Class 7 Mathematics Chapter 13: Connecting the Dots

Two kinds of question

Ask how tall the classroom door is. One measurement settles it, and anybody who measures again will get the same answer.

Now ask how tall the pupils in the class are. There is no single answer, because they differ from one another.

A question that needs many answers

That second question can still be answered, but only by collecting many measurements and looking at them together.

A question like that is called a statistical question. It expects variety in its answers.

What makes a question statistical

The test is whether the answers vary. If everybody who answers gives the same reply, the question is not statistical.

If the answers differ from person to person, or day to day, then it is, and data is needed.

Vague questions cannot be answered at all

Some questions are neither kind, because they are too vague to settle. Asking whether pupils travel far to school is one.

Far is not a measurement, and nobody can say when an answer would be right.

Sharpening a question

So a vague question has to be made sharp before any data is collected. Instead of asking whether pupils travel far, ask how each pupil travels to school.

That has definite answers, and we can tell when we have collected them all.

Who to ask

The next decision is who to ask. If the question is about one class, the whole class can be asked, and nobody is left out.

If it is about the whole school, asking one class is not enough.

A sample and the whole group

Asking a part of a group is called taking a sample. A sample saves work, and it can give a good answer if it is chosen fairly.

The whole group is what the question is really about.

When a sample misleads

A sample has to resemble the group. Asking only the pupils who arrive last about how they travel would be a poor sample.

They may travel differently from the others, so the answer would be about them rather than about the school.

Collecting the answers

Now the data can be collected. For the travel question, ask each of the 30 pupils in the class and record every answer as it comes.

Recording as you go matters, because remembering 30 answers is not realistic.

A tally

A tally is the usual way. Write the four travel methods down the page and put one mark beside a method for each pupil who gives it.

Marks are grouped in fives, which makes them quick to count and hard to miscount.

Counting the tally

Counting the marks gives the numbers. In this class, 12 walk, 9 cycle, 6 come by bus and 3 come by car.

Those four numbers are the data, and the question can now be answered.

Checking the total

There is a check worth making at once. The four counts should add to the number of pupils asked.

Twelve and 9 and 6 and 3 make 30, which matches the class. Had they made 29 or 31, a mark has been missed or counted twice.

Data answers one question only

Notice what this data can and cannot do. It tells you how the class travels, and it tells you that walking is the commonest method here.

It does not tell you how far anybody lives, or how long the journey takes. Those are different questions needing different data.

Say what you want before you collect

That is why the question is written down first. Collecting data and then deciding what to ask of it usually leaves you with the wrong data.

A sharp question tells you exactly what to record and when to stop.

Two habits worth keeping

The first is to write the question at the top of the page before anything else, so that it is in front of you while you collect.

The second is to make the total check the moment the counting is done, rather than after the drawing and the conclusions. A missed mark found early costs nothing, and one found late undoes everything built on it.

What carries forward

A statistical question is one whose answers vary, and it needs data rather than a single measurement. A vague question must be sharpened first, since far and often are not measurements.

Decide who to ask: a whole group if you can, or a fair sample that resembles it. Record answers as they come, usually in a tally, and check that the counts add to the number asked.

Scenario:

A class of 30 is asked to find out about travel to school. One pupil proposes the question of whether people travel far. A second says the whole school should be described, and offers to ask the pupils who arrive last, since they are easy to find together. The class then asks its own 30 members how each one travels, tallying the answers, and counts 12 walking, 9 cycling, 6 by bus and 2 by car. A third pupil uses the result to say that most pupils live within walking distance.

Analysis:

One question is too vague, one sample would describe the wrong people, one tally does not balance, and one conclusion goes beyond the data.

Activity:

Design a sharp question about your own class, tally the answers, check the total, and write down one question your data cannot answer.

Title:

Run a survey properly

Safety:

Nothing hazardous here, though ask permission before surveying another class.

What You Notice:

The counts should add exactly to the number of people asked. Any other total means a mark was missed or double counted, which is why the check is made before anything else is done with the data.

A question is only statistical if its answers vary, so the height of a classroom door is not one.

A convenient sample is not the same as a fair one: whoever is easiest to find may be exactly the wrong group.

Tally counts that do not add to the number asked reveal a missed or doubled mark at once, before anything else is done.

If the answers vary, you need data.

Write the question before you collect.

One number to stand for many

Five pupils read 4, 7, 7, 9 and 13 books over a holiday. Somebody asks how much the class read.

