Mean of data
Five friends score 6, 8, 7, 9 and 10. If they shared the marks equally, each would get the same number: the mean.
The mean (average) = sum of all values ÷ number of values. It is the "balance point" of the data. The sum of the differences from the mean is zero.
Formula: Mean = (x₁ + x₂ + … + xₙ) ÷ n
In the figure: Total = 6 + 8 + 7 + 9 + 10 = 40, so the mean = 40 ÷ 5 = 8.
Rule 1: Find the mean — Add everything, then divide by how many.
- Easy: 2, 4, 9: mean = 15 ÷ 3 = 5.
- Medium: 10, 20, 30, 40: mean 25.
- Exam-type: Temperatures 28, 30, 32, 26 °C: mean 29 °C.
Rule 2: Missing value — Total needed = mean × count; subtract the known sum.
- Easy: Mean of 4 numbers is 10; three are 8, 9, 13: fourth = 40 − 30 = 10.
- Medium: Mean of 5 is 12: total 60.
- Exam-type: Marks 70, 80 and x with mean 80: x = 90.
The mean always lies between the smallest and the largest value.
Dividing the sum by the number of distinct values instead of all values.
"Total over count."
Practice questions
- 1The numbers 11, 27, 7, 30, 25 are the daily steps (in hundreds) recorded for five days. What is their mean?
- 2The mean of 6 numbers is 9. What is the sum of the 6 numbers?
- 3Find the mean of the 10 numbers 58, 19, 39, 26, 45, 6, 47, 39, 24, 12.
- 4The mean of 6 numbers is 14. 5 of the numbers are 12, 6, 10, 11, 18. What is the remaining number?
- 5In a quiz, the scores 3, 4 and 8 were obtained by 5, 3 and 12 students respectively. What is the mean score of the 20 students?
- 6What is the mean of the 7 equally spaced numbers 20, 30, 40, …, 80?
- 7The mean of 9 numbers is 24. If every number is first increased by 5 and the result is then multiplied by 4, what is the new mean?
- 8The mean of six numbers is 17. Four of them are 20, 8, 20, 10. The other two numbers are equal. What is each of them?
Show answers (try them yourself first)
- 120
- 254
- 331.5
- 427
- 56.15
- 650
- 7116
- 822
Mean of data
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Open in the appHow the mean changes
One very high salary in a small office pulls the average pay up, even if most people earn little. A single big value drags the mean.
Adding the same number k to every value adds k to the mean. Multiplying every value by k multiplies the mean by k. Adding one new value moves the mean towards that value.
Formula: New mean = (old total + new value) ÷ (n + 1)
In the figure: Mean 8 for five values. Adding 40 gives (40 + 40) ÷ 6 ≈ 13.3.
Rule 1: Shift and scale — The mean follows the same operation as the data.
- Easy: Mean 20; add 5 to each: 25.
- Medium: Mean 12; double each value: 24.
- Exam-type: Mean 50; subtract 10 from each: 40.
Rule 2: Add or remove a value — Work with totals, not with means.
- Easy: Mean of 4 is 10 (total 40); adding 20 gives 60 ÷ 5 = 12.
- Medium: Mean of 5 is 8 (total 40); removing 8 leaves mean 8.
- Exam-type: Class mean 60; a new student scoring 90 raises it.
Adding a value equal to the mean leaves the mean unchanged.
Averaging the old mean and the new value directly, ignoring how many values there were.
"Use totals, not means."
Practice questions
- 1The mean of 5 numbers is 27. When one more number, 33, is included, what is the new mean?
- 2The mean of 5 numbers is 26. When one more number, 56, is included, what is the new mean?
- 3The mean of 9 numbers is 40. One of the numbers, 56, is removed. What is the mean of the remaining 8 numbers?
- 4A student's mean score in 4 tests is 73. What score must the student get in the next test so that the mean of all 5 tests becomes 78?
