Data handlingInvestigateabout 45 min

What happens if…? Experiments with averages

Predict, change the data, and test: outliers, shifts, missing values and datasets built to order

Treat averages like a science experiment. Predict what adding a value, an outlier, or a change to every value does to the mean, median, mode and range, then test it in the labs. Build data sets to order, hunt missing values and compare real Indian data.

Start at chapter 1

In this part you’ll

Predict and test how adding or removing a value changes the mean, median, mode and range.
Explain why an outlier moves the mean a lot but the median very little.
Predict the effect of adding a constant to every value, or multiplying every value.
Build a data set that has a given mean, median, mode and range, and find missing values.
Decide whether statements about averages are always, sometimes or never true, with examples.

Scientists do not just learn facts; they poke things to see what happens. In this layer you will poke data. What if a new value joins? What if one value is enormous? What if everyone gets 5 more marks? What if I double everything?

For each experiment: predict first (commit to an answer), then test in a lab or by calculating, then explain what you saw. Some results will surprise you, and those surprises are exactly where the understanding is hiding.

Chapter 01

A new value joins the data

Priya, a school cricketer, scored 34, 12, 56, 8, 42, 26, 32 in her last 7 innings (she was out every time). Her mean is 210 ÷ 7 = 30 runs, and her median is 32 (ordered: 8, 12, 26, 32, 34, 42, 56).

She is about to bat again. Let us experiment with what her next score does to her averages.

Predict first

In her 8th innings Priya scores exactly 30, the same as her mean. What happens to her mean and median?

Both stay the sameMean stays 30; median changesMean changes; median stays 32Both change

TableWhat Priya's 8th score does to her averages (starting mean 30, median 32)
8th score	New total	New mean	New median	Range
0	210	26.25	29	56
30	240	30	31	48
38	248	31	33	48
100	310	38.75	33	92

The pattern in the table is a rule you can rely on:

A new value above the mean pulls the mean up.
A new value below the mean pulls it down.
A new value equal to the mean leaves it unchanged.

And how far it pulls depends on how far the new value is from the mean, shared over the new count. A score of 38 is 8 above the mean of 30, so the mean rises by 8 ÷ 8 = 1, to 31. A score of 100 is 70 above, so the mean rises by 70 ÷ 8 = 8.75, to 38.75.

The median moves at most a little: it only shifts by half a step along the ordered list, however big or small the new value is.

Lab

Change Priya's 8th innings and watch how far each score pulls her mean and median.

Priya's scores in 8 innings (runs)

Round 1 / 4★ 0 ptsBest: 0

Challenge 1Change her 8th score (30) so that her mean becomes 33.

Target: mean = 33. Right now the mean is 30. Add or remove dots below — it checks as you go.

Tap the number line to add a value; tap a dot to remove it. Dashed long line = mean (●), dotted line = median (▲).

The values (8)

Mean (share it out equally)30 runs

sum ÷ count = 240 ÷ 8 = 30

Median (the middle value)31 runs

812263032344256

8 values (even), so take the two middle ones: (30 + 32) ÷ 2 = 31.

Mode (most common)No mode

Every value appears only once. The usual convention: when nothing repeats, we say there is no mode.

Range (spread)48 runs

max − min = 56 − 8 = 48

Text version of this activity

A dot plot of 8 innings: 34, 12, 56, 8, 42, 26, 32 and a changeable 8th score, starting at 30. Mean 30, median 31, range 48.

Challenges:

Mean 33: the total must be 33 × 8 = 264, so the 8th score must be 264 − 210 = 54.
Mean 27: total 27 × 8 = 216, so the 8th score must be 216 − 210 = 6. Try asking for a mean of 26: the 8th score would have to be −2, which is impossible, so you would need to change an earlier score too.
Median 33: with the 8th score at 34 or more, the middle two are 32 and 34, so the median is 33.
Range 70: the smallest score is 8, so make a score of 78.

Each extra run in one innings lifts the mean by 1 ÷ 8 = 0.125.

Need a different angle?

