20×60+30×70=1200+2100=3300. Total students=50. Mean=3300/50.
Final answer: 66.
Question 24. Explain why a steep line graph segment does not always imply the same physical rate of change across two different graphs.
Step-by-step solution: Slopes depend on both horizontal and vertical scales and the units. A 5-unit rise over 1 day differs from a 5-unit rise over 1 year. Compare changes per unit of horizontal variable after normalising scales.
Final answer: Compare rates with consistent units and axis scales, not appearance alone.
Quick revision, MCQs and exam traps
MCQ: Which measure resists a single extreme outlier more strongly? (A) mean (B) median. Answer: B. HOTS: Adding a value equal to the existing mean leaves it unchanged. Competency: Before trusting a graph, read both axes and the data source. Flashcards: Mean=total/count; median=central sorted value; frequency mean=weighted total/frequency total; a line graph is especially useful for trends over ordered observations.
Official NCERT exercises and source transparency
Official source: Ganita Prakash Part II Chapter 5, Tales by Dots and Lines. The original PDF was too large for direct retrieval in this session. The textbook has three Figure it Out sets around pages 113–116, 122–123 and 127–132, with additional Math Talk and Try This tasks. The exact full question lists and diagram-specific answers have not been fully audited; do not mistake the original exercises on this page for a complete official NCERT answer key.
Frequently asked questions
What is Tales by Dots and Lines about?
Mean, median, dot plots, frequency tables, spreadsheets, line graphs, infographics and data interpretation.
What is the mean?
The sum of numerical observations divided by their count.
Why is mean called a balance point?
The total distance of observations below the mean equals the total distance above it.
What happens when a value above the mean is added?
The mean increases.
What happens when the same number is added to every observation?
The mean increases by that number.
What is the median?
The middle sorted observation, or the average of the two central observations for an even count.
How do you find an unknown observation from the mean?
Multiply the mean by the count and subtract the sum of known observations.
How do you find mean from a frequency table?
Divide the sum of value×frequency by the sum of frequencies.
What does =AVERAGE(B3:G3) mean?
Calculate the arithmetic mean of numerical entries in the spreadsheet range B3 through G3.
What is a line graph best used for?
Showing how values change across ordered observations, often over time.
Why can graphs be misleading?
Selective scales, missing context, biased samples and unsupported conclusions can distort interpretation.
Does this page reproduce all official NCERT exercises?
No. It includes textbook-aligned examples and original worked practice; full Figure it Out diagram-level verification is pending.