- 5A group of 10 friends has a mean age of 37 years. A friend aged 38 years leaves and a new friend aged 58 years joins. What is the new mean age?
- 6The mean weight of 5 boys is 39 kg. When a new boy joins, the mean rises by 1 kg. What is the weight of the new boy?
- 7A batter has a mean of 45 runs after 9 innings. How many runs must the batter score in the next innings to raise the mean to 48?
- 8The mean of 4 numbers is 33. Two more numbers are added and the mean of all 6 numbers becomes 31. What is the sum of the two new numbers?
Show answers (try them yourself first)
- 128
- 231
- 338
- 498
- 539
- 645
- 775
- 854
How the mean changes
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Open in the appCombined and weighted mean
A school with a big Class A and a small Class B cannot simply average the two class means. The bigger class counts more.
Combined mean = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂). A weighted mean multiplies each value by its weight (importance or frequency) and divides by the total weight.
Formula: Weighted mean = Σ(value × weight) ÷ Σ weights
In the figure: Totals 30 × 60 = 1800 and 20 × 70 = 1400 give a combined mean 3200 ÷ 50 = 64, not 65.
Rule 1: Combine two groups — Rebuild each group total from its mean, add, then divide.
- Easy: 10 pupils mean 50, 10 pupils mean 70: 60.
- Medium: 20 with mean 40 and 30 with mean 60: 52.
- Exam-type: Combined mean lies nearer to the larger group.
Rule 2: Weighted mean — Multiply each value by its weight first.
- Easy: Test 80 (weight 2) and exam 90 (weight 3): (160 + 270) ÷ 5 = 86.
- Medium: Grades 5 (3 credits), 7 (1 credit): 22 ÷ 4 = 5.5.
- Exam-type: Frequency table mean: Σfx ÷ Σf.
A weighted mean with equal weights is the ordinary mean.
Averaging two group means without weighting by group size.
"Big group, big say."
Practice questions
- 1In a test, the mean score of 20 students of Team X is 33 and the mean score of 20 students of Team Y is 63. What is the mean score of all 40 students together?
- 2Two groups have the same number of members, 20 each. Their means are 76 and 60. What is the mean of the combined group?
- 3A shopkeeper mixes 2 kg of dal costing ₹70 per kg with 8 kg of the same dal costing ₹40 per kg. What is the mean cost per kg of the mixture (in ₹)?
- 4A final grade counts the project for 60% and the exam for 40%. A student scores 82 in the project and 65 in the exam. What is the final grade?
- 5The mean mark of 30 students is 56. The first 10 students have a mean of 70. What is the mean mark of the remaining 20 students?
- 6Two kinds of mixture costing ₹180 per litre and ₹150 per litre are mixed in the ratio 1 : 3. What is the cost per litre of the blend (in ₹)?
- 7Three sections have 10, 10 and 30 students with mean scores 65, 33 and 68 respectively. What is the mean score of all 50 students?
- 8Group A has 20 members with mean 40. Group B has a mean of 60. When both groups are combined the mean is 50. How many members does Group B have?
Show answers (try them yourself first)
- 148
- 268
- 346
- 475.2
- 549
- 6157.5
- 760.4
- 820
Pie charts
A round roti cut into slices shows how a family's monthly spending is shared: a big slice for food, a small one for travel.
A pie chart shows parts of a whole as sectors of a circle. Sector angle = (value ÷ total) × 360°. The sector angles add up to 360° and the percentages add up to 100%.
Formula: Angle = (value ÷ total) × 360°
In the figure: Sleep takes 50% of the circle, which is 180°; school is 25% = 90°.
Rule 1: Draw the angles — Convert each value into a fraction of 360°.
- Easy: 20 of 80: (20/80) × 360° = 90°.
- Medium: 30% of the pie: 108°.
- Exam-type: Half the pie: 180°.
Rule 2: Read the chart — Use angle ÷ 360 × total to get the value.
