Compute and Check a Dataset from Start to Finish

You now hold all four moves of the method: choose the quantity, compute it, read what it means, and check it before you trust it. So far you have practiced them one at a time. This final chapter runs the whole method end to end on a single messy dataset, the way a real question actually arrives: as a pile of raw numbers with a mistake hiding somewhere inside. Follow the four moves in order and watch them turn that pile into answers you can defend.

Meet the Data

You have volunteered to keep the records for a small community center’s spring evening classes. Over April you logged how many people attended each class, and now a few questions have come in from the center’s coordinator. Here is the month exactly as you recorded it.

You also have two extra facts for later: the Dance class drew 231 attendances the following month (May), and you will want to compare Yoga and Choir fairly even though they run a different number of sessions.

The coordinator asks four things:

• How big was April overall, and which class drew the biggest share?

• What does a typical Dance night look like?

• Is Dance attendance growing?

• Yoga pulls more people in total than Choir, but is it really the more popular class night for night?

You can, but the first move is not arithmetic. It is deciding, for each question, which quantity actually answers it. Get that right and every calculation below falls into place.

Step 1: Choose the Right Quantity for Each Question

Translate each plain-language question into the quantity that answers it, exactly as you learned at the start of the course.

• “How big overall, and which class drew the biggest share?” is two asks: a total (how much in all) and a set of shares of the whole (each class against the total).

• “A typical Dance night” asks for a typical value, so it is an average, and because one night might be unusual you will compute both the mean and the median.

• “Is Dance growing?” asks how a value moved over time, so it is a percentage change from April to May.

• “More popular night for night” asks for a fair comparison between classes that run different numbers of sessions, so it is a rate: attendances per session.

With the four questions routed, you can compute.

Step 2: Compute and Read the Core Numbers

Work through the quantities in turn, reading what each one says as you go.

Total and shares. Add the attendances as recorded: 210 + 174 + 900 + 72 + 54 = 1,410. Now each class’s share of that total, part divided by total times 100:

That Choir share is striking, nearly two thirds of everyone who came all month. Hold that thought; you will test it before trusting it.

A typical Dance night. The eight Dance sessions drew these attendances:

20, 22, 23, 24, 25, 26, 27, 43

The mean is 210 ÷ 8 = 26.25. The median, the middle of the eight sorted values, sits between the fourth and fifth (24 and 25), so it is 24.5. The two differ because one showcase night drew 43, well above the rest. That 43 is a real, correctly recorded night, not a mistake, so you keep it, but for a “typical” night the median of 24.5 is the honest figure; the mean is nudged upward by the single big evening.

Is Dance growing? Dance drew 210 in April and 231 in May. The base is April, the month you are growing from: (231 - 210) ÷ 210 × 100 = 10%. Dance attendance rose 10% from April to May. Note what this does not say: it covers this one class at this one center over two months, and it says nothing about why.

Step 3: Run the Check Catalog Before You Trust Anything

You have several figures in hand, but the method is not finished until you have checked them. Run the catalog.

• Unit and definition consistency. Every value is attendances counted the same way, per class, over the same month. Consistent.

• Parts sum to the whole. The shares add to 64 + 15 + 12 + 5 + 4 = 100%. This check passes, which is worth pausing on: a clean 100% did not catch the problem, because a wrong total still divides into shares that sum to 100%. Passing a check is reassurance, not proof.

• Sanity and order of magnitude. Here the trouble surfaces. Choir supposedly drew 900 across 4 sessions, which is 225 people per night, in a room that seats a few dozen. One class accounting for 64% of all attendance is equally hard to believe. The figure is the wrong order of magnitude.

• Error and outlier scan. Scanning the raw column, 900 towers over every other class (54 to 210). It is almost certainly a data-entry slip, an extra zero on a 90-attendance month.

• Recomputation with the fix. Correct Choir to 90 and re-add: 210 + 174 + 90 + 72 + 54 = 600. A quick second route agrees: five classes averaging somewhere around 120 should land near 600, and it does.

With Choir corrected to 90 and the true total at 600, the shares are worth recomputing:

Now the month reads honestly: Dance drew the biggest share at 35%, and Choir, far from dominating, sits in the middle.

