Averages and Statistics

Mean, median, and range appear in roughly 15% of QR questions. The maths itself is straightforward - the UCAT makes it hard by asking you to work backwards (given the mean, find a missing value) or by burying numbers in frequency tables. The delta shortcut in this lesson is the single biggest time-saver for these problems.

The Three Rearrangements

The mean formula has three variables. Any question gives you two and asks for the third. Base formula: `Mean = Sum / n`.

The key rearrangement: `Sum = Mean × n`. This is the one most students forget and the one the UCAT tests most often.

Finding a Missing Value

Question type: "The mean of 5 numbers is 12. Four of them are 10, 14, 8, 15. What is the fifth?"

1. Sum = Mean × n = 12 × 5 = 60
2. Sum of known values = 10 + 14 + 8 + 15 = 47
3. Missing value = 60 − 47 = 13

This works every time. But there's a faster way when the question asks about changes to the mean rather than a specific missing value.

This is the biggest time-saver in QR statistics. When a question asks what happens to the mean after adding, removing, or changing values, you don't need to calculate the old mean or the new mean separately.

The Formula

`Change in Mean = Change in Sum / n`

Worked Example

Question: A warehouse has 5 crates. 6 kg is added to each of 4 crates. By how much does the mean weight of the 5 crates change?

Delta method:
Change in sum = 6 × 4 = 24 kg (added 6 to each of 4 crates)
n = 5 crates (total number stays the same)
Change in mean = `24 / 5 = 4.8 kg`

Done. No original values, no old mean, no new mean. One multiplication, one division.

Verification

Say the original crate weights are 10, 20, 30, 40, 50 kg. Mean = 150/5 = 30 kg.
After adding 6 kg to 4 crates: 16, 26, 36, 46, 50. Sum = 174. Mean = 174/5 = 34.8 kg.
Change = 34.8 - 30 = 4.8 kg. Matches.

When to Use It

Adding or Removing an Item

When items are added or removed, you go through the sum. Mean × n = Sum is the bridge.

Adding an item: "Mean of 8 values is 25. A 9th value of 34 is added. New mean?"
Current sum = 25 × 8 = 200. New sum = 200 + 34 = 234. New mean = 234 / 9 = 26.

Removing an item: "Mean of 10 values is 50. Value of 20 is removed. New mean?"
Current sum = 50 × 10 = 500. New sum = 500 − 20 = 480. New mean = 480 / 9 = 53.3.

Question: The table below shows how many patients visited a clinic each day over 20 days. What's the mean number of daily visits?

1. Total visits: (5 × 3) + (8 × 7) + (10 × 6) + (14 × 4) = 15 + 56 + 60 + 56 = 187
2. Total days: 3 + 7 + 6 + 4 = 20
3. Mean: 187 / 20 = 9.35 visits per day

Calculator with memory: 5 * 3 = (P) → 8 * 7 = (P, memory 71) → 10 * 6 = (P, memory 131) → 14 * 4 = + C = 187 → / 20 = 9.35. Time: ~25 sec.

Trap: don't just average 5, 8, 10, 14 → that gives 9.25. Wrong, because each value appears a different number of times. You need the weighted mean.

What It Is

The median is the middle value when all values are arranged in order. Odd count: the single middle value. Even count: the average of the two middle values.

The Elimination Method

Don't sort the entire list. Cross off the highest and lowest in pairs until the middle remains.

Values: 23, 17, 31, 8, 42, 15, 29

Round 1: remove 8 (lowest) and 42 (highest) → 23, 17, 31, 15, 29
Round 2: remove 15 and 31 → 23, 17, 29
Round 3: remove 17 and 29 → 23

Median = 23. No sorting required.

Even Number of Values

Values: 12, 45, 23, 67, 34, 8

Round 1: remove 8 and 67 → 12, 45, 23, 34
Round 2: remove 12 and 45 → 23, 34

Two values remain: Median = `(23 + 34) / 2 = 28.5`.

Which Position Is the Median?

For large datasets (like reading from a table with 100 entries), you need the position: `Position of median = (n + 1) / 2`.

`Range = Highest value − Lowest value`.

That's it. But under time pressure it's easy to confuse this with mean or median. If a question says "range," you need the biggest number minus the smallest. Nothing else.

The Negative Number Trap

Problem: temperatures across 5 days: 17, 3, −3, 8, 12. What is the range?

Wrong: 17 − 3 = 14 (used 3 instead of −3 as lowest)
Right: 17 − (−3) = 17 + 3 = 20

Subtracting a negative = adding. This trap appears in temperature, elevation, and profit/loss data.

The mode is the most frequently occurring value. Rare in QR but easy marks when it shows up.

Values: 3, 7, 2, 7, 5, 3, 7, 9, 3, 7

7 appears 4 times - mode. 3 appears 3 times. Everything else appears once. Mode = 7.

For tables and charts, the mode is the value with the highest bar or most entries.

A common QR pattern combines both skills:

Problem: the mean salary of 8 employees is £32,000. If salaries increase by 5%, what is the new mean?

Slow: 32,000 × 8 = 256,000, then × 1.05 = 268,800, then / 8 = 33,600.
Fast: 32,000 × 1.05 = 33,600. If every value increases by the same percentage, the mean increases by that same percentage. No need to find the sum at all.

Rule: A uniform percentage change applied to all values changes the mean by the same percentage. A uniform addition changes the mean by the same addition.

Trap 1: Confusing mean and median

Mean uses all values. Median uses only the middle. Read the question word carefully - they give different answers.

Trap 2: Weighted vs simple mean

"Average of 60 and 80" = 70 (simple mean)
"Average of 60 (for 3 items) and 80 (for 7 items)" = (60×3 + 80×7) / 10 = 740/10 = 74 (weighted mean)

If groups have different sizes, you need the weighted mean.

Trap 3: Forgetting to count all items

"Mean of 5 tests is 70. Sixth test is 88. New mean?"
n changes from 5 to 6. New mean = (350 + 88) / 6 = 73, not (350 + 88) / 5 = 87.6.

Trap 4: Range with negative numbers

17 − (−3) = 20, not 14. Subtracting a negative = adding.

Next lesson: 3.5 Geometry - area, perimeter, and volume through decomposition, covering about 8% of QR questions.