Listing all five is honest but awkward. A single number standing for the whole set is easier to use, and there are three of them in common use.

The mean

Add the values and divide by how many there are. Four and 7 and 7 and 9 and 13 make 40, and 40 divided by 5 is 8.

So the mean is 8. It is the figure each pupil would have read if the books had been shared out equally.

The median

Now put the values in order and take the middle one. In order they read 4, 7, 7, 9, 13, and the middle of five values is the third.

So the median is 7. Half the values are at or below it and half at or above.

The mode

The mode is the value appearing most often. Here 7 appears twice and everything else once.

So the mode is 7 as well. It answers a different question: which single result was commonest.

When there is no middle value

Take a set of four: 2, 3, 5 and 8. There is no single middle value, because four values have two in the middle.

So take the mean of those two. The middle pair is 3 and 5, so the median is 4.

And when there is no mode

That same set has no mode at all, because no value appears more than once.

A set can have no mode, or more than one. That is not a fault in the data, it simply means the mode has nothing to say here.

The mean of that set

Two and 3 and 5 and 8 make 18, and 18 divided by 4 is 4.5.

Notice that the mean need not be one of the values, and need not even be a whole number.

A set with one very large value

Now take 5, 6, 7, 8 and 64. Perhaps four pupils read a few books and one read a great many.

The values are in order already, so the median is the middle one, which is 7.

What the mean does here

Add them: 5 and 6 and 7 and 8 and 64 make 90, and 90 divided by 5 is 18.

So the mean is 18. But four of the five pupils read fewer than 9 books, so a mean of 18 describes none of them.

Why that happened

The mean uses every value, so one very large value pulls it a long way. A value far from the rest like this is called an outlier.

The median barely noticed. Moving the 64 to 640 would leave the median at 7 exactly.

So which average to use

Where the values are reasonably close together, the mean is a good summary and uses all the information.

Where there is an outlier, the median is safer, because it reports the middle rather than the balance point.

And where the mode belongs

The mode is the one to use when the values are not numbers at all. The commonest colour or the commonest travel method has no mean.

It is also useful for shoe sizes or shirt sizes, where a shop wants the size most people ask for.

Reporting more than one

Often the honest answer is to give two of them. Saying that the mean is 18 while the median is 7 tells a reader far more than either alone.

A gap between them is itself information: it says the values are not evenly spread.

A check on any average

Two checks are worth making. A mean must lie between the smallest and largest values, so a mean of 18 for data from 5 to 64 is possible while one of 70 would not be.

A median must be one of the values, or halfway between two of them.

What carries forward

The mean is the total divided by the count, the median is the middle value in order, and the mode is the commonest value. A set may have no mode, or more than one.

The mean uses every value, so an outlier pulls it badly: for 5, 6, 7, 8 and 64 the mean is 18 while the median is 7. Use the median when there is an outlier, the mode for things that are not numbers, and report two of them when they disagree.

Scenario:

Five pupils report reading 4, 7, 7, 9 and 13 books, and a teacher wants a single figure for the group. A second class reports 5, 6, 7, 8 and 64. A pupil gives the mean of 18 as the typical number read. A third group of four reports 2, 3, 5 and 8. The same pupil says that data must be faulty, because it has no mode. He also takes the middle of the written list 13, 4, 9, 7, 7 as the median.

Analysis:

One set is straightforward, one has an outlier, one has no mode, and one median was taken without ordering.

Activity:

Collect a set of numbers from your class, find all three averages, and then add one very large value and see which averages move.

Title:

Watch an outlier move the mean

Safety:

Nothing hazardous here.

What You Notice:

The mean starts at 18, with four of the five values below it, and the median is 7. Changing the 64 to 640 lifts the mean to 133 while the median stays at exactly 7, which shows how little the median cares about an outlier.

A mean of 18 can describe nobody at all: for 5, 6, 7, 8 and 64 four of the five values fall below 9.

Changing an outlier from 64 to 640 leaves a median exactly where it was, because a median reports a position rather than a total.

The mode is the only average available for things that are not numbers, since a colour has no total to divide.

Order first, then take the middle.

An outlier moves the mean, not the median.

A table is accurate but slow

Thirty-six pupils name a favourite fruit. Eighteen say mango, 9 say banana, 6 say apple and 3 say orange.

That table is exact. But nothing about it shows at a glance which fruit won, or by how much.

A bar chart

So draw a bar for each fruit. Make the height of each bar match its count.

Now the answer is visible. The mango bar is the tallest, and it is twice the height of the banana bar.

