Part II, Chapter 5Tales by Dots and Lines

Collection A: 6, 7, 8

Mean \(=\dfrac{6+7+8}{3}=\dfrac{21}{3}=7\)

Collection B: 3, 6, 9

Mean \(=\dfrac{3+6+9}{3}=\dfrac{18}{3}=6\)

Collection C: 2, 4, 9

Mean \(=\dfrac{2+4+9}{3}=\dfrac{15}{3}=5\)

Collection D: 4, 11, 15

Mean \(=\dfrac{4+11+15}{3}=\dfrac{30}{3}=10\)

The orange lines show the distance of each value from the mean.

Take collection C: 2, 4, 9 with mean 5.

Values to the left of 5 are 2 and 4.

Their distances from 5 are \(5-2=3\) and \(5-4=1\), a total of 4.

The value to the right of 5 is 9.

Its distance from 5 is \(9-5=4\).

Total distance on the left \(=\) total distance on the right \(=4\).

Take collection D: 4, 11, 15 with mean 10.

Left: \(10-4=6\).

Right: \((11-10)+(15-10)=1+5=6\).

Again the two totals are equal.

So the mean is the point on the number line where the data balances, like the pivot of a see-saw.

This is why it is called the centre of the collection.

Collection E: 11, 13, 17, 19

Mean \(=\dfrac{11+13+17+19}{4}=\dfrac{60}{4}=15\)

Collection F: 5, 6, 15, 16

Mean \(=\dfrac{5+6+15+16}{4}=\dfrac{42}{4}=10.5\)

Collection G: 10, 10, 11, 17

Mean \(=\dfrac{10+10+11+17}{4}=\dfrac{48}{4}=12\)

Collection H: 3, 5, 10, 12

Mean \(=\dfrac{3+5+10+12}{4}=\dfrac{30}{4}=7.5\)

The orange lines show the distance of each value from the mean.

Collection E

Extremes are 11 and 19, so the midpoint \(=\dfrac{11+19}{2}=15\).

The mean is also 15.

Collection F

Extremes are 5 and 16, so the midpoint \(=\dfrac{5+16}{2}=10.5\).

The mean is also 10.5.

Collection G

Extremes are 10 and 17, so the midpoint \(=\dfrac{10+17}{2}=13.5\).

The mean is 12, which is not 13.5.

Collection H

Extremes are 3 and 12, so the midpoint \(=\dfrac{3+12}{2}=7.5\).

The mean is also 7.5.

In E, F and H the values are placed symmetrically, so the two coincide.

In G, three values are bunched near 10 while 17 is far away, so the mean is pulled towards the bunch.

The mean is still the centre: left distance \(=2+2+1=5\) and right distance \(=5\).

For each collection, find the mean.

Then add the distances of the values lying to the left of the mean (LHS) and to the right of the mean (RHS).

In every collection LHS \(=\) RHS.

Take the collection 10, 10, 11, 17 with mean 12.

Remove 17 (greater than the mean)

New collection: 10, 10, 11.

New mean \(=\dfrac{31}{3}\approx10.33\), which is less than 12.

Remove 10 (less than the mean)

New collection: 10, 11, 17.

New mean \(=\dfrac{38}{3}\approx12.67\), which is more than 12.

Remove a value equal to the mean

Take 10, 12, 14 with mean 12 and remove 12.

New collection: 10, 14, with mean \(\dfrac{24}{2}=12\).

Why

The mean is the balancing point.

Removing a value above it takes weight away from the right, so the balance point moves left.

Removing a value below it takes weight away from the left, so the balance point moves right.

A value sitting at the balance point has distance 0, so removing it changes nothing.

Take 10, 10, 11, 17 with mean 12 (sum 48, 4 values).

Include 12: new mean \(=\dfrac{48+12}{5}=\dfrac{60}{5}=12\).

Remove 12 from 10, 12, 14: new mean \(=\dfrac{24}{2}=12\).

Fair-share meaning

The mean is the amount each person would get if the total were shared equally.