- Easy: Sector 72° of 500 students: 100 students.
- Medium: Two sectors of 90° and 60° leave 210° for the rest.
- Exam-type: A pie chart shows shares, not exact numbers, unless the total is given.
The largest sector means the largest share.
Using 100° or 100 as the total instead of 360°.
"Share times 360."
Practice questions
- 1In a pie chart of a family's monthly spending, 30% is spent on rent. What is the central angle of this sector?
- 2A sector of a pie chart has a central angle of 126°. What percentage of the whole does it represent?
- 3A pie chart shows how 720 students come to school. The sector for 'cycle' has a central angle of 45°. How many students come by cycle?
- 4Out of 72 people surveyed, 14 prefer tea. In a pie chart of the survey, what is the central angle of the 'tea' sector?
- 5A pie chart has four sectors. Three sectors (mango, apple, banana) have central angles 30°, 70° and 100°. What is the central angle of the fourth sector?
- 6A pie chart shows three items in the ratio 1 : 2 : 3. What is the central angle of the largest sector?
- 7In a pie chart of 240 students' favourite subjects, Maths has a sector of 36° and Science a sector of 108°. How many more students prefer Science than Maths?
- 8A pie chart has sectors with central angles 80°, then two sectors in the ratio 3 : 5 which together fill the remaining circle. What is the bigger of those two angles?
Show answers (try them yourself first)
- 1108°
- 235%
- 390
- 470°
- 5160°
- 6180°
- 748
- 8175°
Stacked and 100% stacked bars
A school plots boys and girls on one bar each year: the whole bar is the total, the coloured parts are the groups.
In a stacked bar the parts are piled on top of each other, so the full height is the total. In a 100% stacked bar every bar is scaled to 100%, which shows shares clearly even when totals differ.
Formula: Share % = (part ÷ bar total) × 100
In the figure: Each bar splits into girls (blue) and boys (green); the whole bar height is the total pupils.
Rule 1: Read a stacked bar — Part = top edge minus bottom edge of its colour.
- Easy: Blue 40 and green 30: total 70.
- Medium: Blue 50, green 35 next year: both groups grew.
- Exam-type: Total in year 3: 55 + 45 = 100.
Rule 2: Convert to 100% — Divide each part by the bar total and multiply by 100.
- Easy: 40 of 70 girls: ≈ 57%.
- Medium: 55 of 100: 55%.
- Exam-type: 100% bars cannot show how big the total is.
Use stacked bars for totals and 100% stacked bars to compare shares.
Reading the top number as the value of the top colour instead of the total.
"Pile for totals, percent for shares."
Practice questions
- 1In a stacked bar chart, the column for Shop P has two parts: adults = 27 and children = 17. What is the total height of this column?
- 2A stacked column for Shop Q has a total height of 23. The adults part is 14. What is the height of the children part?
- 3A stacked column for Village A has total height 80. Its passed part is 60. What percentage of the column is passed?
- 4In a 100% stacked bar for Village B, the passed segment is 45% of the bar. The total for Village B is 500. How many are passed?
- 5Class 9A: total 60, boys = 17. Class 9B: total 80, boys = 46. In a 100% stacked bar chart, which column has the longer boys segment?
- 6Stacked columns: Shop P has adults = 9, children = 9; Shop Q has adults = 11, children = 24. What is the total adults in both columns together?
- 7In 100% stacked bars, boys is 75% of Class 9A (total 100) and 10% of Class 9B (total 100). What percentage of the combined total is boys?
- 8Class 9A total = 200, boys = 40. Class 9B total = 400, boys = 300. By how many percentage points does the boys share change from Class 9A to Class 9B?
Show answers (try them yourself first)
- 144
- 29
- 375%
- 4225
- 5Class 9B
- 620
- 742.5%
- 855 points up
Stacked and 100% stacked bars
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Open in the app