MathsCBSE Class 8 · Tales by Dots and Lines

Mean, Median and How They Change with the Data for Class 8

Updated 17 October 2026 · Fact-checked

The mean is the sum of the values divided by the number of values; it is the balancing centre where the total distance of the lower values equals that of the higher values. Adding a value above the mean raises the mean, adding a value equal to the mean leaves it unchanged, and if every value is increased by k or multiplied by k the mean also increases by k or is multiplied by k. The median is the middle value of sorted data.

Understand Mean, Median and How They Change with the Data

Mean. The mean (average) of $n$ values $x_{1}, x_{2}, \dots, x_{n}$ is $\overset{x}{ˉ} = \frac{x _{1} + x _{2} + \dots + x _{n}}{n}$ . For two numbers the mean is exactly halfway between them: the mean of 3 and 7 is 5.

Mean as a centre of balance. On a dot plot, the mean is the point where the total distance of the values below it equals the total distance of the values above it. It is not always the midpoint of the two extremes. For the data 10, 10, 11, 17 the mean is 12: the distances to the left are $2 + 2 + 1 = 5$ and to the right $5$ . There is only one such centre: if you move to a larger value, all the left distances grow and all the right distances shrink, so the balance is lost.

Fair-share interpretation. If you pool all the quantities and share them equally among the $n$ people, each gets the mean.

Including or removing values.

Including a value greater than the mean raises the mean; including a value less than the mean lowers it; including a value equal to the mean leaves it unchanged. The reverse holds when a value is removed (removing a value above the mean lowers it).
It is possible to include two (or three) values together so that the mean stays the same, as long as the extra distances above the mean balance the extra distances below it.
In terms of sums: new mean $= \frac{old sum + new value}{n + 1}$ .

Changing every value.

If a fixed number $k$ is added to every value, the mean increases by $k$ : $\frac{( x _{1} + k ) + \dots + ( x _{n} + k )}{n} = \overset{x}{ˉ} + k$ . Subtracting $k$ lowers the mean by $k$ . Their relative position stays the same.
If every value is multiplied by $k$ , the mean is multiplied by $k$ : $\frac{k x _{1} + \dots + k x _{n}}{n} = k \overset{x}{ˉ}$ (distributive property).
Combined: the mean of $a x_{i} + b$ is $a \overset{x}{ˉ} + b$ . For example, converting a mean of $2 5^{\circ}$ C by $F = 1.8 C + 32$ gives $7 7^{\circ}$ F.

Median. Sort the data. The median is the middle value if the number of values is odd, and the mean of the two middle values if it is even. There are equal numbers of values below and above it. The median is not affected by how large the extreme values are.

Including a value greater than the median moves the median up (to the next value or between values); including a value equal to the median keeps it; including one smaller moves it down.
To keep the median unchanged when two values are added, add one value at or below it and one at or above it.

Summary measures. Describe data with the minimum, maximum, mean and median.

Spreadsheets and line graphs are on the next topic page of this chapter.

Mean, Median and How They Change with the Data: formulas to remember

Mean: $\overset{x}{ˉ} = \frac{x _{1} + x _{2} + \dots + x _{n}}{n}$; Sum divided by the number of values.
Balance property: $\sum (x_{i} - \overset{x}{ˉ}) = 0$; Distances below the mean equal distances above it.
Adding a new value: $new mean = \frac{n x ˉ + v}{n + 1}$; v is the included value.
Add or multiply every value: $mean (x_{i} + k) = \overset{x}{ˉ} + k, mean (k x_{i}) = k \overset{x}{ˉ}$; Median behaves the same way.
Median: $median = middle value (odd n), mean of the two middle values (even n)$; Sort first.

How to solve Mean, Median and How They Change with the Data questions

Use these routines for mean and median problems.

1Write the sum of the values as n times the mean; this is the key link between mean and sum.
2For a changed data set, update the sum (add or subtract values, or shift or scale all values) and divide by the new number of values.
3To decide whether the mean rises or falls, compare the new value with the old mean.
4For the median, sort the data, count the values, and locate the middle position(s).
5When adding values to keep the median, check which positions become the middle in the new data.
6Check by recomputing the mean or the median of the final data.

Mean, Median and How They Change with the Data in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Correcting a mean

Suppose the marks of 30 students in a test were entered with an average of 62, and later the teacher finds every mark was entered 3 too high.

What is the correct average, and what is the average if the corrected marks are all multiplied by 1.5 to change the maximum from 100 to 150?

Try it first, then show the solutionSolution

Subtracting 3 from every value lowers the mean by 3: $62 - 3 = 59$ .
Multiplying every value by 1.5 multiplies the mean: $59 \times 1.5 = 88.5$ .

Answer: 59, and then 88.5

Memory hook: Shifting or scaling all values shifts or scales the mean the same way, so no need to re-add 30 numbers.

Adding a new value

Imagine five friends whose pocket money averages Rs 120 per week; a sixth friend who gets Rs 180 joins the group.

Find the new average.

Try it first, then show the solutionSolution

Old sum $= 5 \times 120 = 600$ .
New sum $= 600 + 180 = 780$ , so the new mean $= 780 \div 6 = 130$ .

