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.