Tales by Dots and Lines

Why This Matters

Open any newspaper. You will see numbers everywhere. The average rainfall this year. The middle income of a town. A graph of cricket scores match by match. A chart of who likes which subject in a class.

These are not just numbers. They are stories. A line that climbs tells the story of something growing. A short bar next to a tall one tells you one thing is far less common than another. A single dot sitting far away from the rest tells you something unusual happened.

But here is the catch. The same data can be told in honest ways and misleading ways. A shop owner could say “our customers spend ₹500 on average” — and that could be true even if almost everyone spends ₹100 and one rich customer spent ₹5000. The “average” hid the real story.

This chapter teaches you to read these stories carefully and tell them clearly. You will understand what an average really means (it is a balance point, not just a sum-divided-by-count). You will learn when the average lies and the median tells the truth. And you will learn to draw and read the pictures — dot plots, bar graphs and line graphs — that turn a pile of numbers into something you can see in one glance.

The Big Idea

The mean (average) of data is a balance point: it sits at the spot where the total distance to the values on its left exactly equals the total distance to the values on its right. The median is simply the middle value when the data is lined up in order. The mean uses every value, so one far-away value (an outlier) can drag it; the median only watches the middle, so it stays steady. To see data instead of just listing it, we draw dot plots (a dot for each value), bar graphs (bar height shows a count) and line graphs (a line shows how something changes over time).

Let’s Break It Down

Last year you met the mean and the median. We will not just repeat them. We will look at them in a fresh way and learn why they behave the way they do. Let us first jog your memory.

The mean is a balance point

Take just two numbers, say 3 and 7. Their mean is (3 + 7) ÷ 2 = 5. Notice something: 5 is exactly halfway between 3 and 7. It is 2 away from 3 on the left and 2 away from 7 on the right. The two distances are equal.

This is the real secret of the mean. The mean is the point where the data “balances”. Imagine the values as little weights placed along a ruler, and imagine the ruler resting on a single support, like a see-saw. The mean is exactly where you must put the support so the see-saw does not tip either way.

But be careful. People sometimes think the mean is just the middle of the smallest and largest value. It is not. What is really equal is the total distance on each side. Let us see this with the data 10, 10, 11, 17, whose mean is 12. Figure 5.1 below shows the see-saw.

Tap to zoom

Let us check that arithmetic, because it is the whole point. To the left of 12: the value 10 is 2 away, the second 10 is also 2 away, and 11 is 1 away. Add those distances: 2 + 2 + 1 = 5. To the right of 12: only 17, which is 17 − 12 = 5 away. Both sides total 5. They balance.

Total distance to the values on the left = total distance to the values on the right.

This also explains why there is only one mean. Suppose you tried a centre bigger than 12, say 13. Then every left value is now further away (their distances all grow) and the right value 17 is now closer (its distance shrinks). The two sides no longer match — 13 is not a balance point. The same goes for any value below 12. So 12 is the only place that balances, which is why every data set has exactly one mean.

What happens to the mean when data changes?

Because the mean is a balance point, we can predict how it moves without redoing the whole sum. This saves a lot of work.

Adding a value bigger than the mean pushes the mean up. Think of the see-saw. If you drop a new weight far out on the right, the right side gets heavier, so the balance point must shift right to even things out. So the mean increases. In the same way, adding a value smaller than the mean pulls the mean down.

Adding a value exactly equal to the mean changes nothing. A weight placed right on the support does not tip the see-saw at all. So the mean stays the same.

Add the same number to every value, and the mean shifts by that same number. If every score goes up by 10, the whole picture slides 10 to the right — so its balance point slides 10 to the right too. Let us prove this neatly with algebra.

Double every value, and the mean doubles. If you stretch every weight’s position to twice its distance from zero, the whole arrangement stretches out, and so does the balance point. By the same algebra (multiply each value by 5, factor the 5 out of the sum), multiplying every value by a number multiplies the mean by that number too.

This idea is genuinely useful. Here is a classic example.

Finding a missing value using the mean

The mean links the sum and the count. If you know the mean and the count, you know the sum. That lets you fish out a missing value.

When the median beats the mean: outliers

Here is where the mean can mislead. An outlier is a value that sits far away from all the others — a giant or a tiny one.

Because the mean uses every value in its sum, a single outlier can drag it a long way. The median, on the other hand, only watches the middle position. It does not care how far out the extreme value is. So an outlier usually pulls the mean but leaves the median calm.

Let us see it with five test marks: 18, 19, 20, 21, 22. Both the mean and median are 20. Now imagine the last mark was actually 90 (maybe a typing error, or one student who did spectacularly). Figure 5.2 below compares the two cases.

