Connecting the Dots
Why This Matters
Numbers are all around you. The runs a cricketer scores. The price of onions at the market. The heights of the students in your class. The marks in a test.
But a long list of numbers is hard to make sense of. If someone hands you the runs a batter scored in twenty matches, can you tell at a glance whether they batted well? Not really. Your eyes just see a jumble.
This chapter is about connecting the dots — turning a messy pile of numbers into a clear story. You will learn two big tricks.
First, you will squeeze a whole group of numbers into one single number that stands for the group. Second, you will turn numbers into pictures so the story jumps out instantly.
This is the heart of statistics — collecting data, organising it, and reading the story inside it. Newspapers, sports channels, weather reports, even your school report card all use these ideas. Once you learn them, you will never look at a table of numbers the same way again.
The Big Idea
A big group of numbers is hard to understand on its own. So we do two things. We pick a representative value — one number that stands in for the whole group (like the mean or the median) — and we draw the data as a picture (like a dot plot or a bar graph). The single number tells us the “centre” of the data. The picture shows us how the data is spread out. Together, they let a pile of numbers tell its story.
Let’s Break It Down
Questions that data can answer
Some questions have one fixed answer. “What is 7 × 8?” The answer is 56, always. You don’t need to collect anything.
But some questions are different. “How tall are the Grade 7 students in our school?” Here there is no single answer. Some students are tall, some short. To answer it, you must collect data — go and measure heights — then study what you collected.
A question like this is called a statistical question. A statistical question is one you can only answer by gathering data, because the answer varies from case to case.
A statistical statement is a claim built from such data. For example: “On average, Grade 7 students are about 150 cm tall.” It uses numbers to summarise something that varies.
Here is the simple test. If the answer would be the same every time, it is not statistical. If the answer changes from person to person, day to day, or place to place — so you need data — then it is statistical.
Is 'What is the price of a tennis ball in India?' a statistical question?
Yes. Prices are not the same in every shop or every city, and they change over time. To answer it honestly you must collect prices from many places. Because the answer varies, you need data — so it is a statistical question. Compare it with “What is 12 × 5?”, which has one fixed answer and needs no data at all.
One number for the whole group: the mean
Suppose Shubman scored 23, 26, 10, 52, 18 runs across five matches, and Yashasvi scored 53, 02, 15 across three matches. Who batted better?
You could add up each player’s runs. But that is unfair — Shubman played more matches, so of course he has a bigger total. We need a number that does not depend on how many matches each played.
That number is the average, also called the arithmetic mean, or just the mean. Here is the rule:
Mean = (Sum of all the values) ÷ (Number of values)
So divide the total by how many numbers there are. The mean is the same kind of number whether you played 3 matches or 30.
The best way to picture the mean is as a fair share. Imagine everyone in a group throws all their things into one big pile, and then the pile is split equally. The equal share each person gets is the mean. Figure 5.1 below shows this with guavas.
Notice what happened. The mean evened out the highs and lows. The child with 10 gave some away; the child with 3 received some; everyone ended up at 6. That is exactly what the mean does — it balances out the differences.
Let’s see the rule at work on the cricket scores.
Shubman scored 23, 26, 10, 52, 18 in 5 matches. Yashasvi scored 53, 02, 15 in 3 matches. Find each player's mean runs per match. Who has the higher average?
- Add up Shubman’s runs. 23 + 26 + 10 + 52 + 18 = 129. Wait — let me add carefully: 23 + 26 = 49, 49 + 10 = 59, 59 + 52 = 111, 111 + 18 = 129. So Shubman’s total is 129.
- Shubman played 5 matches, so his mean = 129 ÷ 5 = 25.8 runs per match.
- Now add Yashasvi’s runs. 53 + 02 + 15 = 70.
- Yashasvi played 3 matches, so his mean = 70 ÷ 3 = 23.3 runs per match (rounded).
- Compare the means: 25.8 is more than 23.3. So by average runs per match, Shubman performed slightly better, even though they played a different number of matches. This is why the mean is fair — it does not reward a player just for playing more games.
One small but important point. When we count “the number of matches”, we count only the matches a player actually played. If a player was absent for a match, that match does not count. But if a player played and scored 0, that match does count, and the 0 goes into the sum. Zero is a real score; “did not play” is not a score at all.
Seeing the data: the dot plot
The mean gives us one number. But it hides a lot. Two very different groups can have the same mean. So we also want to see the data spread out. One simple picture for this is the dot plot.
A dot plot is just a number line with a dot placed above each value. If a value appears twice, you stack two dots above it. If it appears three times, three dots, and so on. Figure 5.2 below shows how it works.
