Class 8 Mathematics Chapter 12: Tales by Dots and Lines

Two familiar measures

To find the mean of some data, total its values and share that total out over however many values went into it. The median is instead whichever value lands in the middle once the data is sorted. Both are already familiar, and this topic looks at the mean from a new angle.

Start with two numbers

Take 3 and 7. Their mean is 10 over 2, which is 5. Take 8 and 9 instead and the mean is 17 over 2, which is 8.5. Drawn as dots on a line, the mean sits exactly halfway between the two values each time.

A tempting generalisation

That suggests the mean is always the midpoint between the smallest and the largest value. It is a reasonable guess from two numbers, and it is wrong as soon as there are three.

What is actually true

The mean is called a measure of central tendency because it marks the centre of the data. But the centre is not the midpoint of the extremes. It is the place where the total distances on either side are equal.

Seeing it on a set

Take 10, 10, 11 and 17. Their total is 48 and there are 4 values, so the mean is 12. Now measure the distances. Below the mean: 2, then 2, then 1, which is 5 in all. Above it: 5. The two totals match exactly.

Why the midpoint guess fails

The midpoint of the extremes here would be halfway between 10 and 17, which is 13.5. That is nowhere near 12. The three small values pull the balance point down, and the mean records that pull while the midpoint ignores it.

Could there be a second centre?

Suppose some value larger than 12 also balanced the data. Moving right from 12, every distance on the left grows and every distance on the right shrinks, so the left total now outweighs the right. The balance is lost.

The same argument the other way

Moving left from 12 does the reverse: the left distances shrink and the right ones grow. So no value above or below 12 can balance the data either. There is exactly one centre.

Adding a value

What happens to the mean when a new value joins? If the new value is above the mean, it adds to the total on the high side, and the balance point must move up to restore equality. So the mean increases.

And the other cases

A new value below the mean pulls the other way, so the mean decreases. And a value exactly equal to the mean adds nothing to either side, so the mean does not move at all.

Checking that on real numbers

Take 4, 6, 8, 10 and 12, whose mean is 8. Add 20 and the mean becomes 60 over 6, which is 10. Add 8 instead and the mean is 48 over 6, which is still 8. Both predictions hold.

Two values that cancel

This suggests a trick. Add two values, one as far above the mean as the other is below, and the two pulls cancel. Adding 4 and 12 to that same set gives a total of 56 over 7 values, which is 8 again.

Three values that cancel

It works with three as well, so long as the pulls still balance. Two values below the mean can be offset by one far enough above it, which is worth trying out on a dot plot.

Shifting every value

Now a different question. What if every value in the data goes up by the same fixed amount? Raising 4, 6, 8, 10 and 12 each by 5 gives 9, 11, 13, 15 and 17, whose mean is 13. The mean has gone up by 5 as well.

Proving the shift

Let the n values be x1 up to xn, with mean a, so their total is n times a. Adding 3 to every value adds 3n to the total. Dividing the new total by n gives a plus 3. So the new mean is 3 more than the old one.

Scaling every value

What if every value is multiplied instead? Trebling 4, 6, 8, 10 and 12 gives 12, 18, 24, 30 and 36, whose mean is 24. The old mean was 8, so the mean has trebled too.

Proving the scaling

The same algebra works. Multiplying every value by 5 makes the new total 5 times x1 plus 5 times x2 and so on, which the distributive property turns into 5 times the old total. Dividing by n gives 5 times a.

What the two results mean

So shifting the data shifts the mean by the same amount, and scaling the data scales the mean by the same factor. In both cases the mean keeps its relative position among the values, which is exactly what a centre should do.

Scenario:

A teacher has marked a test. The five marks are 4, 6, 8, 10 and 12 out of 15, and she notes the mean. A pupil says the mean must be 8 because that is halfway between 4 and 12. A second pupil arrives late, scores 8 himself, and the first pupil predicts the mean will rise because there is now more data. The teacher then decides to award every pupil 5 bonus marks, and a third pupil claims the mean will be multiplied by 5. All three want to know who is right.

Analysis:

One got the right answer for the wrong reason, one has forgotten the equal case, and one has swapped the shift result for the scale one.

Title:

Shift and scale a set of data

Safety:

Nothing here is risky, but write the first mean down before you change anything.

What You Notice:

The first change adds 7 to the mean and the second multiplies it by 4. Both predictions can be made without adding the new values up at all, which is the point of proving them.

The mean is the only value in a set of data with the balance property. Shift away from it in either direction and one side immediately outweighs the other.

Two values can be added to a set without moving its mean at all, so long as one sits as far above the mean as the other sits below.