Tap to zoom

Check the mean in case (b): (18 + 19 + 20 + 21 + 90) ÷ 5 = 168 ÷ 5 = 33.6. That “average” of 33.6 is higher than four of the five marks! It does not describe a typical student at all. The median, 20, still does. This is why, for things like incomes or house prices where a few huge values exist, people often report the median — it is not fooled by the giants.

The median follows its own version of the “include a value” rule too. If you add a value bigger than the current median, the middle shifts up a little, so the median rises (or stays). Add a value smaller than the median, and it falls (or stays). But it shifts gently, by position — never leaping the way the mean can.

Organising data with frequencies

When the same value appears many times, writing it out again and again is wasteful. Instead we make a frequency table: one row of values, one row saying how many times (the frequency) each value occurs.

Say we collect the family size of 36 students. The table might look like this:

A tempting mistake is to do (3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) ÷ 8. But that pretends each value happened once! The number 4 happened eleven times — it should count eleven times. To find the mean, multiply each value by its frequency, add those up, and divide by the total count.

Notice the trick in Step 4: we did not write out all 36 numbers. By adding up frequencies, we found which value sits in the middle positions quickly. This is the everyday way to read a median off a frequency table.

Visualising data: dot plots

Numbers in a list are hard to feel. A picture makes the shape of the data jump out. The simplest picture is a dot plot: draw a number line, and put one dot above a value for each time it occurs. A tall stack means that value is common.

Here is the same family-size data as a dot plot (using a smaller class so it is easy to see). Figure 5.3 shows it.

Tap to zoom

A dot plot is great for small data. You can instantly spot the most common value (the tallest stack), any gaps, and any lonely dot far from the rest (an outlier). And because you can see each value’s distance from the centre, a dot plot is the natural way to picture the balance-point idea.

Visualising data: bar graphs

When you are comparing separate groups — not a number line, but categories like fruits or subjects — a bar graph is the right picture. Each group gets a bar, and the bar’s height shows its count. Taller bar, bigger count.

The most important rule: the heights must be drawn to scale. If 12 is the value, the bar must reach the 12 mark on the vertical axis — not “a bit taller than the others”. Figure 5.4 shows a bar graph of favourite fruits, drawn carefully to scale.

Tap to zoom

To read a bar graph, look across from the top of a bar to the vertical scale. To draw one: choose a scale that fits your biggest value, mark the axis evenly, then draw each bar up to its value. Keep all bars the same width and leave equal gaps.

Visualising data: line graphs

Now the most powerful picture for one special job: showing how something changes over time. For that, we use a line graph. We mark each value as a point, then join the points in time order with straight line segments. The line itself tells the story — climbing, falling, flat.

Figure 5.5 below shows the monthly maximum temperature of two states across a year.

Tap to zoom

Read a line graph in two steps. Step 1 — identify what is given: what do the axes measure, what scale is used, which line is which (here, circles vs squares and a legend tell you). Step 2 — infer and interpret: read the trend. In Figure 5.5, Punjab’s temperature varies a lot (steep rise to June, steep fall after), while Kerala’s is almost constant. The steeper a segment, the faster the change.

Why a line graph and not 24 separate bars? Because the line connects the points, your eye follows the trend smoothly. A clustered bar graph of 12 months × 2 states would be 24 bars crowded together — hard to read the pattern. Line graphs are made for change over time.

Common Mistakes

These slips trip up students every year. Read them once and you will dodge them.

Quick Check

Try these. Each one checks an idea from the chapter.

Practice Problems

Try each one yourself first. Only then tap to see the full solution.

Easy

Medium

Challenge

Summary

• The mean (average) = sum of all values ÷ number of values. Deeper still, the mean is a balance point: the total distance to the values on its left equals the total distance on its right. That is why each data set has exactly one mean.
• Because the mean balances, you can predict its moves: adding a value above the mean raises it, a value below lowers it, and a value equal to it changes nothing. Add a fixed number to every value and the mean shifts by that number; double every value and the mean doubles.
• The median is the middle value of the sorted data (the average of the two middle values if the count is even). You can find it from a frequency table by adding up frequencies until you reach the middle position.
• An outlier (a far-away value) drags the mean but usually leaves the median steady, so the median often describes a typical value better when extremes are present.
• To find a mean from a frequency table, multiply each value by its frequency, add the products, and divide by the total count.
• Dot plots show each value as a dot (good for small data and spotting outliers). Bar graphs compare separate groups by bar height, drawn to scale. Line graphs join points in time order and are best for showing change over time — the steeper the line, the faster the change.

What’s Next

You can now read a pile of numbers as a story — where its centre sits, when an average can mislead, and which picture tells the tale best. That habit of asking “what is this data really saying?” is a life skill, not just an exam topic.

Next we return to the world of symbols and patterns. The next chapter is Algebra Play, where you will use letters to stand for numbers and uncover the rules that hide behind them. See you there!