A dot plot is powerful because it shows three things at once: the smallest value, the largest value, and where the data clusters. You also see gaps — values that never happened.
One thing a dot plot gives up: it forgets the order the data came in. Once you sort the dots onto the line, you can no longer tell which value was the first, second, or third. That is a fair trade — you lose the order but gain a clear view of the spread.
In a dot plot, why do we stack dots on top of each other instead of placing them side by side?
Because stacking keeps each value above its own spot on the number line, while the height of the stack shows how many times that value occurred. If we spread repeated values sideways, they would drift away from their number and the picture would lie about where the data sits. Stacking lets the position show the value and the height show the count — two pieces of information in one neat picture.
When the mean misleads: outliers and the median
The mean is wonderful, but it has a weak spot. One strange value can drag it far off.
Look at the heights of one family: 118 cm, 165 cm, 170 cm, 173 cm, 175 cm. Four of the five people are 165 cm or taller. But one is a young child, only 118 cm. What is the mean?
The sum is 118 + 165 + 170 + 173 + 175 = 801. Divide by 5: the mean is 160.2 cm. But that is less than four of the five heights! The mean says “about 160 cm”, yet almost everyone is taller than that. The mean does not describe this family well.
The villain here is the 118. A value that sits far away from the rest of the data is called an outlier. An outlier is an unusually high or unusually low value that does not fit with the crowd. Because the mean adds up everything, a single outlier can pull it strongly in its direction. Figure 5.3 below shows the pull.
So we need a backup — a number that ignores a stray outlier. That number is the median.
The median is the middle value once you sort the data from smallest to largest. “Sort” means line them up in order. The median is the one standing in the centre of the line.
Why is the median tougher than the mean? Because it only cares about position, not size. It looks for whoever is standing in the middle. It does not matter how far away the outlier is — moving it further left does not change who is in the middle. So the outlier cannot drag the median around.
But there is a catch. If you have an odd number of values, there is a clear middle one. If you have an even number, two values share the middle — so you take their mean (their average) as the median. Figure 5.4 below shows both cases.
Let’s find a median step by step.
Find the median height of this family: 170 cm, 173 cm, 165 cm, 118 cm, 175 cm.
- First, sort the heights from smallest to largest. We get: 118, 165, 170, 173, 175.
- Count how many values there are. There are 5 — an odd number. So there is one clear middle value.
- Find the middle. With 5 values, the 3rd one is in the centre (2 below it, 2 above it). The 3rd value is 170.
- So the median height is 170 cm. Notice it sits right among the tall family members, unlike the mean of 160.2 cm. Here the median describes the family far better, because it shrugs off the outlier 118.
The mean and median together are called measures of central tendency. “Central tendency” just means the tendency of the data to pile up around a centre. Both try to find that centre — they just do it differently. The mean shares everything out; the median finds the middle.
How the outlier tilts the mean
Here is a neat pattern worth knowing. When data is balanced and has no outlier, the mean and median land close together. When there is an outlier, the mean shifts toward it while the median stays put. Figure 5.5 below shows the three cases side by side.
This gives you a quick clue. If someone tells you the mean is much bigger than the median, suspect a high outlier. If the mean is much smaller than the median, suspect a low outlier. If they are close, the data is probably balanced.
The spread: minimum, maximum, and range
The centre is only half the story. We also care about how spread out the data is. Two cricketers can both average 30 runs, yet one is steady (28, 31, 30, 29, 32) and the other is wild (0, 90, 5, 60, 5). The mean hides this; the spread reveals it.
The simplest measure of spread is the range. The range is the largest value minus the smallest value.
Range = Maximum value − Minimum value
A small range means the numbers are close together and steady. A large range means they are scattered. For onion prices in Yahapur, the highest was 59 and the lowest was 24, so the range is 59 − 24 = 35. In Wahapur the range was 60 − 17 = 43 — a bigger spread, meaning Wahapur’s prices swung more wildly through the year.
Two batters both have a mean of 30 runs. Batter A's scores range from 28 to 32. Batter B's scores range from 0 to 90. What does the range tell you that the mean cannot?
The mean (30) is the same for both, so it cannot tell them apart. But the range can. Batter A’s range is 32 − 28 = 4, which is tiny — A is very consistent, scoring close to 30 every time. Batter B’s range is 90 − 0 = 90, which is huge — B is wildly unpredictable, sometimes failing and sometimes huge. So the range exposes the difference in steadiness that the mean completely hides.
Comparing groups: the double bar graph
Often we want to compare two sets of data side by side — like onion prices in two towns across the months. A single bar graph shows one set. To compare two, we use a double bar graph (also called a clustered or double column graph).
