Aggregation Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Aggregation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Aggregation is the process of combining many individual data values into a single summary statistic such as a sum, mean, count, or proportion.

Going from individual values to totals, averages, or other summaries.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Aggregation combines a pile of individual data values into a single number like a sum, count, mean, or proportion.

Common stuck point: The procedure for aggregation is the easy part; the trap is aggregating groups of different sizes by raw totals. Asking "Am I collapsing many individual values into a single summary number for the group?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I collapsing many individual values into a single summary number for the group?

Worked Examples

Example 1

medium

Simpson's Paradox: Hospital A has a 90% recovery rate overall. Hospital B has 85%. However, for severe cases: A has 70%, B has 75%; for mild cases: A has 98%, B has 95%. Explain the paradox.

Answer

Simpson's Paradox: aggregated rates can reverse when a confounding variable (case severity mix) is ignored.

First step

Hospital B is better for BOTH severe (75%>70%) and mild (95% vs 98%... wait: A=98>95=B) — let's check: for mild cases A=98%, B=95%, so A is better for mild

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Example 2

easy

Daily temperatures: Mon=20°, Tue=22°, Wed=19°, Thu=25°, Fri=21°. Calculate the weekly mean and explain what aggregation loses and preserves.

Example 3

easy

Worked example: a store has

4

checkouts. In one hour they process

20,30,25,25

customers. Total processed and average per checkout?

Example 4

medium

Worked example: Hospital X cures

90/100

of mild cases and

40/100

of severe; Hospital Y cures

80/100

of mild and

30/100

of severe. Compare each per case type.

Example 5

hard

Worked example (Simpson's paradox): School Old has

80\%

acceptance for STEM (

90/100

Liberal Arts ignored) wait — show that combining acceptance rates from two departments can reverse the comparison.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Monthly sales ($thousands): Jan–Mar: 50, 60, 55; Apr–Jun: 80, 90, 85; Jul–Sep: 40, 35, 45. Calculate quarterly totals and annual total. What pattern does aggregation reveal?

Example 2

hard

A school reports average SAT scores improved 20 points. But when broken down by income group, scores dropped for every income group. Explain this as Simpson's Paradox and identify the mechanism.

Example 3

easy

The scores

80, 90, 70

are combined into a single class average. What aggregation operation is this?

Example 4

easy

A store records

5

sales of $10 each. What is the aggregated total revenue?

Example 5

easy

A survey has

200

'yes' out of

500

responses. What is the aggregated proportion of 'yes'?

Example 6

easy

Replacing every student's exact score with the class average loses what kind of information?

Example 7

easy

Two groups: group A has

3

items summing to

30

; group B has

2

items summing to

10

. What is the combined (pooled) mean of all

5

items?

Example 8

easy

Is the average of two group averages always equal to the overall average? Yes or no.

Example 9

easy

A dashboard shows total website visits per month. This single number is an example of what?

Example 10

easy

Why might combining 'age 5–10' and 'age 60–70' survey results into one average be a poor aggregation?

Example 11

medium

Hospital A treats

100

patients,

90

survive (

90\%

). Hospital B treats

100

patients,

80

survive (

80\%

). But A took

90

mild +

10

severe cases, B took

10

mild +

90

severe. For mild cases A saved

87/90

and B saved

10/10

; for severe A saved

3/10

and B saved

70/90

. Which hospital is better per case type?

Example 12

medium

A weighted average: group of

40

scores averages

90

; group of

10

scores averages

60

. Find the overall mean.

Example 13

medium

Batting: Player X has

1

hit in

2

at-bats early (

.500

) and

48

hits in

98

later (

\approx .490

). Player Y has

40/100

early (

.400

) and

1/2

late (

.500

). Compute each player's overall average.

Example 14

medium

A company reports average employee salary rose from $50k to $55k year over year, yet every employee took a pay cut. How is this possible through aggregation?

Example 15

medium

Sales by region: North

\$300

(3 reps), South

\$200

(1 rep). The CEO compares 'average sales per region' vs 'average sales per rep.' Compute both.