In the first group everyone would get 12.

A new person who already holds exactly 12 neither needs anything from the others nor gives anything to them.

So the fair share of every person is still 12.

Likewise, if a person holding exactly 12 leaves, the rest still share equally at 12 each.

Algebra

Let the mean of \(n\) values be \(a\), so their sum is \(na\).

Including \(a\) gives \(\dfrac{na+a}{n+1}=\dfrac{(n+1)a}{n+1}=a\).

The data has 17 values with sum 153, so the mean is \(\dfrac{153}{17}=9\).

Including 2 values

Include 7 and 11.

7 is 2 below the mean and 11 is 2 above it, so the distances cancel.

New mean \(=\dfrac{153+7+11}{17+2}=\dfrac{171}{19}=9\).

Removing 2 values

Remove 8 and 10.

New mean \(=\dfrac{153-8-10}{17-2}=\dfrac{135}{15}=9\).

Remove 5 and 13.

New mean \(=\dfrac{153-5-13}{15}=\dfrac{135}{15}=9\).

Rule

The two values must lie at equal distances on opposite sides of 9.

That is, the two values must add up to \(2\times9=18\).

Use the data above: 17 values, sum 153, mean 9.

The 3 values must have sum \(3\times9=27\).

Then their distances above and below 9 cancel.

Including 3 values

Include 7, 7 and 13.

Sum \(=27\).

New mean \(=\dfrac{153+27}{20}=\dfrac{180}{20}=9\).

Removing 3 values

Remove 7, 8 and 12.

Sum \(=27\).

New mean \(=\dfrac{153-27}{14}=\dfrac{126}{14}=9\).

The mean is 9.

Choose two values above 9, say 11 and 11.

Their distances above the mean are \(2+2=4\).

To balance, the value below 9 must be 4 less than 9, that is, 5.

The sum of the 3 values is \(11+11+5=27=3\times9\).

New mean \(=\dfrac{153+27}{20}=9\).

Other choices work too, for example 10, 12 and 5, since \(1+3=4\).

Let the values be \(x_1, x_2, x_3, \dots, x_n\) with average \(\dfrac{x_1+x_2+x_3+\dots+x_n}{n}=a\).

After subtracting 2 from every value, the new average is

\(\dfrac{(x_1-2)+(x_2-2)+\dots+(x_n-2)}{n}\)

\(=\dfrac{x_1+x_2+\dots+x_n-2n}{n}\)

\(=\dfrac{x_1+x_2+\dots+x_n}{n}-\dfrac{2n}{n}\)

\(=a-2\)

The average is the equal share each person gets when the total is divided equally.

Suppose 5 friends have an average of 20 stickers each, so the total is 100.

Now every friend gets 3 more stickers.

The new total is \(100+5\times3=115\).

Shared equally, each gets \(\dfrac{115}{5}=23=20+3\).

In general, if every one of \(n\) people gets \(k\) more, the total grows by \(nk\), so each equal share grows by \(\dfrac{nk}{n}=k\).

In the same way, if every person loses \(k\), the average falls by \(k\).

B7 is not a column.

It is a single cell.

Column B holds the marks in Odia.

Row 7 is the row of Gowri.

The cell where column B and row 7 meet holds Gowri's marks in Odia.

Ashwin's marks are in row 4 (cells B4 to G4).

Odia 29: not more than 30.

Telugu 31: more than 30.

English 33: more than 30.

Maths 34: more than 30.

Social Science 30: not more than 30, since it is exactly 30.

Science 28: not more than 30.

Science is in column G.

The 22 students are in rows 2 to 23.

Science marks (G2 to G23): 37, 39, 28, 44, 36, 30, 20, 46, 40, 18, 43, 26, 33, 22, 34, 40, 39, 35, 18, 30, 42, 29.

The formula is =AVERAGE(G2:G23).

Check by hand: sum of the Science marks \(=729\), so the average \(=\dfrac{729}{22}\approx33.14\).

