The hammer rang once. A spark jumped off the anvil and did not go out. It waited. They followed.

The Green Gazette's summer special came out in the first week of August, when school was over and everyone had time to read it properly. It was twice as thick as a normal issue, and it had a pull-out section on the middle pages called THE GREEN'S BEST, with the best ice cream, the best bench, the best dog walk, and, at the bottom of the second page, a small table that Tara had almost cut for space.

SWIM CLUBS: WHO PASSES MOST?

Sam in Year 8 had made it. It was his first piece for the paper. There were two swimming clubs on the Green. Hillside met at the new leisure centre on the hill. Riverside met at the old baths by the river, which had a cracked tiled roof and a café that sold the best chips in town. Every summer, both clubs tested their swimmers for the next badge.

Sam had asked both clubs for their badge results and done the sums carefully. He had even checked them twice, on a calculator, in front of Tara.

Hillside: 56 of 80 swimmers passed their badge this year. 70%.

Riverside: 48 of 80 swimmers passed. 60%.

Underneath, Sam had written one line: If you want your child to pass, Hillside is the place to go.

Tara had let the line go in. It was a summer special. It was meant to be helpful.

The following week, there was a letter in the Gazette box. It was written in blue biro on the back of a Riverside club leaflet, the one with the drawing of a duck in goggles.

Dear Gazette, it said. I coach the Riverside club. Since your summer special, three families have taken their children out and moved them to Hillside. One of them told me it was because of your table. I would like to explain something about our club to your reporter, if he will let me. Best wishes, Abena Osei.

Tara pinned it to the board above the back table at the Whetstone café. She read it twice.

She had been pleased with the summer special. Now she was uneasy. She did not know why yet.

"I was going to say both numbers are right," Tara said.

She had brought the letter to the back table. Danya was there, still damp from swimming, with his hair sticking up and a towel round his neck. He swam at Riverside, as it happened, on Tuesdays. Across from him sat Opal, a quiet sixth-former who designed the school's surveys and was exact about words in a way that made other people check their sentences before they said them.

"I was going to say it like this," said Tara. "Sam got the badge results from both clubs. The clubs gave him the numbers themselves. Fifty-six out of eighty is seventy per cent. Forty-eight out of eighty is sixty. He checked it twice. Nothing's wrong with the maths. I can write back to Ms Osei and say so, kindly."

She had the sentence half written. Thank you for your letter. Our figures were provided by both clubs and carefully checked.

It felt good. It felt solid. You cannot argue with arithmetic.

Opal read the table, and then the letter, and then the line underneath the table.

"What was the number standing in for?" said Opal.

"How many people passed," said Tara.

"That's what it is," said Opal. "What did people read it as? Say what we actually wanted, in words first."

Tara thought. "Which club's better," she said slowly. "Which one teaches better. That's why Mrs Lin moved her daughter."

"So the pass rate was standing in for good teaching," said Opal. "A measure is a stand-in, not the thing. Nobody said which thing. And Sam's line underneath says it outright. If you want your child to pass, Hillside is the place."

"But more people pass at Hillside," said Danya. "That's true. Says who that isn't the same as better?"

"Nobody says it," said Opal. "That's the problem. Nobody asked what kind of swimmers go to each club."

Tara looked at the letter. I would like to explain something about our club.

"How do we know who swims at each club?" Danya asked.

Nobody knew. That was the problem.

Deed went to the baths the next morning, because Deed believed in going to the original document, and the original document was a ring binder in Ms Osei's office, next to a box of lost goggles.

Ms Osei gave her the binder, and a cup of tea, and then left her alone with it, which Deed said afterwards was the nicest thing anyone had done for her all summer. Then Deed went up the hill to the leisure centre and asked Mr Grant for his records too. Mr Grant printed them out from a computer, very efficiently, and gave her a leaflet about aqua aerobics.

"Go to the original document," said Deed, putting both on the back table that afternoon. "Not Sam's table. These. Read it where it was first written down."

In both clubs, every swimmer was listed by name, and by group, and by whether they had passed.

There were two groups. Beginners, who were learning their first strokes and their first badge. And improvers, who could already swim and were working towards longer distances.

Prei, who built spam filters for fun and liked counting things, took both and began to count them apart.

"Let's put it on paper with real numbers," said Prei. "Count the flags, then count the right ones. Riverside first."

