ReasoningCampus · Online SHL preparation
Weighted Averages: Our SHL Method and 6 Practice Questions
If one branch handles 80 applications and another handles 20, their percentages should not have equal influence on the combined result. That is the idea behind weighted averages. ReasoningCampus offers SHL numerical reasoning preparation that teaches you to identify the relevant base, rebuild the totals and explain the result—not just remember a formula.
Numerical reasoning is the use of numbers to answer a practical question: which team is faster, what has changed, or what a table actually tells us. Weighted averages are one part of that skill. In this focused lesson, you will work with unequal group sizes, success rates and profit margins, then see why an improving headline figure can hide a decline inside every group.
View full preparationWhere weighted averages fit in SHL numerical reasoning
SHL’s public numerical reasoning material describes working with numerical information in practical contexts. The teaching categories below explain different uses of the same weighted-average relationship. They are not an official list of every item in SHL’s question bank.
| What the question asks | Recover first | Divide by |
|---|---|---|
| Average handling time | Minutes in each group = calls × minutes per call | All calls |
| Combined success rate | Successful cases = cases × success rate | All cases |
| Overall profit margin | Profit and revenue for every unit | Total revenue |
| A changing overall rate | Both the category rates and their changing weights | The relevant population in each period |
| A result with missing group sizes | Which denominator information is absent | Do not assume equal weights |
Before using a calculator, predict which group should pull the answer towards its rate. If 90 of 100 cases come from the lower-rate group, an answer close to the higher rate should make you pause. This estimate will not replace the calculation, but it gives you a useful independent check.
Weighted averages: the ReasoningCampus denominator-first method
We ask you to name the denominator before writing a formula. For minutes per call, the weight is calls; for a profit margin, it is revenue. Write the contribution and its base in two columns, then total each column. This keeps the reasoning visible when a question changes the labels or presents the same idea in a table.
A group average describes one group. A combined average describes all the observations together. If the groups contain different numbers of observations, treating each group as one equally important item can give the wrong result.
Combined average = total contribution ÷ total weight.
For two groups: (group 1 size × group 1 average + group 2 size × group 2 average) ÷ combined size.
- Name the quantity. Is it minutes per call, revenue per order, or a success rate?
- Recover each contribution. Multiply the group average or rate by its relevant base.
- Add comparable contributions and bases. Use the same unit and a clear population.
- Divide and sense-check. With positive weights, the result must lie between the smallest and largest group averages.
The weight depends on the question
| Measure | Contribution to add | Weight to add |
|---|---|---|
| Minutes per call | Total call minutes | Number of calls |
| Successful applications | Number of successes | Number of applications |
| Profit margin | Profit | Revenue |
| Average order value | Sales value | Number of orders |
The weight is not automatically the largest number in the table. For a combined profit margin, weighting by staff headcount answers a different question. Write the denominator in words before entering numbers.
Combined profit margin: check what receives the weight
A useful explanation has three parts: the measure, its denominator and the reconstructed total. “I weighted by revenue because the question asks for profit divided by revenue” is more informative than “I used a weighted average”. If the answer is wrong, first check that choice of base, then the arithmetic. Use the profit-margin question below to practise this distinction.
Two worked examples: combine groups, then inspect the mix
Example 1: two branches with unequal volumes
| Branch | Applications | Successful | Success rate |
|---|---|---|---|
| North | 80 | 60 | 75% |
| South | 20 | 10 | 50% |
The combined result is (60 + 10) ÷ (80 + 20) = 70%. The simple mean, (75% + 50%) ÷ 2 = 62.5%, gives the smaller South branch the same weight as North. That is not the overall application success rate.
You can also calculate (80 × 75 + 20 × 50) ÷ 100 = 70%. Here the rates are expressed as percentage numbers, so the result is a percentage number too. Use one convention consistently.