Distances balance, not endpoints — The mean is not halfway between the smallest and the largest. It is where the pull from each side is equal.

Back to the median

The median is the middle value once the data has been sorted, so that as many values lie below it as above it. Unlike the mean, it answers to counts rather than to distances, and that makes it move differently.

Adding a value above it

Take a set whose median is 8 and add the value 11. The new value is above the old median, so there are now more values above 8 than below it, and 8 can no longer be the middle. The median must rise.

Where it lands

With an even number of values there is no single middle one, so the median is the average of the two middle values. Here those are 8 and 11, so the new median is 9.5. A value below the median would have pulled it down in the same way.

A value nobody wrote down

Coach Balwan recorded the weights of ten wrestlers and their mean, but one weight got smudged. The nine readable weights are 42, 40, 39, 33, 48, 38, 42, 35 and 32, and the mean was 39.2 kg.

Running the mean backwards

The mean is the total divided by the number of values, so the total is the mean times that number: 39.2 times 10, which is 392. The nine readable weights total 349, so the missing weight is 392 less 349, which is 43 kg.

When a value turns out to be wrong

Venkayya averaged his coconut harvest at 25.6 per tree across 15 trees. His son then found that one tree had been written down as 3 more than it really yielded. What is the correct average?

Solving it without the data

The individual counts are not given, and they are not needed. The total must be 25.6 times 15, which is 384. Three of those were never picked, so the true total is 381, and the correct average is 381 over 15, which is 25.4.

Why that worked

Notice what made it possible. A mean carries all the information about the total, so any question about the total can be answered from the mean alone. The separate values only matter when the question asks about one of them.

Data that repeats

Now a different shape of problem. A class records how many people live in each pupil’s family, and the answers run from 3 to 10. Rather than a long list, the data is collected into a table of frequencies.

A tempting wrong average

The table lists eight different family sizes, from 3 to 10, and adding those and dividing by 8 gives 52 over 8, which is 6.5. That is wrong, and it is wrong for one clear reason.

What the frequencies are for

Finding a mean means adding all the VALUES in the data, not all the different values. Three occurs three times over, 4 occurs eleven times, and so on. A calculation that ignores the frequencies has thrown most of the data away.

Doing it properly

Multiply each value by its own frequency and add those products: 3 times 3, plus 4 times 11, plus 5 times 9, and so on. That gives 9 plus 44 plus 45 plus 42 plus 21 plus 8 plus 9 plus 10, which is 188.

Dividing by the right number

The divisor must be the number of values in the data, which is the total of the frequencies. That total is 3 plus 11 plus 9 plus 7 plus 3 plus 1 plus 1 plus 1, which is 36. So the mean family size is 188 over 36, which is 5.22.

The median of the same table

There are 36 values, an even number, so the median is the average of the 18th and 19th values in order. Writing out all 36 numbers would work, but there is a much quicker way.

Counting up the frequencies

Add the frequencies from the smallest value upwards. Three gives 3, so positions 1 to 3 hold the value 3. Adding the 11 fours gives 14, so positions 4 to 14 hold 4. Adding the 9 fives gives 23, so positions 15 to 23 all hold 5.

Reading off the answer

Positions 18 and 19 both fall inside that range, so both of those values are 5, and the median is 5. The whole thing took one column of addition and no sorting at all.

A median with room to move

One last point about the median. Take the eleven values 12, 47, 8, 73, 18, 35, 39, 8, 29, 25 and p. The median is 29 when p is 29, 30, 40, 47 or 100, but not when p is 10 or 25.

Why that is

With eleven values the median is the sixth in order. Without p, the sorted list puts 29 sixth already, so any p at or above 29 leaves it there. A p below 29 pushes an extra small value in front of it, and the sixth place falls back to 25.

Scenario:

A games teacher is checking three records. Ten shot-put throws have a mean of 8.4 metres, but one throw is illegible, and the nine readable ones total 75 metres. A table of press-up counts lists the values 5 to 9 with frequencies 4, 6, 8, 5 and 2, and an assistant has averaged the five values and reported 7. Finally the teacher notes that last term’s median height was 142 cm, and a new pupil of 150 cm has just joined.

Analysis:

One record needs the mean run backwards, one ignores the frequencies, and one asks which way a median moves.

Title:

Hide a value and recover it

Safety:

Keep the scissors pointing away from you and cut on a flat table.

What You Notice:

One hidden value can always be recovered, because the mean fixes the total exactly. Two cannot: the mean fixes only their sum, and many pairs share a sum.

A mean can be run backwards even when none of the individual values are known. Fifteen trees averaging 25.6 coconuts must have yielded 384 between them, whatever each tree did.