Try it

Chapter 02

One giant value: outliers

Ten children in a class get this much pocket money per week (₹): 40, 40, 50, 50, 60, 60, 60, 70, 80, 90. The mean is ₹60, the median is ₹60 and the mode is ₹60. Very tidy.

Then a new child, Rohan, joins the class. Rohan gets ₹400 a week.

Predict first

After Rohan joins (₹400), roughly what happens to the mean and median?

Mean ≈ ₹91, median still ₹60Mean ≈ ₹91, median ≈ ₹91Mean still ₹60, median ≈ ₹230Both jump to about ₹230

Lab

Drag the outlier and the other values to compare how strongly each moves the mean and the median.

Weekly pocket money of 11 children, including Rohan (₹)

Round 1 / 4★ 0 ptsBest: 0

Challenge 1Suppose Rohan's ₹400 was a typing mistake. Fix it so the mean is back to ₹60.

Target: mean = 60. Right now the mean is 90.91. Add or remove dots below — it checks as you go.

Tap the number line to add a value; tap a dot to remove it. Dashed long line = mean (●), dotted line = median (▲).

The values (11)

Mean (share it out equally)90.91 ₹

sum ÷ count = 1000 ÷ 11 ≈ 90.91

Median (the middle value)60 ₹

40405050606060708090400

11 values (odd), so the middle one — number 6 in order — is the median.

Mode (most common)60

60 appears 3 times — more than any other value.

Range (spread)360 ₹

max − min = 400 − 40 = 360

Text version of this activity

A dot plot of 11 amounts: 40, 40, 50, 50, 60, 60, 60, 70, 80, 90, 400. With Rohan's ₹400 the mean is 1,000 ÷ 11 ≈ ₹90.9, the median is ₹60, the mode is ₹60 and the range is 400 − 40 = ₹360.

Challenges:

Mean 60: total must be 60 × 11 = 660; change 400 to 60.
Mean 100: total must be 1,100, i.e. 100 more. Change one ₹40 child to ₹140.
Median 70: the 6th ordered value must be 70. Changing Rohan does nothing, because he is already at the top. Raising one ₹60 child is not enough (the 6th value is still 60); raise two of the ₹60 children to ₹70.
Range 50: change 400 to 90 (90 − 40 = 50).

Moving Rohan all the way from 400 to 0 changes the mean by 400 ÷ 11 ≈ ₹36, yet the median stays at ₹60.

Need a different angle?

Try it

Chapter 03

Change every value at once

Predict first

A teacher adds 5 bonus marks to every child's test score. Before the bonus: mean 12, median 11, mode 10, range 9. What are the new values?

Mean 17, median 16, mode 15, range 14Mean 17, median 16, mode 15, range 9Mean 17, median 11, mode 10, range 9Only the mean changes

Predict first

Now every value in a data set is doubled. The mean was 6 and the range was 8. What happens?

Mean 12, range 8Mean 12, range 16Mean 6, range 16Mean 8, range 10

TableWhat shifting and stretching do (data 2, 4, 6, 8, 10)
Change	New data	Mean	Median	Range
None	2, 4, 6, 8, 10	6	6	8
Add 5 to each	7, 9, 11, 13, 15	11	11	8
Subtract 2 from each	0, 2, 4, 6, 8	4	4	8
Double each	4, 8, 12, 16, 20	12	12	16
Halve each	1, 2, 3, 4, 5	3	3	4
Multiply by 10	20, 40, 60, 80, 100	60	60	80

Lab

Predict whether each change to a small data set moves the mean, the median, both or neither.

Start with 2, 4, 6, 8, 10 (mean 6, median 6). Sort each change by what it does.

14 cards, 4 bins. Tap a card, then tap its bin. You can also drag, or press a bin’s number key.

Text version of this activity

A sorting game starting from 2, 4, 6, 8, 10 (mean 6, median 6), with 14 changes to sort.