Type =AVERAGE(B2:B23) for Odia and =AVERAGE(C2:C23) for Telugu.

Odia marks (B2 to B23): 25, 41, 29, 47, 33, 27, 16, 31, 39, 12, 48, 25, 29, 23, 34, 35, 42, 37, 15, 29, 41, 29.

Telugu marks (C2 to C23): 39, 43, 31, 46, 35, 29, 19, 38, 37, 17, 45, 28, 36, 25, 36, 42, 45, 32, 18, 24, 44, 30.

Odia: sum \(=687\), average \(=\dfrac{687}{22}\approx31.23\).

Telugu: sum \(=739\), average \(=\dfrac{739}{22}\approx33.59\).

\(31.23\lt 33.59\)

The last student is in row 23, so the averages go in row 24.

In B24 type =AVERAGE(B2:B23).

Copy the formula to the right, up to G24.

The column letter changes automatically in each cell.

Type Total in cell H1.

In H2 type =SUM(B2:G2), which adds the six subject marks of Ratna.

Copy the formula down to H23.

The row number changes automatically, for example H3 becomes =SUM(B3:G3).

(i) First 50 natural numbers: 1, 2, 3, …, 50

Sum \(=\dfrac{50\times51}{2}=1275\)

Mean \(=\dfrac{1275}{50}=25.5\)

(ii) First 50 odd numbers: 1, 3, 5, …, 99

Sum \(=50\times50=2500\)

Mean \(=\dfrac{2500}{50}=50\)

(iii) First 50 multiples of 4: 4, 8, 12, …, 200

Sum \(=4\times1275=5100\)

Mean \(=\dfrac{5100}{50}=102\)

Observations

In (i), the mean \(25.5=\dfrac{1+50}{2}\), the average of the first and the last number.

In (ii), the mean \(50=\dfrac{1+99}{2}\).

In (iii), the mean \(102=\dfrac{4+200}{2}\).

The numbers in each list are equally spaced, so the mean is exactly the middle of the list.

The mean of

(iii) is 4 times the mean of (i), because every number in

(iii) is 4 times the number in (i).

The list is already in order and has 16 values.

The median is the average of the 8th and 9th values.

8th value \(=41\) and 9th value \(=41\).

Median \(=\dfrac{41+41}{2}=41\).

(i) Including one value

There will be 17 values, and the median is the 9th value.

If the new value is 41 or less, it goes before the 9th place, so the old 8th value 41 becomes the 9th.

If the new value is 41 or more, it goes after the 9th place, so the 9th value stays 41.

In both cases the 9th value is 41.

(ii) Including two values

There will be 18 values, and the median is the average of the 9th and 10th values.

To keep both middle values equal to 41, one new value must be 41 or less and the other 41 or more.

For example, 20 and 60, or 41 and 41.

If both new values are less than 41, the median becomes less than 41.

If both are greater than 41, the median becomes more than 41.

(iii) Removing one value

There will be 15 values, and the median is the 8th value.

Remove a value below 41: the 8th value is the old 9th value, 41.

Remove a value above 41: the 8th value is still the old 8th value, 41.

Remove one of the 41s: the 8th value is the remaining 41.

The earlier clustered-column graph shows the monthly maximum temperature of Kerala and Punjab in 2023, using two bars for each month.

The line graph shows the same 12 values for each state.

Each value is marked by a point, and the points are joined by lines.

The information is the same, but the line graph makes the change from month to month easier to follow.

It shows at a glance that Punjab rises to a peak in June and then falls, while Kerala stays almost flat.

What the graph shows

In Punjab, the monthly maximum temperature rises from about 19 °C in January to about 38 °C in June.

It then dips to just under 35 °C in July, stays almost flat till September, and falls to about 23 °C in December.

In Kerala, the temperature stays between about 29 °C and 33 °C all year.

The two lines cross around April and again around November.

Punjab varies by about \(38-19=19\) °C over the year, Kerala by only about \(33-29=4\) °C.

Questions that come up

Why are the trends so different in the two states?