Example 2: an overall rise despite lower rates in both groups
The following fictional service handles easy and difficult cases. Each period contains 100 cases, but the mix changes.
| Period | Easy cases and success rate | Difficult cases and success rate | Total successes |
|---|---|---|---|
| 1 | 20 at 90% | 80 at 60% | 18 + 48 = 66 |
| 2 | 80 at 80% | 20 at 50% | 64 + 10 = 74 |
The combined rate rises from 66% to 74%, although the easy-case rate falls from 90% to 80% and the difficult-case rate falls from 60% to 50%. Period 2 contains many more easy cases, which retain the higher success rate.
The supported conclusion: the aggregate rose while both category rates fell. It does not prove that staff became better at either kind of case. Nor does this table alone explain why the mix changed.
When is the ordinary mean safe?
Equal weights make the simple average of group averages valid. Unequal weights can also happen to produce the same result in special cases, such as identical group averages, but this is not a reliable shortcut. Reconstruct totals when the actual bases are supplied.
Missing-base check: if you only know two rates, 40% and 80%, their overall rate is not uniquely determined. Equal group sizes would produce 60%; a larger 40% group would pull it lower.
Explore the effect of changing group sizes
Enter two positive whole-number group sizes and rates from 0% to 100%. Keep the rates fixed, then change just one group size. The result moves towards the rate of the larger group. Displayed values are rounded to at most 12 significant digits. Rates in this demonstration are mathematical inputs; if actual rates were rounded, the combined rate is an estimate.
Weighted averages: six explained practice questions
Try all six questions before opening the worked answers below. Submit your choices to see your score and individual feedback. You can change your choices before submission, or go straight to the answers and explanations for self-study.
This is an original ReasoningCampus learning check. One point is awarded per correct answer. The total is not an SHL score, a percentile, a pass prediction or a personality profile.
Answers and complete worked explanations
Open any question below to study the correct answer, the alternatives, the method and the common trap. These explanations are available without completing the quiz.
Review the reasoning in any missed or uncertain answer. The score describes this small set only.
Question 1: Average handling time
Correct answer: B. 14 minutes
An average handling time means total minutes divided by total calls. Recover the minutes before combining the teams: the first team contributes 60 × 20 = 1,200 minutes and the second contributes 40 × 5 = 200 minutes. Across 100 calls, that is 1,400 ÷ 100 = 14 minutes per call. B is correct. The result is closer to 20 than to 5 because more calls come from the slower team.
A comes from (20 + 5) ÷ 2 = 12.5, which gives the two teams equal weight despite their different call volumes. C would imply 1,500 total minutes, but only 1,400 are supplied. D adds the two averages to get 25, which is not an average and even exceeds both group averages. Keep “minutes per call” in your working: it tells you exactly what belongs in the denominator.
Question 2: Unequal success volumes
Correct answer: D. 54%
Convert each percentage into the number of successful cases. Group A contributes 10 × 0.90 = 9 successes. Group B contributes 90 × 0.50 = 45. There are 54 successes among 100 cases, so the combined rate is 54 ÷ 100 × 100 = 54%, answer D.
The 70% in A is the unweighted mean of 90% and 50%. It treats the ten-case group as if it were as large as the ninety-case group. B reports only A’s rate and ignores the much larger group B. C reports only B’s rate; adding A’s higher-rate cases must raise the combined figure above 50%. Before using a calculator, expect a result quite close to 50%, because nine out of ten cases belong to B.
Question 3: A valid equal-weight shortcut
Correct answer: A. The groups contain equal numbers of observations
Let the two group means be a and b and let each group contain n observations. The combined mean is (n × a + n × b) ÷ (n + n). Cancelling the common n gives (a + b) ÷ 2, which is exactly the simple mean. Equal group sizes therefore guarantee the shortcut, so A is correct.
B does not help: two percentages may be based on ten cases and a thousand cases. C concerns the labels, which do not affect contributions or weights. D says which mean is larger but tells us nothing about how many observations support it. Identical group means are another special case where weights do not change the result, but that is not one of these options. The question asks which stated condition guarantees the equality, not which makes it look plausible.
Question 4: Combined profit margin
Correct answer: C. 12.73%
Read the requested measure literally: combined profit divided by combined revenue. The profits add to 30 + 40 = 70 and the revenues to 150 + 400 = 550. Calculate 70 ÷ 550 × 100 = 12.7272…%, then round only the final result to 12.73%. That selects C.
