8.4 Averages, Weighted Means & Measures of Spread

8.4 Averages, Weighted Means & Measures of Spread

Every Quantitative Reasoning data set — the ward registry, the clinic activity table, the epidemiology chart — invites summary statistics, and UCAT exploits the fact that most candidates have a fluent but shallow grasp of them. The arithmetic here is GCSE-level. The traps are not.

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The Three Averages and When Each Is Being Tested

The exam-relevant point is why a data set was built the way it was. Consider five patients' lengths of stay in days:

   2,  3,  3,  4,  38

• Mean = (2 + 3 + 3 + 4 + 38) ÷ 5 = 50 ÷ 5 = 10 days
• Median = the 3rd of 5 ordered values = 3 days
• Mode = 3 days

A mean of 10 describes none of these patients. When a UCAT data set contains one conspicuous outlier, the question is nearly always probing whether you notice that the mean and median diverge — and which one the stem actually asked for. Read the stem word: mean, median, average, typical, most common. They are not interchangeable.

Finding the median position

For an ordered list of $n$ values, the median sits at the $\frac{n+1}{2}$ th position.

• Odd n (n = 7): position (7 + 1) ÷ 2 = 4th value.
• Even n (n = 8): position (8 + 1) ÷ 2 = 4.5th — meaning the mean of the 4th and 5th values.

Forgetting to average the two middle values on an even-sized set is one of the most reliable ways to lose a QR mark.

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Weighted Means: The Trap That Catches Almost Everyone

Rule: You cannot average two averages unless the groups are the same size.

A clinic runs two sessions:

• Morning: 30 patients, mean consultation 12 minutes
• Afternoon: 70 patients, mean consultation 22 minutes

The tempting answer is (12 + 22) ÷ 2 = 17 minutes. It is wrong, because the afternoon session contributes more than twice as many consultations.

$\text{Weighted mean} = \frac{\sum (\text{value} \times \text{weight})}{\sum \text{weight}}$

$= \frac{(30 \times 12) + (70 \times 22)}{30 + 70} = \frac{360 + 1540}{100} = \frac{1900}{100} = \mathbf{19 \text{ minutes}}$

The correct answer of 19 sits closer to 22 than to 12, exactly as it should, because the larger group pulls the combined mean towards itself. Note that the naive answer, 17, is almost always present among the options.

The sanity check that costs two seconds

A weighted mean must lie between the two group means, and must sit nearer the mean of the larger group. If your answer is outside 12–22, you have made an arithmetic error. If it is nearer 12, you have weighted the wrong group.

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Back-Calculating from a Known Mean

UCAT frequently supplies the mean and asks for a missing element. Rearrange to totals — the mean is only ever a disguised total.

$\text{Total} = \text{mean} \times n$

Worked example. A ward's mean daily admissions across 6 days is 14. On the 7th day, 28 patients are admitted. What is the new 7-day mean?

• Total for 6 days = 14 × 6 = 84
• New total = 84 + 28 = 112
• New mean = 112 ÷ 7 = 16 admissions per day

Worked example (reverse). A student's mean score across 4 practice tests is 620. What must they score on a 5th test to raise the mean to 650?

• Current total = 620 × 4 = 2,480
• Required total = 650 × 5 = 3,250
• Required 5th score = 3,250 − 2,480 = 770

That second pattern — "what is needed to move the mean to X" — appears in QR and is worth drilling until it is automatic: required total minus current total.

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Measures of Spread

Two data sets can share a mean and describe completely different realities.

   Clinic A waits (min):  28  30  31  32  34   → mean 31, range 6
   Clinic B waits (min):   8  15  31  47  54   → mean 31, range 46

   Same mean. Completely different patient experience.

You are not asked to compute a standard deviation by hand on the UCAT. You are asked to interpret one: if a question compares two wards with equal means and different standard deviations, the correct conclusion concerns consistency or predictability, never average performance.

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The Base Trap: Mean Of What, Per What?

The most expensive averaging error in QR is not arithmetic — it is denominator selection. From a single hospital table you can compute:

• Mean cost per patient = total cost ÷ number of patients
• Mean cost per bed-day = total cost ÷ total bed-days
• Mean cost per admission = total cost ÷ number of admissions

These are three different numbers, and a patient may account for several bed-days and more than one admission. Before dividing, name the denominator out loud: "per what?" Then check that the stem asked for that one.

Timing note. At 43.3 seconds per QR item you have time for exactly one calculator chain. Identify the denominator before you start typing, because re-entering a five-step chain because you divided by the wrong total costs more than the question is worth.