Only the mean changes: change 10 → 20 (mean 7); change 2 → 0 (mean 5.6); change 10 → 100 (mean 24). Changing an end value never moves the middle.
Only the median changes: 6 → 7 and 10 → 9 (total still 30, middle now 7); 6 → 5 and 2 → 3 (middle now 5).
Both change: add 1 to every value; 6 → 7; double every value; add a new value 30 (mean 10, median 7); remove the 10 (mean 5, median 5).
Neither changes: add a new 6; reorder the values; 4 → 5 and 8 → 7; remove the 6 (mean 24 ÷ 4 = 6, median (4 + 8) ÷ 2 = 6).

Key ideas: the mean follows the total; the median follows the middle position.

Try it

Chapter 04

Always, sometimes or never?

Mathematicians love statements that are always true, because you can build on them. Here are some claims about averages. For each, the investigator's job is to find either a reason it must always hold, or a single counterexample that breaks it.

TableClaims about averages, tested
Claim	Verdict	Evidence
The mean lies between the smallest and largest values.	Always	A fair share cannot be more than the richest or less than the poorest has.
The mean is one of the data values.	Sometimes	3, 7, 4, 6, 5 → 5 (yes). 1, 2, 6 → 3 (no).
The median is one of the data values.	Sometimes	Always with an odd count; with an even count 2, 4 → 3 (no).
A data set has a mode.	Sometimes	5, 7, 9 has no mode.
Mean, median and mode are all equal.	Sometimes	4, 5, 5, 6 → all 5. But 1, 1, 7 → mean 3, median 1, mode 1.
The range is bigger than the mean.	Sometimes	1, 9 → range 8, mean 5 (yes). 50, 52 → range 2, mean 51 (no).
The range can be negative.	Never	Max − min, and max is never below min.
If all values are equal, the range is 0.	Always	Max and min are the same number.
More than half the values are above the mean.	Sometimes	1, 1, 1, 9: mean 3, only one value above. 1, 9, 9, 9: mean 7, three above.

Predict first

Can every value but one in a data set be below the mean?

No, about half must be above the meanYes, if one value is very largeOnly if there are exactly two valuesOnly if all values are equal

Try it

Chapter 05

Build a data set to order

Now turn the problem around. Instead of finding the averages of given data, design data with averages you choose. This is how puzzle-setters, exam writers and data scientists testing their programs think.

Worked example

0 / 7 steps shown

Five numbers with mean 7, median 5, mode 4 and range 8

Find five whole numbers with mean 7, median 5, only one mode, 4, and range 8.

Lab

Design five numbers step by step to hit a target mean, median, mode and range together.

Five numbers to design

Round 1 / 5★ 0 ptsBest: 0

Challenge 1Make the mean 7.

Target: mean = 7. Right now the mean is 6. Add or remove dots below — it checks as you go.

Tap the number line to add a value; tap a dot to remove it. Dashed long line = mean (●), dotted line = median (▲).

The values (5)

Mean (share it out equally)6

sum ÷ count = 30 ÷ 5 = 6

Median (the middle value)6

246810

5 values (odd), so the middle one — number 3 in order — is the median.

Mode (most common)No mode

Every value appears only once. The usual convention: when nothing repeats, we say there is no mode.

Range (spread)8

max − min = 10 − 2 = 8

Text version of this activity

Five values start at 2, 4, 6, 8, 10 (mean 6, median 6, no mode, range 8). Each challenge adds a condition.

Mean 7: the total must be 35, five more than now; e.g. change 10 to 15.
Median 5: the middle ordered value must be 5, with the total still 35.
Mode 4 only: two values must be 4, both below the median.
Range 8: biggest − smallest = 8. One full solution is 4, 4, 5, 10, 12.
Range 0 with mean 7: every value must be 7: 7, 7, 7, 7, 7.

The lab checks each target separately, so check the earlier conditions yourself as you go.

Try it

Chapter 06

Monsoon figures put to the test

Back to Mumbai's monthly rainfall from Discover (IMD normals for 1991–2020 at Mumbai (Santacruz), rounded, in mm): 0, 0, 0, 0, 7, 526, 920, 561, 384, 91, 11, 2. Total ≈ 2,502 mm.

Predict before you calculate: will the mean month and the median month be close?

Predict first

For Mumbai's 12 monthly rainfall values, which is true?