What decides a place’s temperature, such as how far it is from the equator, the sea and the mountains?

Do other states show trends like Punjab or like Kerala?

What would a graph of the monthly minimum temperatures look like?

Countries and agencies report the objects they launch, such as satellites, probes, landers, crewed spacecraft and parts of space stations.

These reports are collected by the United Nations Office for Outer Space Affairs.

For each year, the objects reported for a country are counted to get that country’s line.

The counts of all countries added together give the line for the whole world.

Since the lines of the four countries do not add up to the world line, the remaining countries are not drawn.

Read the graph (values are approximate).

Statement 1: the worldwide count increased every year

Not valid.

The line dips from 2017 to 2018 (about 500 to 470) and again from 2023 to 2024 (about 2920 to 2870).

Statement 2: the USA launched about \(\dfrac{3}{4}\) of the world count in 2022–24

Valid.

The ratio is about 0.77 to 0.79, which is close to \(\dfrac{3}{4}=0.75\).

Statement 3: Nepal launched nothing in 2012–24

Not valid.

Nepal is not shown in the graph, so we cannot say anything about it.

A country that is not shown may still have launched objects.

Statement 4: China and Russia together launched about 400 in 2024

Valid, approximately.

China is about 300 and Russia about 100 to 150 in 2024, so together they are about 400 to 450.

Find the ratio of each year’s count to the previous year’s count (world line).

The line rises most steeply between 2016 and 2017, and between 2019 and 2020.

In every other pair of consecutive years the count grows by less than 2 times.

The six cities are Kovalam, Udupi and Mumbai (first graph) and Rameswaram, Chennai and Puri (second graph).

Grouping

Kovalam, Udupi and Mumbai lie on the west coast.

Rameswaram, Chennai and Puri lie on the east coast.

Observations

The west-coast cities Udupi, Mumbai and Kovalam get their peak rainfall in June to August.

Rameswaram gets most of its rain in October to December.

Chennai starts getting rain from June, peaks in November and continues till December.

Puri, though on the east coast, gets its peak rainfall in July to September.

January to March are dry months for all six cities.

Inference

The regions along the west coast receive more rain.

The two coasts get their rain in different seasons: the south-west monsoon (June to September) mainly brings rain to the west coast,

and the north-east monsoon (October to December) brings rain to the south-east coast.

The south-west monsoon blows from the Arabian Sea and the Bay of Bengal towards India from about June to September.

It brings most of the yearly rain to the west coast (Kovalam, Udupi, Mumbai), the Western Ghats, central India and the north-east.

That is why Kovalam, Udupi and Mumbai have their peak rainfall in June to August in the graph.

The north-east monsoon blows from the land towards the sea from about October to December.

It brings rain to the south-east coast: Tamil Nadu (Chennai, Rameswaram), south coastal Andhra Pradesh and Puducherry.

That is why Rameswaram and Chennai have their peak rainfall in October to December in the graph.

Puri, on the east coast of the north, gets its rain mainly in July to September, from the south-west monsoon.

Observations

The states in the north-west (Rajasthan, Haryana, Punjab, Madhya Pradesh, Delhi) have negative values, so they prefer wheat.

The states in the south, the east and the north-east have positive values, so they prefer rice.

The red line separates the two regions, running from the north-east of the country to the south-west.

The national value is +13.48, so on the whole rice wins, but only by a small margin.

(i) Karnataka

Karnataka is shaded in the same dark purple as its neighbours Tamil Nadu (+85) and Andhra (+92).

Goa (+57) and Kerala (+79) nearby are slightly lighter, and Maharashtra (−15) is pale.

So a reasonable guess is a value in the +80s, say about +85.

(ii) Top 5 states for rice

Manipur (+100), Nagaland (+99), Mizoram (+97), Tripura (+96) and Meghalaya (+95).

(iii) Top 5 states for wheat

Rajasthan (−93), Haryana (−81), Punjab (−78), Madhya Pradesh (−60) and Delhi (−45).