Answer: Rs 130

Memory hook: A value above the mean pulls the mean up.

Tricks to remember Mean, Median and How They Change with the Data

Short memory aids. Say each one aloud once, then use it on a question the same day.

Mean times n is the sum
When the mean and the count are known, the sum is mean x n; use it before adding or removing values.
Use it when: Every changed-data problem.
Above pulls up, below pulls down, equal does nothing
A new value above the mean raises it, below lowers it, equal leaves it.
Use it when: Direction questions.
Shift shifts, scale scales
Add k to all values: mean adds k. Multiply all by k: mean multiplies by k.
Use it when: Corrections and unit changes.
Median: sort, then middle
Sort first; odd count: middle value; even count: average the two middle values.
Use it when: Median questions.

Mean, Median and How They Change with the Data: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Mean after including a value · 1 mark · MCQBoard level

The mean of five numbers is 12. A sixth number, 18, is included. The new mean is: (a) 12.5 (b) 13 (c) 14 (d) 15

Try it first, then show the solutionSolution

Old sum $= 5 \times 12 = 60$ .
New sum $= 78$ , new mean $= 78 \div 6 = 13$ .

Answer: (b) 13

Example 2: Mean and median of a list · 2 marks · Short answerBoard level

Find the mean and the median of 12, 15, 9, 21, 17, 11, 25, 19.

Try it first, then show the solutionSolution

Sum $= 129$ , number of values $= 8$ , mean $= 129 \div 8 = 16.125$ . (1 mark)
Sorted: 9, 11, 12, 15, 17, 19, 21, 25. Median $= \frac{15 + 17}{2} = 16$ . (1 mark)

Answer: Mean 16.125; median 16

Example 3: Correcting a recorded average · 3 marks · Short answerHarder board level

The average mark of 30 students was recorded as 62, but every mark was entered 3 too high. Find the correct total and average, and the average if all corrected marks are then multiplied by 1.5.

Try it first, then show the solutionSolution

Recorded total $= 30 \times 62 = 1860$ ; the error in the total is $30 \times 3 = 90$ , so the correct total is $1770$ . (1 mark)
Correct average $= 1770 \div 30 = 59$ . (1 mark)
Multiplying every mark by 1.5 multiplies the average by 1.5: $59 \times 1.5 = 88.5$ . (1 mark)

Answer: Total 1770; average 59; then 88.5

Example 4: Removing two values · 3 marks · Short answerHarder board level

The mean of 12 numbers is 40. Two of them, 32 and 56, are removed. Find the mean of the remaining 10 numbers. Does the mean go up or down, and why?

Try it first, then show the solutionSolution

Total $= 12 \times 40 = 480$ . (1 mark)
Remaining total $= 480 - (32 + 56) = 392$ , mean $= 392 \div 10 = 39.2$ . (1 mark)
It goes down because the removed values have mean $\frac{88}{2} = 44$ , which is greater than 40. (1 mark)

Answer: 39.2; the mean decreases because the removed values average more than 40.

Example 5: Keeping the median unchanged · 4 marks · Long answerHarder board level

The data 7, 11, 14, 18, 22, 29, 35 has median 18. (i) What single value can be included without changing the median? (ii) Give two values that can be included together without changing the median. (iii) Which value can be removed without changing the median?

Try it first, then show the solutionSolution

(i) With 8 values the median is the mean of the 4th and 5th values. Including 18 gives 7, 11, 14, 18, 18, 22, 29, 35 and median $\frac{18 + 18}{2} = 18$ . Any other value moves the middle pair away from 18, so only 18 works. (1 mark)
(ii) Include one value at or below 18 and one at or above 18, for example 12 and 30: the 9 values are 7, 11, 12, 14, 18, 22, 29, 30, 35, and the 5th is 18. (1 mark)
(iii) Remove 18 itself: the six values 7, 11, 14, 22, 29, 35 have median $\frac{14 + 22}{2} = 18$ . (1 mark)
(iii) works here because 14 and 22 are equally far (4) from 18; removing another value would shift the median. (1 mark)

Answer: (i) 18 only; (ii) for example 12 and 30; (iii) 18.

Where students lose marks in Mean, Median and How They Change with the Data

The slips examiners see most often, and what to write instead.

Mistake: Dividing by the old number of values after adding one
The count changes to n + 1 (or n - 1).
Wrong working
"New mean = (60 + 18) / 5."
Right working
$\frac{60 + 18}{6} = 13$ .
Do this: Write the new count beside the new sum.
Usually costs: The answer.
Mistake: Taking the median without sorting
The median is the middle of the sorted data.
Wrong working
"12, 15, 9, 21, 17, 11, 25, 19: the middle values are 21 and 17, median 19."
Right working
Sorted: 9, 11, 12, 15, 17, 19, 21, 25; median $= \frac{15 + 17}{2} = 16$ .
Do this: Rewrite the list in order every time.
Usually costs: 2 marks.
Mistake: Thinking the mean must be the midpoint of the extremes
The mean balances distances; it is not always halfway between the smallest and largest.
Wrong working
"3, 5, 6, 14: the mean is (3 + 14)/2 = 8.5."
Right working
Mean $= 28 \div 4 = 7$ .
Do this: Always add all values and divide by the count.
Usually costs: 1 mark.
Mistake: Adding k to the median when values are multiplied by k
Multiplying every value by k multiplies the mean and median by k; adding k adds k.
Wrong working
"All scores doubled: new median = 15 + 2 = 17."
Right working
$15 \times 2 = 30$ .
Do this: Match the operation: add gives add, multiply gives multiply.
Usually costs: 1 mark.