In a double bar graph, each category gets two bars standing together, one for each group, drawn in two different colours. This lets you compare the two groups in every category at a glance. Figure 5.6 below compares onion prices in two towns over four months.
To read a double bar graph well, follow two simple steps. Step 1: identify what is given. Notice the categories along the bottom, the scale up the side, and which colour is which group. Step 2: infer from what you see. Compare the heights — within each cluster (which town is higher this month?) and across clusters (is the price rising or falling over time?).
In Figure 5.6, why is it useful to draw the two towns' bars side by side in the same colour scheme for every month, instead of drawing two separate graphs?
Because placing the bars side by side lets your eye compare the two towns directly, month by month, without flicking between two pictures. In a single cluster you instantly see which bar is taller, so you know which town was costlier that month. Using one fixed colour per town across the whole graph means you never have to re-check which colour is which. Two separate graphs would force you to line them up in your head, which is slow and error-prone.
Common Mistakes
To find the mean, divide the total by the number of matches the player could have played, counting matches they missed.
It feels fair to count every match in the series, since the player was 'part of the team' the whole time and the schedule had that many games.
Divide only by the number of values you actually have. If a player did not play a match, that match gives no score, so it is not counted. (But a match where they played and scored 0 does count — 0 is a real value.)
The median is just the middle number of the data as it was written down.
The word 'middle' makes it sound like you simply pick whatever sits in the centre of the list, and the data already looks like a tidy row.
You must sort the data from smallest to largest first, then take the middle of the sorted order. The middle of the unsorted list is meaningless.
The mean is always the best single number to describe a group of values.
The mean is the first average everyone learns and it uses every value, so it feels like the most complete and trustworthy summary.
When the data has an outlier, the mean gets dragged toward it and can misrepresent the group. In that case the median often describes the data better.
With an even number of values, there is no median because no single value sits exactly in the middle.
Every example with an odd count had one neat middle value, so it seems like an even count simply has no centre to point at.
With an even count, two values share the middle. The median is the average (mean) of those two middle values.
Quick Check
The data 4, 9, 4, 2, 6 has a sum of 25. What is its mean?
To find the median of 17, 3, 9, 12, 5, what is the very first thing you must do?
A family's heights are 120, 168, 170, 172, 174 cm. One value is an outlier. Which number describes this family better, the mean or the median?
A batter's scores are 0, 5, 90, 8, 2. What is the range?
Practice Problems
Easy
Shreyas bounced a ball on a bat and counted the bounces over 8 tries: 6, 2, 9, 5, 4, 6, 3, 5. Find the mean number of bounces.
First add all the values. 6 + 2 + 9 + 5 + 4 + 6 + 3 + 5 = 40. There are 8 values, so divide by 8. Mean = 40 ÷ 8 = 5 bounces on average.
Find the median of these 5 marks: 85, 76, 90, 39, 75.
First sort from smallest to largest: 39, 75, 76, 85, 90. There are 5 values (odd), so there is one clear middle. The 3rd value is in the centre. The 3rd value is 76. So the median is 76.
The number of pages in a newspaper from Monday to Sunday was 16, 18, 20, 22, 26, 16, 10. What is the range?
Range = maximum − minimum. The maximum is 26 and the minimum is 10. Range = 26 − 10 = 16 pages. This tells us the page count swung by 16 pages across the week.
Medium
Find the median of these 6 heights (cm): 169, 173, 155, 165, 160, 164.
First sort from smallest to largest: 155, 160, 164, 165, 169, 173. There are 6 values — an even number. So no single value is in the middle; two values share it. The two middle values are the 3rd and 4th: 164 and 165. Take their average (mean): (164 + 165) ÷ 2 = 329 ÷ 2 = 164.5. So the median is 164.5 cm.
A player scored 57, 13, 0, 84, did-not-play, 51, 27 across the listed matches. Find the mean runs per match.
Be careful with the two special cases. The 0 is a real score (the player played and got out for a duck), so it counts. The “did not play” match gives no score, so it does NOT count. So we use these 6 scores: 57, 13, 0, 84, 51, 27. Sum = 57 + 13 + 0 + 84 + 51 + 27 = 232. Number of matches actually played = 6. Mean = 232 ÷ 6 = 38.67 runs per match (rounded). Notice we divided by 6, not 7 — the missed match is left out entirely.
Nikhil's running times (seconds) are 17, 18, 17, 16, 19, 17, 18. Sunil's are 20, 18, 18, 17, 16, 16, 17. A faster runner has a lower time. Who is quicker on average?
