Collecting data: tally and frequency
To know the class's favourite fruit, you ask everyone and record the answers with tally marks.
Data is information collected to answer a question. A tally mark counts one item (every fifth mark crosses the four before it). The frequency of a value is how many times it occurs, and a frequency table lists values with their frequencies.
In the figure: Three full groups of five and two bars make 17.
Collecting data: tally and frequency
In the app you can move this figure with your finger.
Open in the appRule 1: Tally to number — Count 5 for every crossed group, then add the extra bars.
- Easy: 4 groups and 3 bars = 20 + 3 = 23.
- Medium: 5 groups and 3 bars = 28.
- Exam-type: Frequency of 3 in 3, 5, 3, 2, 3, 5 is 3.
Rule 2: Reading a frequency table — Total of frequencies = number of observations; value × frequency summed = total of the data.
- Easy: Table 0:2, 1:5, 2:3 → 10 observations.
- Medium: Table 1:3, 2:4, 3:5 → sum = 3 + 8 + 15 = 26.
- Exam-type: 30 students: 12 cricket, 8 football, rest kabaddi → 10.
Rule 3: Good questions for data — Ask something that has different answers from person to person.
- Easy: "How many hours do you sleep?" gives data.
- Medium: "What is the height of our flagpole?" has one answer.
- Exam-type: Ask the whole group, or a fair sample.
Group tally marks in fives to count fast.
Counting a crossed group as 4. It is 5.
Cross the group on the fifth mark so it is easy to count later. Memory clue: "Four lines and a gate make five."
Spot the difference
| Term | Meaning |
|---|---|
| Tally | one count |
| Frequency | total count of a value |
Practice questions
- 1Tally: 4 groups of five and 3 extra bars. Total?
- 2Data 5, 2, 5, 7, 5, 2. Frequency of 5?
- 3Frequencies are 6, 9 and 5. Total observations?
- 4Table 1:2, 2:5, 3:4. Sum of all values?
- 5Table 0:3, 1:4, 2:4. Observations?
- 640 students: 15 bus, 12 walk, 5 cycle, rest car. Car?
- 7Frequencies 6, x, 8, 5 total 30. Find x.
- 8Tally: 6 groups of five and 4 bars. Total?
Show answers (try them yourself first)
- 123
- 23
- 320
- 423
- 511
- 68
- 711
- 834
Mean (average)
If three friends have ₹4, ₹6 and ₹8, sharing equally gives each ₹6: the mean is the equal share.
The mean = (sum of all values) ÷ (number of values). It is the value every item would have if the total were shared equally. So sum = mean × number.
In the figure: Bars 4, 6, 8 level out to 6 each.
Mean (average)
In the app you can move this figure with your finger.
Open in the appRule 1: Find the mean — Add everything, divide by how many.
- Easy: 4, 6, 8 → 18 ÷ 3 = 6.
- Medium: 10, 20, 30, 40 → 100 ÷ 4 = 25.
- Exam-type: 14, 18, 22, 26, 30 → 110 ÷ 5 = 22.
Rule 2: Use sum = mean × number — Find the total first when the mean is known.
- Easy: Mean 12 of 5 numbers → total 60.
- Medium: Mean of 12, 15, x, 21 is 17: total 68, so x = 20.
- Exam-type: Mean of 8 marks 62: total 496; 7 marks total 430 → eighth 66.
Rule 3: Adding or removing a value — Change the total, change the count, then divide again.
- Easy: Mean of 6 numbers 8, add 22: (48 + 22) ÷ 7 = 10.
- Medium: Mean of 5 numbers 20, remove 30: (100 – 30) ÷ 4 = 17.5.
- Exam-type: Mean of the first 5 whole numbers 0…4 is 2.
Mean always lies between the smallest and largest value.
Dividing by the wrong count. Count the values carefully.
Total = mean × count; use it backwards for a missing value. Memory clue: "Sum up, share out."
Spot the difference
| Known | Find |
|---|---|
| values | mean = sum ÷ n |
| mean and n | sum = mean × n |
Practice questions
- 1Mean of 3, 5, 7, 9, 11.
- 2Mean of 12 and 18.
- 3Mean of 8, 12, 16, 20.
- 4Five numbers have mean 9. Their total?
- 5Mean of 6 numbers is 7. Add 14. New mean?
- 6Mean of 9, x, 13, 15 is 12. Find x.
- 7Mean of 10 marks is 62; nine marks total 560. The tenth?
- 8Mean of the first 10 natural numbers?
Show answers (try them yourself first)
- 17
- 215
- 314
- 445
- 58
- 611
- 760
- 85.5
Median and mode
Line up students by height: the one in the middle is the median height. The most common shoe size in the class is the mode.
The median is the middle value of the sorted data (for an even count, the mean of the two middle values). The mode is the value that occurs most often; there can be no mode, one mode, or more than one.
In the figure: Six values: the two middle ones are 9 and 12, so the median is 10.5.
Median and mode
In the app you can move this figure with your finger.
Open in the appRule 1: Median — Sort first. Odd count: middle one. Even count: average the middle two.
- Easy: 7, 2, 9, 4, 5 → 2, 4, 5, 7, 9 → median 5.
- Medium: 4, 6, 8, 10 → (6 + 8) ÷ 2 = 7.
- Exam-type: 12, 5, 18, 7, 9, 14 → median 10.5.
Rule 2: Mode — Count how often each value appears.
- Easy: 2, 3, 3, 4, 5, 3 → mode 3.
- Medium: 10, 12, 12, 15, 15, 15, 18 → mode 15.
- Exam-type: 5, 8, 11, 14: all appear once → no mode.