(iv) Balanced preference

Values close to 0 mean both are eaten about equally.

Bihar (+3) is the most balanced.

Maharashtra (−15), Uttarakhand (−18) and Himachal Pradesh (−19) are also fairly balanced.

(i) What each colour stands for

How to tell: the long light-blue block at night is sleeping.

Orange fills the school hours 9:30 am to 4:30 pm on a school day, so it is attending classes, studying and homework.

Green is a short block near 9 am, 12:30 pm and 7 pm, which are meal times, so it is eating.

Grey appears just before or after going somewhere, so it is travelling.

Purple comes soon after waking up, so it is showering, getting dressed and exercise.

Yellow is the remaining time, which is for friends, hobbies, media and family.

(ii) Which day is which

Strip 1 is Friday: orange from 9:30 am to 12:30 pm and from 1:30 pm to 4:30 pm matches a school day of 9:30 am to 4:30 pm.

Strip 2 has the most yellow (about 7 hours of friends, hobbies and media) and the least study, so it is the free day, Sunday.

Strip 3 has a leisure block in the morning and a long study block from 3 pm to 6:30 pm, so it is most likely Saturday.

This is a reasonable reading, but the strips alone do not prove it.

(iii) The long movie

Look for a long yellow block with grey (travelling) on both sides.

Strip 3 has grey from 8:30 to 9:00 am, then yellow from 9:00 am to 12:30 pm (3½ hours), then grey from 1:00 to 1:30 pm.

So the movie was most likely on the weekend morning of Strip 3, from about 9:00 am to 12:30 pm.

(iv) Lunch break at school

On Strip 1, classes run from 9:30 am to 12:30 pm.

Then there is green (eating) from 12:30 pm to 1:00 pm and grey from 1:00 pm to 1:30 pm.

Classes start again at 1:30 pm.

So the lunch break begins at about 12:30 pm and lasts about one hour, till 1:30 pm.

(v) What more the strips tell us

Adding up the boxes gives the time spent on each activity:

He sleeps about 10½ hours on every day, goes to bed by about 8:30 pm and wakes up between 6:30 and 7:00 am.

On the school day he spends about 7 hours in classes and about 1½ hours more on homework;

on the weekend he spends about 7 hours on friends and hobbies on Sunday.

On Strip 3 he also takes a 1-hour nap (light blue) at 2:00 pm.

This is a personal question, so the answer is your own data.

How to make your strip

Take a strip of 48 boxes, one for each half hour from midnight.

Recall what you did in each half hour and colour it using the same six colours as Manoj.

How to compare

Count the boxes of each colour and divide by 2 to get hours.

Compare with Manoj’s school day (Strip 1): about 10½ hours of sleep, 7 hours of classes and 1½ hours of homework, about 3 hours for friends, hobbies and media.

On a vacation day, expect less orange (classes and homework) and more yellow (friends, hobbies, media), like Manoj’s Sunday.

Ask an adult at home to tell you what they did in each half hour of a day.

Colour a strip for them in the same way.

Place it next to your own strip and compare.

Things to look for

Who sleeps longer, and who wakes up earlier?

An adult’s strip usually has a different colour for work in place of classes, and more grey for travelling to work.

Is there a long block of orange (work or study) in both strips?

How much yellow (free time) do you have, compared with them?

Reading the graph

Sleep is highest for the youngest children: about 9.5 hours a day at age 6.

It then decreases through the teenage years and early adulthood.

Between ages 30 and 50 it is lowest, at about 8 hours a day.

After age 50 it increases again, reaching about 8.5 hours near age 70 and a little more after that.

The two graphs

Graph 1 starts at 0, so the curve looks almost flat and the pattern is hard to see.

Graph 2 is zoomed in to 7 to 12 hours, so the fall and the rise are easy to see.

The same data can look very different when the scale is changed.

Why a smooth curve?

The graph has about 80 data points placed very close to each other, one for each age, so the line looks like a smooth curve.

A column graph would need about 70 columns and would look cluttered.