Find each runner’s mean time. Nikhil’s sum = 17 + 18 + 17 + 16 + 19 + 17 + 18 = 122. He has 7 times, so mean = 122 ÷ 7 = 17.43 seconds. Sunil’s sum = 20 + 18 + 18 + 17 + 16 + 16 + 17 = 122. He also has 7 times, so mean = 122 ÷ 7 = 17.43 seconds. The means are equal! On average, both run equally quickly — about 17.43 seconds each. The mean alone cannot separate them; you would need to look at their spread (range) to say more about consistency.
Challenge
In a cricket match, a team scored a big total, yet the median runs scored by its players was 0. How is that possible? (Hint: a team has 11 batters, and many may get out for 0.)
The median is the middle value of the sorted scores. A team has 11 players, so the middle is the 6th score in sorted order. Suppose 6 or more of the 11 players scored 0 (got out without scoring). When you sort all 11 scores from smallest to largest, the first six or more are all 0. The 6th value — the middle — is therefore 0. So the median is 0 even though a few players at the top scored heavily and pushed the total high. This is a perfect example of why the median ignores extremes: a couple of big scores form a high cluster on the right, but they cannot move the middle value if the lower half is full of zeros. The mean, by contrast, would be much higher because it adds those big scores in.
The daily water usage (litres) was 5.6, 8, 3.09, 12.9, 6.5, 12.1, 11.3, 20.5, 7.4. Without finding the exact mean or median, explain why neither can possibly be greater than 30, and why neither can be less than the smallest value.
Think about what the mean and median mean. The mean is a fair share of the total. When you share equally, no one can end up with more than the biggest pile started with, and no one ends up with less than the smallest. So the mean must lie somewhere between the minimum (3.09) and the maximum (20.5). Since 30 is bigger than the maximum, the mean cannot reach 30. And the mean cannot drop below the minimum 3.09 either. The median is one of the values (or the average of two of them), and it sits in the middle of the sorted data. So it must also lie between the smallest and largest values — between 3.09 and 20.5. It cannot be 30 (too high) and cannot be below 3.09 (too low). Conclusion: both the mean and the median always sit between the minimum and the maximum value of the data. They can never be larger than the biggest value or smaller than the smallest. So neither can be 30, and neither can be below 3.09.
Summary
- A statistical question is one you can only answer by collecting data, because its answer varies (heights, prices) rather than being one fixed number.
- The mean (arithmetic mean) is Sum of values ÷ Number of values. Picture it as a fair share: pool everything and split it equally. Count only the values you actually have (a played match scoring 0 counts; a missed match does not).
- A dot plot places a dot above each value on a number line and stacks repeats; it shows the minimum, maximum, and where data clusters, at a glance.
- The median is the middle value of the sorted data. With an odd count there is one middle; with an even count, average the two middle values.
- An outlier is a value far from the rest. It drags the mean toward it but barely moves the median, so the median often describes outlier-heavy data better.
- The mean and median are measures of central tendency — they find the “centre” of the data in different ways.
- The range = maximum − minimum measures how spread out the data is; a small range means steady data, a large range means scattered data.
- A double bar graph draws two coloured bars side by side in each category, letting you compare two groups at a glance, both within a category and over time.
What’s Next
You have learned to turn a pile of numbers into a clear story — squeezing it into a single representative value, drawing it as a dot plot or a double bar graph, and spotting when an outlier is hiding. These tools let data speak.
Next we return to shapes, but now you get to build them yourself. The next chapter is Constructions and Tilings. There you will use a ruler and compass to construct exact figures, and discover how shapes fit together to tile a flat surface with no gaps. See you there!
Frequently Asked Questions
How do you calculate the mean of a set of numbers?
Add all the numbers together and then divide by how many numbers there are. For example, the mean of 4, 6, and 8 is (4 + 6 + 8) ÷ 3 = 18 ÷ 3 = 6. The mean is like a 'fair share' — if all values were equal, each would be the mean.
What is the median and how is it different from the mean?
The median is the middle value when all numbers are arranged in order. For example, in the set 3, 5, 9, 12, 20 the median is 9. The mean uses all values equally, but the median is not pulled by very high or very low values, so it is often more useful when there are outliers.
What is an outlier and how does it affect the mean?
An outlier is a value that is much higher or much lower than the rest of the data. Because the mean adds everything up, a single very large or very small value can pull the mean far away from where most data sits. The median is not affected the same way, so it is often a better representative when outliers exist.
What is a dot plot and how do you read one?
A dot plot is a number line where each data value is shown as a dot above its position. If a value appears three times, there are three dots stacked above it. You read a dot plot by looking at where the dots are clustered (the centre) and how spread out they are (the range).
What is a double bar graph used for in statistics?
A double bar graph compares two sets of data side by side for the same categories. For example, you might compare boys' and girls' scores in four subjects. Each category has two bars — one for each group — so you can see both the individual values and how the two groups differ at a glance.