Mean and median are both about 200 mmThe mean is about 210 mm but the median is only about 9 mmThe median is much bigger than the meanThe mode is the best description

Rainfall in millimetres covers a huge range (0 to 920), which is hard to plot. A friendlier measure is the number of rainy days in each month. For Mumbai (Santacruz) the IMD normals for 1991–2020, rounded to whole days, are: Jan 0, Feb 0, Mar 0, Apr 0, May 1, Jun 14, Jul 23, Aug 21, Sep 14, Oct 4, Nov 1, Dec 0. That makes about 78 rainy days a year, nearly all from June to September.

Lab

Explore how a seasonal pattern gives a mean far above the median, and try changing the season.

Mumbai (Santacruz): rainy days in each month, Jan to Dec (days)

Round 1 / 4★ 0 ptsBest: 0

Challenge 1Imagine a year with a longer monsoon. Change months so the median becomes 3 rainy days.

Target: median = 3. Right now the median is 1. Add or remove dots below — it checks as you go.

Tap the number line to add a value; tap a dot to remove it. Dashed long line = mean (●), dotted line = median (▲).

The values (12)

Mean (share it out equally)6.5 days

sum ÷ count = 78 ÷ 12 = 6.5

Median (the middle value)1 days

0000011414142123

12 values (even), so take the two middle ones: (1 + 1) ÷ 2 = 1.

Mode (most common)0

0 appears 5 times — more than any other value.

Range (spread)23 days

max − min = 23 − 0 = 23

Text version of this activity

A dot plot of 12 monthly counts: 0, 0, 0, 0, 1, 14, 23, 21, 14, 4, 1, 0. Sum 78; mean 78 ÷ 12 = 6.5; ordered, the 6th and 7th values are 1 and 1, so the median is 1; the mode is 0 (five months); the range is 23 − 0 = 23.

Challenges: for median 3, the 6th and 7th ordered values must average 3, e.g. change two dry months to 3 days each; for mean 7 the year needs 84 rainy days, 6 more; for mode 1 only, make more months equal to 1 than to 0, e.g. change two of the 0s to 1 (then 1 appears 4 times, 0 three times); for range 25, set July to 25.

The mean (6.5) is more than six times the median (1): a sign of a lopsided, skewed data set.

Chapter 07

Comparing two groups

The school team needs one more batter. Two players have the same mean over their last 8 innings:

Arjun: 40, 45, 50, 40, 45, 50, 40, 50
Bhavna: 0, 120, 5, 90, 0, 75, 60, 10

Both average 45 runs. So they are equally good? Not so fast. Look at the spread.

TableTwo batters, same mean
Measure	Arjun	Bhavna
Total runs	360	360
Mean	45	45
Median	45	(10 + 60) ÷ 2 = 35
Range	50 − 40 = 10	120 − 0 = 120
Scores under 20	0	4

Lab

Compare two batters with the same mean but very different spreads.

Tap the number line to add a value; tap a dot to remove it. Dashed long line = mean (●), dotted line = median (▲).

The values (8)

Mean (share it out equally)45 runs

sum ÷ count = 360 ÷ 8 = 45

Median (the middle value)45 runs

4040404545505050

8 values (even), so take the two middle ones: (45 + 45) ÷ 2 = 45.

Mode (most common)40, 50

A tie! 2 values each appear 3 times, so there are 2 modes.

Range (spread)10 runs

max − min = 50 − 40 = 10

Text version of this activity

Two dot plots. Arjun (40, 45, 50, 40, 45, 50, 40, 50): all dots packed between 40 and 50. Mean 45, median 45, range 10. Bhavna (0, 120, 5, 90, 0, 75, 60, 10): dots scattered from 0 to 120. Mean 45, median 35, range 120.

Arjun is consistent: you can count on about 45. Bhavna is explosive: she might win a match on her own or be out for 0. Half her innings were 10 or less. Which to choose depends on the situation: a steady opener who protects the start of the innings, or a big hitter to chase a large total quickly. The mean alone cannot make that decision; you need the spread too.

Reflect

You are the captain. Your team needs 180 runs from 20 overs and is 30 for 4. Would you rather send in Arjun or Bhavna next? Would your answer change if you only needed 60 runs to win with plenty of overs left? Use the words mean, median and range.

