UCAT Quantitative Reasoning Averages: Mean, Median, Mode and Weighted Averages
At TheUKCATPeople, I am Dr Akash, and averages are one of the most reliable sources of marks in UCAT Quantitative Reasoning. The arithmetic is GCSE level, but the marks are lost in three predictable places: the even-numbered median, the combined average of two unequal groups, and questions that give you the average and ask you to work backwards. Get those three right and averages become free marks.
In the 2026 UCAT, Quantitative Reasoning is 36 questions in 26 minutes, which works out at about 43 seconds per question, with a basic on-screen calculator and no negative marking (UCAT Consortium).
This guide sits alongside the complete Quantitative Reasoning guide and the mental maths shortcuts for percentages. Read those for the wider section strategy; this one drills averages to the point where you never lose a mark on them.
How averages appear in UCAT Quantitative Reasoning
Averages rarely appear as a bare list of numbers. They almost always sit inside a data set, a table or a chart, and the challenge is reading the right values before you calculate. The UCAT Consortium lists averages, including their use across combined samples and in prediction, as a core Quantitative Reasoning skill.
There are four averages you need, and they are not interchangeable:
- Mean: the total divided by how many values there are.
- Median: the middle value once the data are in order.
- Mode: the value that occurs most often.
- Range: the largest value minus the smallest. This is a measure of spread, not an average, and mixing it in is a classic trap.
Use this table to pick the right measure quickly and to spot which one the question is really testing.
Measure | How to find it | Where it catches people out |
|---|---|---|
Mean | Total divided by the number of values | Combining unequal groups; averaging two means directly |
Median | Middle value in order; average the two middles if the count is even | Forgetting to re-order; taking one middle value when the count is even |
Mode | The most frequent value | Bimodal data (two modes) or no mode at all |
Range | Largest minus smallest | Treating it as an average; adding instead of subtracting |
The mean
The mean is the sum of the values divided by the number of values. The formula is mean = total ÷ number of values. The single most useful habit is to think in totals: almost every hard mean question becomes easy once you convert an average back into a total.
Worked example 1: combining the means of two groups
Ward A has 12 patients with a mean age of 40. Ward B has 8 patients with a mean age of 65. What is the mean age across both wards combined?
A) 45
B) 50
C) 52.5
D) 55
Working:
- Convert each mean to a total: Ward A total = 12 × 40 = 480.
- Ward B total = 8 × 65 = 520.
- Combined total = 480 + 520 = 1000, across 12 + 8 = 20 patients.
- Combined mean = 1000 ÷ 20 = 50.
Answer: B) 50
The trap answer is C) 52.5, the simple average of 40 and 65. You can only average two means directly when the groups are the same size. Here Ward A is larger, so it pulls the combined mean below the midpoint.
The combined mean of two groups is (n1 x mean1 + n2 x mean2) divided by (n1 + n2). Never average the two averages unless the groups are equal in size.
The assumed-mean method for speed
When values are large but close together, subtracting a convenient round number first keeps the arithmetic small. To find the mean of 208, 211, 214 and 207, take 210 as an assumed mean. The differences are minus 2, plus 1, plus 4 and minus 3, which sum to zero, so the mean is exactly 210. You have replaced dividing a four-digit total by working with single digits.
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Reverse questions: working backwards from the mean
These are the questions most students under-practise, and they are pure marks once you think in totals.
Worked example 2: finding a missing value
Five clinics have a mean of 18 daily admissions. Four of them record 15, 20, 12 and 22 admissions. How many admissions did the fifth clinic record?
A) 18
B) 19
C) 21
D) 24
Working:
- Total across all five clinics = mean × number = 18 × 5 = 90.
- Total of the four known clinics = 15 + 20 + 12 + 22 = 69.
- Fifth clinic = 90 − 69 = 21.
Answer: C) 21
Worked example 3: the new mean after adding a value
A trial of 10 patients has a mean systolic blood pressure of 130 mmHg. An eleventh patient with a reading of 152 mmHg joins the trial. What is the new mean, to the nearest whole number?