Cumulative frequencies find a median without writing out the data. Thirty-six family sizes can be put in order on one line of arithmetic.

Mean times how many gives the total — Whenever a value is missing, turn the mean back into a total first. Everything else follows from that total.

Another way to draw data

Pictographs, bar graphs and dot plots are already familiar. A line graph is one more: the values are marked as points and then joined up by lines. That joining is the whole point of it.

What the joining is for

Lines between points suggest movement from one to the next, which makes a line graph the natural picture for data across TIME. A clustered column graph can carry the same numbers, but the line shows the direction of travel at a glance.

Two steps, kept apart

Reading a graph well means doing two separate things, and not muddling them. The first step is to identify what is given. The second is to infer and interpret. Rushing to the second is where most misreadings come from.

Step one in practice

Take a graph of monthly maximum temperature for Kerala and Punjab. Notice how the data is organised: each state has its own marked points joined by its own line. Notice the axes: months along the bottom, temperature in degrees up the side.

Why the markers differ in shape

The two series are drawn in different colours, but also with different marker shapes. That matters. A graph printed in black and white, or read by someone who finds colours hard to tell apart, still has to be distinguishable.

Step two for Punjab

Now interpret. Punjab’s monthly maximum climbs from about 19 degrees in January to a high of 38 in June. It then drops to just under 35 in July, stays near there until September, and falls steadily to about 23 by December.

Step two for Kerala

Kerala behaves quite differently. Its line stays nearly flat all year, peaking around 33 degrees in April and dipping to around 29 in July. Its monthly maximum in summer is much the same as in winter.

Putting the two together

The comparison is the useful part. Punjab reaches both colder and warmer figures than Kerala, swinging about 19 degrees across the year against Kerala’s 4. Punjab’s spread is close to five times as wide.

Where the numbers came from

It is worth asking how such a figure is produced. A state has several weather stations recording local temperature, and the monthly maximum is the largest value recorded anywhere in the state that month.

Why that question matters

Knowing how data was gathered tells you what it actually covers. It also shows up the limits: a missing station, or a bias in where the stations are, affects what the graph can honestly be used to say.

Questions a graph provokes

A good graph raises questions it cannot answer. Why do these two states behave so differently? What else decides a region’s temperature? Which other states look like Punjab, and what do they have in common? None of that is a failing of the graph. Showing a pattern clearly enough to provoke the question is most of what a graph is for.

Infographics

A line graph is not the only compact picture. An infographic can carry a great deal at once. One example maps, state by state, the difference between how much rice and how much wheat each person eats, shading each state by that difference.

Reading its scale carefully

The scale runs from minus 100 to plus 100, with a colour key in the corner. Plus 100 marks the largest difference in favour of rice. It does NOT mean the state eats no wheat at all, which is a quite different claim.

What such a picture can show

Read properly, the map shows something a table would hide: the preferences split along a clear geographical line. States near the middle of the scale are the ones eating roughly as much of one as the other.

Judging a claim against a graph

The last skill is the most useful. Given a graph and a statement, ask whether the graph supports it, contradicts it, or says nothing about it at all. The third case is the one people miss.

An example of the third case

A graph of lighting sources over the years can show that kerosene use fell and electricity use rose. It cannot show that there were no power cuts in 2023. Nothing on those axes records power cuts at all.

And a claim about a share

Equally, a graph showing that 10 per cent of urban households used electricity in some year supports a statement about that share. It supports nothing about how many households there were, unless the graph records that too.

Scenario:

A school magazine prints a line graph of monthly maximum temperature for Kerala and Punjab, with months along the bottom and degrees up the side. Four pupils each pin a claim of their own beneath it. The first of them says Punjab varies far more across the year than Kerala. The second says Punjab is warmer than Kerala in every month. The third says the graph shows why the two states differ. The fourth says the graph proves that July was the wettest month in Kerala.

Analysis:

One claim the graph supports, one it contradicts, one asks for a cause, and one is about a quantity the graph never measured.

Title:

Test four claims against one graph

Safety:

Use a graph that is already published. Ask before cutting up anything that belongs to someone else.

What You Notice:

The disagreements are almost always about the third category. A claim that sounds reasonable can still be one the graph never measured.

Giving each series its own marker shape, not just its own colour, keeps a graph readable in black and white and for readers who find colours hard to tell apart.

On the rice and wheat infographic, a value of plus 100 does not mean a state eats no wheat. It marks the largest difference in favour of rice, which is a different claim altogether.

Identify before you infer — Write down what the graph actually shows first. Only then say what it means.