Equal widths, and gaps

Two rules keep a bar chart honest. Every bar must be the same width, and the gaps between them must be equal.

A wider bar looks bigger even at the same height. Equal widths stop the width saying anything at all.

Reading a value off a bar

To read a bar, follow its top across to the scale. The scale must be marked in equal steps.

Here the steps are in threes, so the mango bar reaches 18 and the orange bar reaches 3.

What a bar chart is good at

A bar chart compares amounts. It answers which is biggest, which is smallest, and how two of them compare.

It is the chart to choose when the question is about comparing.

A pie chart

Now draw a circle instead. Divide it into slices, one for each fruit.

Each slice shows that fruit as a fraction of the whole class, rather than as a count.

Each slice is a fraction

Of the 36 pupils, 18 chose mango. That is 18 out of 36, which is one half.

So the mango slice takes half the circle. Nine out of 36 is a quarter, so the banana slice takes a quarter.

Working out a slice in degrees

A full circle is 360 degrees. So a slice is its fraction of 360.

Half of 360 is 180, and a quarter is 90. The apple slice is a sixth, which is 60 degrees, and the orange slice is a twelfth, which is 30.

The slices must make a full turn

There is a check. The four slices should add to 360 degrees.

Here 180 and 90 and 60 and 30 make 360 exactly. Any other total means a slice has been worked out wrongly.

What a pie chart is good at

A pie chart shows shares of a whole. It answers what fraction chose mango, or whether more than half the class agreed.

It is poor at comparing two small slices, because the eye judges angles badly.

Which chart for which question

So the question decides the chart. For comparing amounts, use bars.

For shares of a whole, use a pie. And a pie chart only makes sense when the parts really do make up one whole.

A scale that does not start at zero

Now a warning. Two pupils score 52 and 56 marks in a test.

Drawn from zero, those two bars look almost the same, which is honest: the scores differ by less than a tenth.

The same data, drawn differently

Now draw the same two bars on a scale starting at 50. The first bar is 2 high and the second is 6.

The second now looks three times the first. Nothing in the data changed at all.

Why that misleads

A bar chart invites you to compare heights. If the scale starts above zero, the heights are no longer proportional to the values.

The chart is not lying about the numbers. It is lying about the picture.

How to spot it

So check the bottom of the scale first, before reading any chart. Look at where the axis begins.

If it does not begin at zero, the bar heights cannot be compared by eye, and the numbers must be read instead.

Why a chart is drawn above zero at all

There is sometimes an honest reason. If every value lies between 50 and 60, a chart from zero wastes most of its height and the differences vanish.

So the cut is allowed, but it has to be declared. A broken axis is usually marked, and a careful chart says plainly where its scale begins.

A habit for reading any chart

Read the title, then the axis labels, then the bottom of the scale. Only after that look at the bars themselves.

Three seconds of reading saves you from a picture that was built to be misread.

What carries forward

A bar chart compares amounts, with equal widths, equal gaps and a scale in equal steps. A pie chart shows shares of a whole, where each slice is its fraction of 360 degrees and the slices must total 360.

Choose the chart from the question. And check where the scale starts: an axis beginning above zero makes a small difference look large, which is the commonest way a true chart misleads.

Scenario:

Thirty-six pupils name a favourite fruit: 18 mango, 9 banana, 6 apple and 3 orange. A pupil draws a bar chart, but makes the mango bar wider than the others. He says it deserves to stand out. He then draws a pie chart. He gives the mango slice 180 degrees, the banana 90, the apple 60 and the orange 20. Separately, two test scores of 52 and 56 are drawn on a scale starting at 50. A reader says the second pupil scored three times as well.

Analysis:

One bar chart breaks a rule, one pie chart fails its own check, and one reading is defeated by the scale.

Activity:

Find two charts in a newspaper or online, check where each scale begins, and write down whether the heights can be trusted.

Title:

Draw it twice

Safety:

Take care with the protractor point. No other hazards.

What You Notice:

The bar chart shows mango at twice banana. The slices come to 180, 90, 60 and 30, adding to 360. The two score charts use the same data, yet the second pair of bars looks three times apart while the first pair looks almost level.

A chart can have every number right and still mislead, because the picture it paints is a separate claim from the figures.

Scores of 52 and 56 drawn from a baseline of 50 give bars of 2 and 6, so one looks three times the other.

The slices of a correct pie chart always total 360 degrees, which makes a mis-worked slice easy to find.

Slices must make a full turn.

Check where the scale begins.