Example 16

medium

Daily temperatures for a week are aggregated to a single mean of

20°

. A meteorologist warns this is insufficient. What two aggregates together would describe the week better?

Example 17

medium

Three classes have means

70, 80, 90

with sizes

10, 20, 70

. Estimate the overall mean and explain why it is closer to

90

than to

80

Example 18

medium

A retailer aggregates 'average order value' as total revenue / number of orders. Revenue

= \$12000

, orders

= 300

. Find the AOV, then state what it hides.

Example 19

medium

A factory aggregates output across two shifts: day made

300

units in

10

hours, night made

120

units in

6

hours. Compute the combined units-per-hour rate.

Example 20

challenge

Construct a minimal Simpson's paradox: two groups where group 1 beats group 2 in EACH of two subcategories, yet group 2 wins overall. Give explicit small fractions.

Example 21

challenge

Prove that the pooled mean of two groups always lies between their individual means

\bar{x}_1

and

\bar{x}_2

(assume

\bar{x}_1 \le \bar{x}_2

Example 22

challenge

A city's overall average commute time fell from

30

28

minutes, yet commute time rose in every neighborhood. Explain the mechanism and what data shift makes this consistent.

Example 23

easy

A class of

25

students has a total score of

2000

on a test. What is the class mean score?

Example 24

easy

In a survey,

120

out of

400

people answered 'yes'. What proportion is this?

Example 25

easy

Daily steps over

5

days:

8000,9000,7000,11000,5000

. What is the total (aggregate sum)?

Example 26

easy

For the daily steps

8000,9000,7000,11000,5000

, find the mean.

Example 27

medium

Group A:

30

students with mean

75

. Group B:

20

students with mean

85

. Find the combined mean.

Example 28

medium

A region's quarterly sales (in $000): Q1

120

, Q2

150

, Q3

90

, Q4

180

. What is the annual mean per quarter?

Example 29

medium

A store sold

300

items at $5,

200

items at $8, and

100

items at $20. What is the average selling price per item?

Example 30

medium

A team scored

4,7,6,8,5

goals over

5

games. What is the mean goals per game and the total?

Example 31

medium

A factory's three lines produce

200

units (line 1),

300

units (line 2),

500

units (line 3) in a day. Defect rates are

5\%

4\%

2\%

respectively. Aggregate defect rate?

Example 32

medium

Two regions: North averages $50 per sale across

10

sales; South averages $80 across

40

sales. Find the overall average sale value.

Example 33

medium

A bookstore tracks

5

-week sales:

40,50,60,80,70

books. Find the weekly mean and the proportion of weeks above the mean.

Example 34

hard

Two classes: A has

20

students mean

70

; B has

30

students mean

80

. After a

5

-point grading curve added to class A only, find the new pooled mean.

Example 35

hard

Company average salary went from $50k to $60k, but the median fell. What composition shift could explain this?

Example 36

hard

A web tool has

1000

free users averaging

5

minutes/day, and

200

paid users averaging

30

minutes/day. What is the aggregated mean usage per user per day?

Example 37

hard

In a 3-month period a delivery service handled

1000

1500

, and

2000

packages with on-time rates

90\%

80\%

70\%

. What is the on-time rate across the period?

Example 38

hard

A statistic is robust if it is not strongly affected by outliers. Which aggregate is more robust: mean or median? Justify briefly.

Example 39

challenge

Construct a Simpson's paradox dataset: two groups, two subcategories, where in each subcategory Group 1's rate beats Group 2's, but Group 2 has a higher overall rate. Use small integer counts.

Example 40

challenge

Show algebraically that for two groups with means

\bar x_1\le\bar x_2

and sizes

n_1,n_2>0

, the pooled mean

\bar x=\frac{n_1\bar x_1+n_2\bar x_2}{n_1+n_2}

satisfies

\bar x_1\le\bar x\le\bar x_2

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Related Concepts

Mean Data (Abstract)

Background Knowledge

These ideas may be useful before you work through the harder examples.

meandata abstract