Method 1: find the mean

Number of values \(=25\) and sum \(=443\).

Mean \(=\dfrac{443}{25}=17.72\)

This is not 17.

Method 2: check the balance

Total distance of the dots to the left of 17: \(3\times2+2\times2+1\times3=13\).

Total distance of the dots to the right of 17: \(1\times4+2\times4+3\times3+4\times1+6\times1=31\).

For 17 to be the mean, the two totals would have to be equal, but \(13\ne31\).

The right side is heavier, so the mean is greater than 17.

17 is the value that occurs most often (the mode), not the mean.

Mean

Change in the total weight \(=-2+1+1=0\) kg.

The number of people has not changed.

So the mean \(=\dfrac{\text{total}}{\text{number of people}}\) is unchanged, at 65.3 kg.

Median

The median depends on the order of the weights, so it depends on who changed.

If the changes happen away from the middle, the median stays at 67 kg.

If the person at the median gains or loses weight, the median may change.

We do not know the weights of these three people, so we cannot say how the median changes.

This is a project based on your own data, so there is no single answer.

Here is the method, worked out on Manoj’s three strips.

(i) Regular times

Make a strip for each of 5 to 7 days, using the same colours.

Check whether the green boxes (eating) and the start and end of the light-blue block (sleeping) fall at about the same time each day.

To find the time outdoors, mark the time you spend outside (playing, walking, travelling outside) and add it up in hours.

(ii) Average day

For each colour, add the boxes on all days and divide by the number of days.

For Manoj’s three days, the average sleep is \(\dfrac{21+21+21}{3}=21\) boxes, which is 10.5 hours.

The hours for all activities are in the table below.

(iii) Adult at home

Make the same strips for an adult for the same days.

Compare the hours of each colour: sleep, work, travel, free time.

This is a project based on data that your group collects, so there is no single answer.

Here is the method.

(i) Sleep time of family members

(a) For each person and each day, colour the sleep boxes (night sleep, naps) on a strip and count them.

Divide by 2 for hours.

(b) Group the people as children, adults and elderly.

For each group, write the daily sleep times of all its members for the whole week as one list.

Average \(=\dfrac{\text{sum of all the sleep times}}{\text{number of values}}\).

Median: arrange the values in order and take the middle value (or the average of the two middle values).

Example of the method: if the daily sleep times of the children are 9, 9.5, 9, 10, 8.5 hours, the average is \(\dfrac{46}{5}=9.2\) hours and the median is 9 hours.

(c) Compare the three groups.

Do younger people sleep longer, as in the sleep graph?

(ii) School timings

Ask for the start time, the end time and the break times of Grade 8 in different schools.

For each school, find the school day in hours: for Manoj it is 9:30 am to 4:30 pm, which is 7 hours.

Subtract the break time to get the class time.

Present the data in a table and a dot plot, and find the average and median length of the school day.

(i) Amavasya and purnima

On amavasya (new moon) the moon is in the same direction as the sun.

So it rises with the sun, at about 6 am, and sets with the sun, at about 6 pm.

On the graph, this happens on the 28th and 29th: moonrise is about 6:15 am and 7:00 am, and moonset about 5:05 pm and 6:10 pm.

On purnima (full moon) the moon is opposite to the sun.

So it rises at about 6 pm, when the sun sets, and sets at about 6 am, when the sun rises.

On the graph, this happens on the 13th and 14th: moonrise is about 5:30 pm and 6:30 pm, and moonset about 6:40 am the next morning.

(ii) What do you notice?

The moonrise and moonset times both shift later every day, by about 50 minutes a day.

When a line reaches midnight it wraps around: moonrise goes from about 11:50 pm on the 20th to 12:40 am on the 22nd, and there is no moonrise at all on the 21st.

In the same way there is no moonset on the 6th.

The new moon and the full moon are about 15 days apart, which is half of the lunar month (about 29.5 days).

What do you wonder?

Why is the moon about 50 minutes later every day, when the sun is at the same time every day?

Is the pattern the same every month?