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Chapter 08

Missing-value detective

A smudge of ink, a torn page, a forgotten entry: real data often has gaps. If you know an average, you can often recover the missing value. The key is always to switch from the mean to the total: sum = mean × count.

Worked example

0 / 4 steps shown

The torn attendance register

A class recorded attendance for 6 days: 32, 35, 30, ?, 34, 33. The mean attendance was 33. How many attended on day 4?

Worked example

0 / 4 steps shown

A value leaves the group

Seven friends have a mean height of 140 cm. One friend, 152 cm tall, moves to another city. What is the mean height of the remaining six?

Worked example

0 / 4 steps shown

A missing value and the median

Five numbers in order are 3, 7, x, 12, 15 and their median equals their mean. Find x.

Try it

Chapter 09

How many should you ask?

In Discover, Class 6B's snack survey of all 30 children gave samosa as the clear winner (9 votes). But suppose the class monitor was in a hurry and stopped after asking only the first few children. Would she have got the same answer?

The survey answers were recorded in the order the children answered. Here is what the leader looked like as more and more answers came in.

TableClass 6B snack survey: the counts after the first n answers
Answers so far	Counts	Mode so far (votes)
5	Samosa 2, Idli 1, Fruit 1, Poha 0, Biscuits 1	Samosa (2)
10	Samosa 4, Idli 3, Fruit 1, Poha 0, Biscuits 2	Samosa (4)
15	Samosa 4, Idli 4, Fruit 4, Poha 0, Biscuits 3	Fruit and Idli and Samosa (4)
20	Samosa 5, Idli 4, Fruit 6, Poha 2, Biscuits 3	Fruit (6)
25	Samosa 7, Idli 5, Fruit 6, Poha 4, Biscuits 3	Samosa (7)
30	Samosa 9, Idli 7, Fruit 6, Poha 5, Biscuits 3	Samosa (9)

Predict first

The monitor stopped after the first 20 answers. Which snack would she have ordered?

Samosa, the true favouriteFruitIdliIt would be a tie

Reflect

You want to know the favourite sport of children in your whole school of 800. You can only ask 50. Describe how you would choose those 50 so the answer is fair. Who might you accidentally leave out if you just asked the first 50 children at the gate?

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Chapter 10

What we found

Here is what our experiments showed, in the language of an investigator's report.

Adding or removing a value moves the mean towards or away from that value, by (distance from the mean) ÷ (new count). The median moves at most half a step along the ordered list.
Outliers drag the mean a long way but barely touch the median. The median is resistant, the mean sensitive.
Adding a constant to every value shifts mean, median and mode by that constant and leaves the range unchanged. Multiplying every value multiplies all of them, range included.
Totals are the key to missing values: sum = mean × count.
Small samples can crown the wrong winner by chance; ask many people, chosen fairly.
Same average, different story: two data sets can share a mean (or a median) and still be very different; always check the spread.

Words to know

All maths vocabulary →

Words from this layer

outlier: A value much larger or smaller than the rest of the data.; Example: ₹400 among amounts of ₹40–₹90.
resistant (robust): Hardly affected by outliers. The median is resistant.
sensitive: Strongly affected by extreme values. The mean and the range are sensitive.
skewed: Lopsided: a few very large (or very small) values stretch the data to one side, pulling the mean away from the median.; Example: Monthly rainfall in Mumbai.
symmetric: Balanced on both sides of the middle, so the mean and median are close.; Example: Heights in one class.
consistent: Having a small spread: values stay close together.; Example: Arjun: 40 to 50 runs every time.
counterexample: One example that shows a general claim is false.; Example: 1, 2, 6 has mean 3, which is not in the data.
assumed mean: A convenient number subtracted from every value to make a mean easier to find, then added back.; Example: Use 100 for 97, 102, 99, 104, 98.