Only the parts that used it. The average, computed from the Dance sessions, and the Dance percentage change never touched the Choir figure, so they still stand. One caught error ripples only as far as the figures that depended on it, which is another reason to catch it before you trust anything.

Step 4: Finish With a Fair Comparison

The last question, whether Yoga is really more popular than Choir night for night, needs a rate, and now you have the clean numbers to compute it. Divide each class’s attendances by its number of sessions:

• Yoga: 174 ÷ 8 = 21.75 attendances per session.

• Choir: 90 ÷ 4 = 22.5 attendances per session.

The raw totals say Yoga wins easily, 174 against 90. But per session the picture flips: Choir draws 22.5 a night against Yoga’s 21.75. Yoga pulls more people overall only because it runs twice as many sessions; measured per session, Choir is marginally the stronger draw. As always, the rate sharpens one comparison and no more: it does not account for room size or the kind of person each class attracts.

The Trusted Answers

After all four moves, here is what you can tell the coordinator and defend:

• April drew 600 attendances in total, with Dance the biggest share at 35%.

• A typical Dance night is about 24 to 25 people (the median), with one showcase evening higher.

• Dance attendance grew 10% from April to May.

• Night for night, Choir (22.5 per session) edges Yoga (21.75), even though Yoga draws more people overall.

Every one of these survived the check that caught the choir’s extra zero, which is what makes them answers rather than guesses.

Your Turn!

You keep the ticket records for a small community theater. Here is one month (September) exactly as you logged it, alongside the number of performances each show ran.

Do the following, writing your answers down before you check:

1. Add up September’s tickets as recorded and compute each show’s share of that total. Then run the check catalog: which check flags a problem, and what is the true value most likely to be?

2. Fix the error and recompute the corrected total and each show’s share.

3. Comedy night sold far more tickets in total than Open-mic. Compute tickets per performance for each show, and say which one actually drew better per performance.

4. In October the theater sold 726 tickets in total. Compute the percentage change from September (corrected) to October, stating clearly which value is the base.

Then check your answer against the model solution found at the end of this chapter.

Let’s Recap!

• The full method runs in four moves on any dataset: choose the quantity, compute it, read what it means, then check it before you trust it.

• Choosing the right quantity first turns a pile of raw numbers into a clear list of calculations.

• Reading a result keeps you from over-claiming what a correct number actually shows.

• The check catalog catches a real error that a flawless calculation would carry straight through, and a passing check is reassurance rather than proof.

• A caught error only ripples as far as the figures that used it, so the untouched results still stand.

That is the whole method, start to finish: you took a messy month of records, chose the right quantity for each question, computed and read each one, and let a simple set of checks catch the mistake hiding inside before it reached anyone. Carry these four moves into your own numbers and you will trust them every time a decision depends on them. A short Part 2 quiz comes next to confirm everything from this second half of the course.

Model Solution

1. As recorded, the total is 240 + 130 + 2,000 + 90 = 2,460 tickets. The shares are Kids’ matinee 2,000 ÷ 2,460 ≈ 81%, Comedy night ≈ 10%, Drama ≈ 5%, and Open-mic ≈ 4%. Running the catalog: unit consistency passes (all are September ticket counts), and parts-sum-to-the-whole passes (the shares total about 100%), so neither catches the problem. The sanity and order-of-magnitude check flags it: 2,000 tickets across 4 performances is 500 a show, and one show making up 81% of the month is not believable for a small theater. The error and outlier scan confirms it, since 2,000 towers over the others (90 to 240), an extra zero on a likely 200.

2. Corrected, the total is 240 + 130 + 200 + 90 = 660 tickets. The shares become Comedy night 240 ÷ 660 ≈ 36%, Kids’ matinee ≈ 30%, Drama ≈ 20%, and Open-mic ≈ 14% (about 100% together).

3. Tickets per performance: Comedy night 240 ÷ 3 = 80, Open-mic 90 ÷ 1 = 90. Per performance, Open-mic drew better (90 against 80), even though Comedy night sold far more tickets in total, because Comedy ran three performances to Open-mic’s one. The raw count and the rate answer different questions.

4. The base is September’s corrected total, 660, the figure you are growing from: (726 - 660) ÷ 660 × 100 = 66 ÷ 660 × 100 = 10% increase from September to October.