Prei wrote it out.

Riverside beginners: 60 swimmers. 30 passed. That's half. 50%.

Riverside improvers: 20 swimmers. 18 passed. 90%.

"Now Hillside," said Prei.

Hillside beginners: 20 swimmers. 8 passed. 40%.

Hillside improvers: 60 swimmers. 48 passed. 80%.

Tara looked at the four lines.

Riverside beginners: fifty per cent. Hillside beginners: forty.

Riverside improvers: ninety per cent. Hillside improvers: eighty.

"Riverside's better," said Danya. "At both. At beginners and at improvers."

"Better pass rate in both groups," said Prei. "Yes."

"Then how is it worse overall?" said Tara.

She felt something shift, the way it had in July with the blueberries, and in June with the poll. It was the feeling of a true number turning out to be standing on the wrong ground. Except this time it felt as if the ground had turned upside down.

Widen kept a notebook of what things cost, from bus fares to bridges, and could not hear a number without asking what it should be compared with. Widen took Prei's four lines and drew a box round each club.

"Compared with what?" said Widen. "That's the whole thing. Let's add them back up."

"Riverside," said Widen, writing. "Thirty beginners passed, and eighteen improvers. Forty-eight. Out of eighty. Sixty per cent. Exactly Sam's number."

"Hillside. Eight beginners passed, and forty-eight improvers. Fifty-six. Out of eighty. Seventy per cent. Also exactly Sam's number."

"So Sam was right," said Tara.

"Sam was right about the totals," said Widen. "And Prei's right about the groups. Both. At the same time. Now look at how many of each kind of swimmer there are."

Tara looked.

Riverside: sixty beginners, twenty improvers. Three beginners for every improver.

Hillside: twenty beginners, sixty improvers. The other way round.

"Beginners pass less often," said Widen. "Everywhere. At both clubs. That's not surprising: they're beginners. So a club with lots of beginners will have a lower total, even if it's better at teaching every single kind of swimmer it has. Riverside wins in both groups and loses overall, because three-quarters of its swimmers are in the group that's hardest to pass."

"So the total isn't comparing the clubs," said Opal quietly. "It's comparing the swimmers they start with."

"Is that a lot, or does it just sound like a lot?" said Widen. "Seventy against sixty sounded like a lot. Compared with what? With swimmers like them. Compare beginners with beginners, and improvers with improvers. Then you're comparing the clubs."

Danya was staring at the numbers as if they were a magic trick.

"That shouldn't be possible," he said. "Better at both and worse at all?"

"It's possible every time the mix is different," said Widen. "It's got a name. I'll tell you later."

Attest came last, with the Gazette's red pen. Attest subbed every story before it went to print and drew a red circle round any claim without a name next to it. The summer special had come back with one circle, round Sam's line. If you want your child to pass, Hillside is the place to go. Next to it Attest had written: whose child?

"Who says so? I wrote that," said Attest. "And then it was a summer special, and it was late, and I let it go."

"I saw it," said Tara. "I thought you meant it was too chatty."

"I meant: which children," said Attest. "Beginners, or swimmers. But I didn't follow it through. Nobody did, until Opal asked what the number was standing in for."

"Let me listen first," Tara said, before anyone could start telling her what to write.

She took her pad and wrote three headings across the top: what we printed, what the groups say, what a pass rate can't tell you.

Under the first she wrote Hillside 70%, Riverside 60%; go to Hillside. Under the second, Riverside better for beginners (50 v 40) and improvers (90 v 80); 3 times as many beginners. The third column: how good the teaching is, then how many children each club takes on, then whether a child would pass at either.

"Hm. So what does a clarification like this have to carry?" Tara asked.

"Every claim names its source," said Attest. "Both clubs' own records, by name. Both totals, which were correct. Then the two groups, for both clubs. Then the plain fact that Riverside's pass rate is higher for beginners and for improvers. Then one sentence saying why the totals point the other way. And then the line we shouldn't have printed."

"Do we say Hillside is worse?"

"No," said Attest. "Hillside's pass rates are good. Eighty per cent of improvers is good. We say we ranked two clubs with different swimmers as if they had the same ones. Put a name next to it."

"Hm. That's a lot for one box," said Tara.

"Five sentences," said Attest. "And a very small table, if Mrs Hallam lets us."

