Weighted averages and combined rates: guide and free practice

Do not average averages blindly

If one team has ten staff and another has thirty, their averages do not carry equal weight. Recover each total by multiplying its average by its group size. Add those totals and divide by the combined headcount.

Choose the correct weight

The weight might be employees, units sold, hours worked or assessment credits. A price weighted by units is not the same as a price weighted by revenue. Label the quantity represented by each weight before calculating.

Use the bounds as a check

A combined average with positive weights lies between the component averages. If your answer is outside that interval, check the denominator and units. When group sizes are equal, the ordinary mean of the two averages is valid; otherwise it generally is not.

Recover a missing group

For a missing average, calculate the known combined total, subtract the known group's total and divide the remainder by the missing group's size. This also works for finding the score needed on a final weighted assessment.

Watch time and distance

Average speed equals total distance divided by total time. Equal-time and equal-distance journeys use different weights. The exercises below include staffing, unit costs, weighted scores and journeys so you can practise choosing the base rather than applying one formula mechanically.

Before this lesson

Know how to multiply a mean by a count and convert percentages into fractions. Start with numerical reasoning or percentage change if either step is unfamiliar.

A repeatable method

1. Name the measure requested: score per person, cost per item, time per call or share of respondents.
2. Identify each group's matching weight. Make sure the groups do not overlap.
3. Recover each total: group mean × group count.
4. Add the totals, then divide by the combined count.
5. Check the units and whether the result lies between the group means.

In symbols written as words: weighted mean = sum of (value × weight) divided by sum of weights. Weights such as 3 and 2 do not need to add to one; the denominator normalises them. Percentage weights totalling 100% already sum to one when written as decimals.

Worked example: unequal groups

Four trainees average 65 points and six average 85. Their totals are 260 and 510, so ten trainees score 770 points together. The combined mean is 77. It lies between 65 and 85 and nearer 85 because that group is larger. The unweighted mean of 75 would incorrectly treat four people and six people as equal-sized groups.

Worked example: missing subgroup

A group of 24 averages 70 points, so its total is 1,680. Nine members average 60, contributing 540. The other fifteen contribute 1,140, giving a mean of 76. Subtracting the two means, 70 minus 60, would not measure the remaining group's average.

Worked example: pooled percentages

A desk resolves 40 of 200 requests immediately; another resolves 36 of 120. Their rates are 20% and 30%. The combined rate is 76 out of 320, or 23.75%. Weight the rates by the numbers of requests, not by how many requests succeeded. A simple 25% mean assumes equal request counts.

When changing group mixes matter

A combined average can rise even when each subgroup average falls. For example, a later period may contain many more long calls and fewer short calls. Calculate the totals and counts separately for each period before claiming that the aggregate proves a change in each group's performance.

What rounded summaries cannot tell you

If the component means were rounded, combining them generally gives an estimate rather than an exact result. Do not invent decimal precision that the original data do not support. Likewise, two means without group sizes may be insufficient to calculate a combined mean.

Common mistakes

• Weighting the wrong quantity: survey satisfaction among respondents uses respondent counts, not everyone invited.
• Changing the denominator incorrectly: a corrected observation changes the total but not the number of observations; removing a person changes both.
• Averaging journey speeds: divide total distance by total time. Equal distance and equal time are different assumptions.
• Ignoring the bounds: with positive weights, a mean cannot lie outside the smallest and largest component values.
• Counting overlapping groups twice: the ordinary pooled formula assumes the totals can be added without duplicating observations.

Continue from the right difficulty

Foundation practice covers totals and basic weights. Intermediate practice adds missing groups, changing headcounts and pooled rates. Advanced practice asks you to recover weights, correct summaries and reason about changing group mixes. These are teaching labels rather than calibrated ability bands. Next, try rates and productivity or table interpretation.

Your first practice session

Use the actual group size, time or unit count as the weight. A calculator is allowed. Do not assume equal weights unless stated.

Try 10 questions with explanations

Choose your practice difficulty

Difficulty labels describe the steps needed in these teaching sets; they are not validated ability bands. Start with foundation practice and move on after you can explain your method.

• Foundation: 10 questions — group totals, missing averages and combined rates.
• Intermediate: 10 questions — Choose the right weights for staffing, prices, service rates and course scores in ten original problems.
• Advanced: 10 questions — Recover missing weights, distinguish pooled and subgroup results, and handle rounded summaries in ten original problems.

Continue learning

• Numerical reasoning
• Rates and productivity
• Interpreting workplace tables
• Ratios and proportion in workplace problems

About this practice

These are independently written teaching questions. A practice score helps you review your answers; it is not a validated hiring prediction or an official employer score.