A) 130
B) 131
C) 132
D) 141
Working:
- Old total = 130 × 10 = 1300.
- New total = 1300 + 152 = 1452, across 11 patients.
- New mean = 1452 ÷ 11 = 132.
Answer: C) 132
The trap answer is D) 141, the average of 130 and 152. Adding one value does not let you average it against the old mean, because the old mean represents ten patients, not one.
The median
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The median is the middle value once the data are ordered. Two things trip students up: forgetting to re-order data that is presented out of sequence, and handling an even number of values.
- Odd number of values: the median is the single middle value, at position (n + 1) ÷ 2.
- Even number of values: the median is the mean of the two middle values. This is the most common median error.
Worked example 4: median with an even number of values
Recorded waiting times in minutes are: 14, 9, 22, 11, 18, 9. What is the median waiting time?
A) 11
B) 12.5
C) 13
D) 14
Working:
- Order the data first: 9, 9, 11, 14, 18, 22.
- There are six values, so the middle two are the third and fourth: 11 and 14.
- Median = (11 + 14) ÷ 2 = 12.5.
Answer: B) 12.5
The trap answers A) 11 and D) 14 are the two middle values taken on their own. With an even count you must average them.
For a large data set or a frequency table, do not list every value. Find the position of the median, (n + 1) ÷ 2, then count through the frequencies until you reach it.
Worked example: median by position from a frequency table
The table shows the number of patients seen per day over 17 days. What is the median number of patients seen per day?
Patients per day | Number of days |
|---|---|
2 | 3 |
3 | 5 |
4 | 7 |
5 | 2 |
A) 3
B) 3.5
C) 4
D) 4.5
Working:
- There are 3 + 5 + 7 + 2 = 17 days, so the median is the (17 + 1) ÷ 2 = 9th value in order.
- Count through the frequencies: values 1 to 3 are 2 patients, values 4 to 8 are 3 patients, values 9 to 15 are 4 patients.
- The 9th value falls in the 4-patients row, so the median is 4.
Answer: C) 4
You never wrote out all 17 values. Finding the position first, then counting through the cumulative frequencies, is far faster and avoids errors.
The mode
The mode is the most frequent value. Three points cover almost every mode question:
- A data set can have two modes (bimodal) if two values tie for most frequent.
- A data set can have no mode at all if every value appears once.
- From a frequency table or a bar chart, the mode is simply the tallest bar or the highest-frequency row. You do not calculate anything.
The range
The range is the largest value minus the smallest value. It describes how spread out the data is, so it is not an average and should never be lumped in with the mean, median and mode. The two errors to avoid are adding the largest and smallest instead of subtracting, and forgetting that a single outlier can inflate the range dramatically.
Weighted and combined averages
This is where the real UCAT marks are, and where competitors give you one sentence. A weighted mean is used when values occur with different frequencies. The formula is the sum of (value × frequency) divided by the sum of the frequencies.
Worked example 5: weighted mean from a frequency table
The table shows the number of prescriptions issued per patient at a clinic. What is the mean number of prescriptions per patient?
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Prescriptions per patient | Number of patients |
|---|---|
0 | 5 |
1 | 8 |
2 | 4 |
3 | 3 |
A) 1.2
B) 1.25
C) 1.5
D) 2
Working:
- Multiply each value by its frequency: (0×5) + (1×8) + (2×4) + (3×3) = 0 + 8 + 8 + 9 = 25.
- Add the frequencies: 5 + 8 + 4 + 3 = 20 patients.
- Weighted mean = 25 ÷ 20 = 1.25.
Answer: B) 1.25
The trap answer is C) 1.5, the plain mean of 0, 1, 2 and 3. That ignores how many patients sit in each row.