Keep this

Cheat sheet

New value above the mean → mean rises; below → falls; equal → unchanged. Change in mean = (new value − old mean) ÷ new count.
The median moves at most half a step when one value is added; an outlier hardly affects it.
Outlier: far from the rest. Check it; it may be real or a mistake. It drags the mean and stretches the range.
Add k to every value: mean, median, mode + k; range unchanged. Multiply by k: all of them × k, range too.
Missing value: use sum = mean × count, then subtract the known values.
Skewed data (rainfall, incomes): mean pulled towards the long tail, away from the median.
Same mean ≠ same data. Compare spreads (range) to judge consistency.
Sample size: a small or unfair sample can give the wrong mode or mean just by chance.
One counterexample disproves an "always"; a reason is needed to prove one.

Related to

Number and shape patterns

Adding the same number to every term of a sequence shifts its mean by that number, just like shifting data.

Helps you understand

Four operations

Missing-value problems are inverse operations: multiply mean by count to undo the division.

Used in

Electricity

Monthly electricity use is seasonal data; a hot summer is like an outlier that lifts the mean bill.

Where this comes from

Sources

Mathematics, Class 7 (withdrawn edition), Chapter 3: Data Handling — archived chapter PDF (opens another website) — NCERT, archived by the Internet Archiveawaiting owner check
Supports representative values, the arithmetic mean and range (§3.2), mode (§3.3), median (§3.4) and double bar graphs (§3.5). NCERT has withdrawn this book and its replacement, Ganita Prakash Class 7, has no data-handling chapter.
Summarizing quantitative data (opens another website) — Khan Academyawaiting check
Supports mean, median and mode as measures of centre, range as a measure of spread, the effect of outliers, and choosing a measure of centre. Not opened: the site answers automated requests with a bot-challenge page.
Finding a Central Value (Mean, Median and Mode) (opens another website) — Math is Funawaiting owner check
Supports the definitions and methods for mean, median (including an even number of values) and mode, bimodal and multimodal data, and the way one outlier can make the mean a poor summary.
India Meteorological Department (opens another website) — Ministry of Earth Sciences, Government of Indiaawaiting owner check
Supports the India Meteorological Department as the body that records daily temperature and rainfall for Indian cities and publishes forecasts and monsoon information.
Climatological Tables of Observatories in India 1991–2020 (opens another website) — India Meteorological Department, Puneawaiting owner check
Supports the monthly normals used in these lessons: mean daily maximum temperature for New Delhi (Safdarjung) and Chennai (Nungambakkam), and monthly rainfall and rainy days for Mumbai (Santacruz), Mumbai (Colaba) and Chennai (Nungambakkam).
Prasanta Chandra Mahalanobis (opens another website) — Wikipediaawaiting owner check
Supports the founding of the Indian Statistical Institute in December 1931, Mahalanobis’s pioneering of large-scale sample surveys in India, and National Statistics Day on 29 June.
Census of India table A-02: Decadal Variation in Population 1901-2011, India (opens another website) — Office of the Registrar General and Census Commissioner, Indiaawaiting owner check
Supports India’s census population at each census from 1951 to 2011 (1,21,08,54,977 in 2011) and the published decadal growth rates: 21.51, 24.80, 24.66, 23.87, 21.54 and 17.70 per cent.

End of Investigate

What you just read

Predict and test how adding or removing a value changes the mean, median, mode and range.
Explain why an outlier moves the mean a lot but the median very little.
Predict the effect of adding a constant to every value, or multiplying every value.
Build a data set that has a given mean, median, mode and range, and find missing values.
Decide whether statements about averages are always, sometimes or never true, with examples.

The web

Explore a connection

Builds on
Number system
Reading, comparing and rounding numbers comes first when you sort data and round a mean.
Builds on
Four operations
Finding a mean means adding every value and dividing by how many there are.
Uses
Angles
In a pie chart each slice's angle shows a share of the data: 360° stands for the whole.

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Revision 1 · release preview-7e1cbbcc4f · accepted 20/09/2026

The values (8)

The values (11)

Five numbers with mean 7, median 5, mode 4 and range 8

The values (5)

The values (12)

The values (8)

The torn attendance register

A value leaves the group

A missing value and the median

Words from this layer

Experiment check

Cheat sheet

Sources

What you just read

Explore a connection