Exam tips

Always compute the sum as mean times count before changing the data.
State whether the new value is above, below or equal to the old mean, and what that means for the direction.
Rewrite data in sorted order before locating the median.
Show a one-line check for a corrected mean by recomputing from the new sum.
Use algebra (x1 + ... + xn) when asked to justify a general statement.

Practice questions on Mean, Median and How They Change with the Data

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): If every value in a data set is increased by 5, the mean increases by 5.
Reason (R): Adding 5 to each of n values adds 5n to the sum, and the mean is the sum divided by n.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Divide the new sum by n.
Answer
(a). New mean $= \frac{old sum + 5 n}{n} = \overset{x}{ˉ} + 5$ .
Q2 · 2 marks · Very short answerBoard level
The mean of 6 numbers is 7.5. A seventh number is included and the mean becomes 8. Find the seventh number.
Hint
Compare the two sums.
Answer
Old sum $= 6 \times 7.5 = 45$ ; new sum $= 7 \times 8 = 56$ ; seventh number $= 56 - 45 = 11$ .
Q3 · 3 marks · Short answerHarder board level
The data 3, 5, 6, 14 has mean 7. (i) Show that the total distance of the values below 7 from 7 equals the total distance of the values above 7. (ii) What happens to the mean if a value 7 is included? (iii) What is the mean if 10 is included?
Hint
Distances: 4, 2, 1 below; 7 above.
Answer
(i) Below: $(7 - 3) + (7 - 5) + (7 - 6) = 4 + 2 + 1 = 7$ ; above: $14 - 7 = 7$ . Equal. (ii) A value equal to the mean leaves the mean unchanged at 7: $\frac{28 + 7}{5} = 7$ . (iii) $\frac{28 + 10}{5} = 7.6$ , which is higher because 10 is above the mean.
Q4 · 4 marks · Long answerHOTS / competency
Let x1, x2, ..., xn have mean m. (i) Show that the sum of (xi - m) is 0. (ii) Show that the mean of (a xi + b) is a m + b. (iii) The mean of a week's maximum temperatures is 25 degrees Celsius. Find the mean in Fahrenheit using F = 1.8 C + 32.
Hint
Use the sum x1 + ... + xn = n m.
Answer
(i) $\sum (x_{i} - m) = \sum x_{i} - nm = nm - nm = 0$ . (ii) $\frac{\sum ( a x _{i} + b )}{n} = \frac{a \sum x _{i} + nb}{n} = am + b$ . (iii) Here $a = 1.8$ , $b = 32$ , $m = 25$ : mean $= 1.8 \times 25 + 32 = 45 + 32 = 7 7^{\circ}$ F.
Q5 · 4 marks · Case-basedHOTS / competency
Case study: 20 students took a test; the mean score was 14 and the median was 15.
(i) After a re-evaluation every score goes up by 2. Find the new mean and median. (1 mark)
(ii) Instead, all scores are doubled. Find the new mean and median. (1 mark)
(iii) A 21st student scoring exactly 14 joins. What happens to the mean? (1 mark)
(iv) Instead, a 21st student scoring 20 joins. Find the new mean to two decimal places. (1 mark)
Hint
Total of 20 scores = 280.
Answer
(i) Mean 16 and median 17. (ii) Mean 28 and median 30. (iii) The mean stays 14, since the new score equals the mean: $\frac{280 + 14}{21} = 14$ . (iv) $\frac{280 + 20}{21} = \frac{300}{21} \approx 14.29$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What happens to the mean if a value equal to the mean is included?Show answer
It stays the same.
2.What happens to the mean when every value is increased by k?Show answer
It increases by k.
3.What happens to the mean when every value is multiplied by k?Show answer
It is multiplied by k.
4.What is the median of an even number of values?Show answer
The mean of the two middle values of the sorted data.
5.What do the distances on either side of the mean satisfy?Show answer
Their totals are equal.
6.Can there be more than one centre with equal total distances on both sides?Show answer
No, only the mean.

Mean, Median and How They Change with the Data: frequently asked questions

Why does a value above the mean raise the mean?

It adds more distance on the high side than the low side, so the balance point shifts upward.

Is the mean always a value in the data?

No. The mean of 3 and 8 is 5.5, which is not in the data.

Does the median change if the largest value becomes bigger?

No, as long as the largest value stays above the middle; the median depends on position, not size.

How many values can I add without changing the mean?

You can add any number of values equal to the mean, or groups of values whose distances from the mean balance (for example one above and one below by the same amount).