Tara looked at the letter again. Three families have taken their children out.

"I think we missed someone," Tara said slowly. "The beginners. Riverside's got sixty of them. Children who've never swum. If they all move to Hillside because of our table, and Hillside only takes twenty beginners, where do the other forty go?"

Nobody answered at once.

"Who you think about is your call," said Attest. "That's your bit."

Mrs Hallam, who advised the Gazette and signed off every issue, came in at four. Sam came with her, looking as if he would like to swim away. Mr Grant came at quarter past, in a Hillside polo shirt, with a stopwatch round his neck. Ms Osei came straight from the baths, with wet hair and her keys on a lanyard shaped like a duck. Mrs Lin came last, with her daughter, who was nine and holding a Hillside badge.

Sam spoke first.

"Both numbers are right," he said. "Fifty-six out of eighty and forty-eight out of eighty. I asked both clubs. I checked it twice. I didn't make anything up. People wanted to know where their children would pass, and I told them."

It was the strongest version of his argument, and every part of it was true.

Mr Grant spoke next.

"I'll be honest," he said. "I liked the table. My swimmers worked very hard this year. Seventy per cent passed. That's real. I'm not going to apologise for children passing their badges." He paused. "But I'll say this. Most of my lot can already swim when they come to me. We're up the hill, it's not on a bus route, and we don't run a beginners' session on a Saturday. That's just how it is."

That was true too.

Ms Osei put the ring binder on the table.

"We take everybody," she said. "That's what Riverside's for. It's been the town baths for ninety years. Half my beginners have never been in water deeper than a bath. Some of them are frightened of it. Some of them are seven, and some of them are thirteen, and nobody ever took them. Half of my beginners passed. That's thirty children who couldn't swim in September." She looked at the table. "I don't mind the table. I mind that it told parents to take the frightened ones somewhere else."

Then Mrs Lin spoke, holding her daughter's hand.

"I just want my daughter to learn to swim," she said. "I read the table and I thought, seventy is more than sixty. I didn't know there were groups. How would I know?" She looked at the badge. "She passed. I'm glad she passed. But she was already an improver. She'd have passed at Riverside too, wouldn't she."

"Ninety per cent of improvers did," said Prei quietly.

Nobody said anything for a while.

"That's not fair," said Danya, finally. "On anybody. Not on Sam, either."

Mrs Hallam took the corrections box out of her bag. It was the small grey panel at the bottom of page two, with a border round it and room for a few lines. She had used it eleven times that year. She set it on the table in front of Tara, with the summer special beside it.

On the table in front of Tara was the summer special, open at THE GREEN'S BEST, with Sam's table and his one line underneath. There was Ms Osei's ring binder and Mr Grant's printout, side by side. There were Prei's four lines, 50, 90, 40, 80, and Widen's two boxes with their totals, 60 and 70, that were both right and both told the wrong story. There was Opal's question, what was the number standing in for?, and Attest's red circle, whose child? There was a duck in goggles on the back of a leaflet, and a nine-year-old holding a badge she would have got anyway.

"You're the editor," said Mrs Hallam. "What does it say?"

She handed Tara the pen.

Tara looked at the empty box.

 

 

(Write the clarification Tara prints under the summer special's league table.)

 

The spark dimmed to a warm dot and settled back on the anvil. The Green was just the Green again.

Back on the Green

That evening Lan, who was seven, said that her class was better at spelling than Danya's, because her class got nearly all their spellings right and his got lots wrong.

Danya asked her how long her spellings were.

Lan said they were three letters long. Cat. Sun. Big.

Danya said his were ten letters long.

Lan thought about it for quite a long time. Then she said that maybe they should both do the same spellings, and then they would know.

They tried. Danya won the long words. Lan won the short ones, because Danya spelled big with two g's, by mistake, on purpose.

Afterword. The idea in this case is Simpson's paradox: a total can point the opposite way from every group inside it, when the groups are mixed in different amounts. Here one club did better with beginners and better with improvers, yet worse overall, because most of its swimmers were beginners, who pass less often wherever they swim. The same thing has happened with real medical treatments, school results and hospital rankings. Before you rank two groups by a single total, ask what the total is standing in for, split it into kinds that can fairly be compared, and compare like with like. You can practise turning a fuzzy question into a fair measure in InquiryForge, and comparing numbers that are not the same kind in InferForge.