The same weighted-mean method applies when the weights are given as percentages rather than counts. If a module mark is 40% coursework at 72, 35% exam at 64 and 25% project at 80, the overall mark is (0.40 x 72) + (0.35 x 64) + (0.25 x 80) = 28.8 + 22.4 + 20 = 71.2. You multiply each value by its weight and add, exactly as with a frequency table.
To combine two group means, turn each mean back into a total first, add the totals, then divide by the combined number. Averaging the two means only works when the groups are the same size.
When to use the calculator, and when not to
Averages are one of the areas where the on-screen calculator earns its place, but only for the final step. Set up the sum in your head, then use the calculator for one clean division of a large total. Opening it to add three two-digit numbers loses you more time than it saves.
- Use mental maths for the totals and for any small, round division.
- Use the calculator for a single large division, or when a slip is likely.
- Keep Num Lock on and type into the calculator with the number pad rather than clicking the on-screen buttons.
Common mistakes and UCAT traps
- Averaging two means without weighting them by group size.
- Taking one of the two middle numbers as the median instead of their average.
- Forgetting to re-order data before finding the median.
- Treating the range as an average, or adding rather than subtracting.
- Reaching for the calculator on sums you could do mentally in half the time.
- Missing a units change in the data set, such as thousands against millions, so the arithmetic is right but the reading is wrong.
Once averages are secure, apply the same read-the-data-first discipline to graphs and tables, where averages often appear inside a chart you have to interpret.
Test yourself
Work these before you read the answers below the list.
1. Class A has 15 students with a mean mark of 62. Class B has 25 students with a mean mark of 70. What is the mean mark across both classes?
2. The mean of six values is 14. Five of them are 10, 12, 15, 16 and 18. What is the sixth value?
3. Goals scored per match were: 0 goals in 4 matches, 1 in 6, 2 in 3, and 3 in 2. What is the mean number of goals per match?
Answers
1. 67. Totals are 15 x 62 = 930 and 25 x 70 = 1750; combined 2680 divided by 40 students = 67. The larger class pulls the mean towards 70.
2. 13. Total = 14 x 6 = 84; the five known values sum to 71; so the sixth is 84 minus 71 = 13.
3. 1.2. The weighted total is (0x4) + (1x6) + (2x3) + (3x2) = 18 across 15 matches, so 18 divided by 15 = 1.2.
Key Takeaway: Think in totals. Convert every average back to a total and most mean, missing-value and combined-average questions collapse into one line of arithmetic. Weight by group size, average the two middles when the count is even, and keep the range out of your average calculations.
Frequently asked questions
Are averages tested in UCAT Quantitative Reasoning?
Yes. Averages, including means, combined samples and use in prediction, are a listed Quantitative Reasoning skill, and they usually appear inside data sets such as tables and charts rather than as bare sums.
What is the difference between mean, median and mode?
The mean is the total divided by how many values there are. The median is the middle value when the data are ordered, and for an even number of values it is the average of the two middle values. The mode is the value that occurs most often.
How do you calculate a weighted average in the UCAT?
Multiply each value by its frequency or weight, add those products together, then divide by the total of the weights. For a frequency table this is the sum of (value multiplied by frequency) divided by the sum of the frequencies.
How do you combine the means of two groups?
Convert each group mean back into a total by multiplying it by the group size, add the two totals, then divide by the combined number of items. You cannot simply average the two means unless the groups are the same size.
How do you find a missing value when you are given the mean?
Multiply the mean by the number of values to get the total, then subtract the values you already know. The remainder is the missing value.
What is the median of an even number of values?
It is the average of the two middle values once the data are in order. For six ordered values, add the third and fourth values and divide by two.
Is the range an average?
No. The range is the largest value minus the smallest and measures spread, not central tendency, so it is not an average. Treating it as one is a common exam mistake.
How long should an averages question take in the UCAT?
Aim for around 30 to 45 seconds. Quantitative Reasoning gives you about 43 seconds per question on average, and averages questions should be quicker than that once you think in totals and reserve the calculator for one final division.
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