Rule 3: Missing value from the median — Put the unknown in the middle position of the sorted list.
- Easy: 3, 7, x, 12, 15 with median 9 → x = 9.
- Medium: 4, 6, x, 10, 15 with median 8 → x = 8.
- Exam-type: Median of the first 10 natural numbers = 5.5.
Sort before finding the median.
Taking the middle of the unsorted list. Sorting comes first.
Cross out smallest and largest in pairs until the middle remains. Memory clue: "Sort, then split the middle."
Spot the difference
| Measure | Idea |
|---|---|
| Mean | equal share |
| Median | middle value |
| Mode | most frequent |
Practice questions
- 1Median of 4, 9, 2.
- 2Mode of 5, 5, 6, 7, 5.
- 3Median of 3, 8, 6, 10.
- 4Median of 15, 3, 9, 21, 12.
- 5Mode of 4, 4, 6, 6, 6, 9.
- 6Median of 18, 5, 12, 7, 9, 25.
- 7Median of 1, 2, 3, 4, 5, 6, 7, 8.
- 84, 6, x, 10, 15 has median 8. Find x.
Show answers (try them yourself first)
- 14
- 25
- 37
- 412
- 56
- 610.5
- 74.5
- 88
Range and choosing the best average
Five families earn ₹10,000 to ₹13,000, and one earns ₹1,00,000. Which "average income" should the newspaper quote?
The range = largest – smallest and shows the spread. The mean uses every value but is pulled by a very large or small value; the median resists such extremes; the mode is best for the most common choice.
In the figure: One value, 102, drags the mean to 24 while the median stays 5.
Rule 1: Range — Largest minus smallest.
- Easy: 4, 9, 15, 2 → 15 – 2 = 13.
- Medium: 20, 35, 28 → 35 – 20 = 15.
- Exam-type: Smallest 12, largest 47 → range 35.
Rule 2: Which average? — Extreme values → median. Most popular → mode. Fair share → mean.
- Easy: 3, 4, 5, 6, 102: mean 24, median 5 → use median.
- Medium: Shop stocking shoe sizes → mode.
- Exam-type: Total cost shared equally → mean.
Rule 3: Changing all values — Add k to each value: mean rises by k. Multiply each by k: mean is multiplied by k.
- Easy: Mean of 2, 4, 6, 8 is 5; each + 3 → 8.
- Medium: Mean of 3, 5, 7 is 5; each × 4 → 20.
- Exam-type: Range does not change when each value gets + k.
A very large or small value pulls the mean but not the median.
Quoting the mean when one value is extreme. Check the median too.
Look at the data first: is there an odd one out? Memory clue: "Odd one out? Use the median."
Spot the difference
| Average | Best when |
|---|---|
| Mean | no extreme values |
| Median | extreme values |
| Mode | most common choice |
Practice questions
- 1Range of 7, 19, 3, 12.
- 2Mean of 2, 3, 4, 5, 86.
- 3Median of 2, 3, 4, 5, 86.
- 4Mean of 4, 6, 8 is 6. Each + 5. New mean?
- 5Mean of 2, 4, 6 is 4. Each × 3. New mean?
- 6Smallest 45, largest 120. Range?
- 71, 2, 3, 4, x (x ≥ 4) has mean equal to median. Find x.
- 8Mean of 5 numbers is 14; four are 10, 12, 16, 18. The fifth?
Show answers (try them yourself first)
- 116
- 220
- 34
- 411
- 512
- 675
- 75
- 814
Reading and interpreting bar graphs
A library notice shows how many books were issued each day as bars. Taller bars mean more books at a glance.
A bar graph shows categories as bars whose heights are the values. Read the scale on the axis, compare heights, and then calculate totals, differences and averages from the readings.
In the figure: Books issued Monday to Friday: 12, 18, 9, 15, 21.
Reading and interpreting bar graphs
In the app you can move this figure with your finger.
Open in the appRule 1: Compare and find extremes — Highest bar = largest value; lowest bar = smallest.
- Easy: Highest day: Friday (21); lowest: Wednesday (9).
- Medium: Difference Friday – Wednesday = 12.
- Exam-type: Tuesday has 9 more than Wednesday.
Rule 2: Totals and averages — Add the heights; divide by the number of bars for the mean.
- Easy: Total = 12 + 18 + 9 + 15 + 21 = 75.
- Medium: Mean = 75 ÷ 5 = 15; days above 15: Tue, Fri.
- Exam-type: Median of 9, 12, 15, 18, 21 is 15.
Rule 3: Reading the scale — Value = number of squares × value of one square.
- Easy: 1 square = 5 units, bar of 6 squares = 30.
- Medium: Class 40: football 14, cricket 18, hockey 3 → kabaddi 5.
- Exam-type: Bar graphs compare categories; they do not need to touch.
Always read the scale before reading a bar.
Reading 6 squares as 6. If one square is 5, the value is 30.
Write each bar's value above it before calculating. Memory clue: "Scale first, bars next."
Spot the difference
| Graph | Best for |
|---|---|
| Bar graph | comparing categories |
| Tally table | collecting counts |
Practice questions
- 1Sales: Mon 20, Tue 35, Wed 15, Thu 30. Highest?
- 2Same sales. Lowest?
- 3Same sales. Total?
- 4Same sales. Mean per day?
- 5Same sales. Tuesday minus Wednesday?
- 61 square = 5 units. A bar of 7 squares shows?
- 7Same sales. Median?
- 8Same sales. How many days are above the mean?
Show answers (try them yourself first)
- 135
- 215
- 3100
- 425
- 520
- 635
- 725
- 82