Unit A’s margin is 30 ÷ 150 = 20%, whereas B’s is 40 ÷ 400 = 10%. Their simple mean is 15%, option A, but it ignores B’s larger revenue. Options B and D each give one unit’s margin instead of the combined margin. Because more revenue comes from the 10% unit, the combined result should be closer to 10% than to 20%. That sense-check supports 12.73% before you inspect the final decimal places.
Question 5: A changing case mix
Correct answer: B. The change in group proportions can explain the aggregate rise
Rebuild the overall result for each period. Initially, the easy cases produce 20 × 0.90 = 18 successes and the difficult cases 80 × 0.60 = 48: 66 successes out of 100. Later, the easy cases produce 80 × 0.80 = 64 and the difficult cases 20 × 0.50 = 10: 74 out of 100. The overall rate rises because the later workload contains many more easy cases, even though both category rates fall. B captures this explanation.
A contradicts those falling category rates. C is disproved by the arithmetic: both totals can be reconstructed from the table. D makes a claim about staff skill that the data cannot establish. Changing the mixture of work can change the headline result without showing an improvement in like-for-like performance. Compare both the category rates and their weights before interpreting an overall trend.
Question 6: Two rates without group sizes
Correct answer: D. It cannot be uniquely determined from these facts
Write the missing information into the formula. If the group sizes are n and m, the combined rate is (0.40n + 0.80m) ÷ (n + m). We know the rates but neither weight, so there is no single numerical answer. For example, equal groups give 60%; groups of 100 at 40% and 300 at 80% give 70%. Both fit the information provided. D is correct.
B assumes equal sizes without evidence. A and C each use one group’s rate as the overall answer and ignore the other group. With both groups having positive size, the combined rate must lie between 40% and 80%, but that interval does not identify one exact value. Recognising missing information is part of numerical reasoning; it is not a failure to find a more complicated calculation.
ReasoningCampus SHL numerical preparation: from averages to data
Our online SHL preparation includes numerical practice, theory and worked explanations. This lesson demonstrates how we teach one calculation skill before combining it with data interpretation. You can study the free examples first, then inspect the numerical materials on the full preparation page.
Choose your next practice step
- You averaged two percentages without checking group sizes. Rebuild the numbers of successes and observations in the branch example.
- Your formula is right but your quantities use different scales. Study the chart-units lesson before repeating the calculation.
- Your combined result changes when the group mix changes. Explain which group gained weight. Then try the changing-case-mix question without assuming that every group improved.
Explore the ReasoningCampus SHL preparation materials and choose the areas named in your assessment invitation.
Weighted averages: practical FAQs
Should I average the percentages shown in a chart?
Only when the question’s relevant weights justify doing so. If percentages describe groups with different sizes, recover the counts or contributions first.
Can an overall rate improve while every subgroup gets worse?
Yes. The original case-mix example shows exactly that. A shift towards the higher-rate group can outweigh lower rates within both groups.
Do I weight profit margin by cost or revenue?
If the question defines margin as profit divided by revenue, use revenue. If it asks about markup on cost, the denominator is cost. Follow the supplied definition.
What if the chart percentages are rounded?
Your reconstructed total may be approximate. Keep full precision during calculation and use the rounding instruction in the question. Do not claim an exact count from rounded rates.
Why is my answer outside both group averages?
With positive weights and comparable units, a combined average stays between the group averages. Check units, missing groups, the denominator and whether you added percentages.
Is the calculator an SHL scoring tool?
No. It demonstrates a mathematical relationship taught by ReasoningCampus. It does not convert practice performance into SHL scores or percentiles.
Official sources and how this lesson was prepared
The SHL references below provide the assessment context. The teaching method, fictional examples, diagrams, tools and explained practice answers are ReasoningCampus learning material; they are not official SHL questions or scoring rules.
AI assistance was used in research, drafting and implementation. Calculations, logic and the page tools were checked during preparation. These lessons are independent educational material; the methods are not SHL scoring rules. Read about ReasoningCampus or send a correction or question.
