Calculate the Daily Average: Analyzing Factory Defective Item Data

Q: Why can't I just average the four daily rates (4, 7, 0, 18)?

Because each rate applies to a different number of days! The 18 defective items happened for 10 days, so it has much more impact than the 0 defective items that only happened for 1 day.

Q: What's the difference between a simple average and weighted average?

A simple average treats all values equally. A weighted average considers how often each value occurs. Here, some daily rates lasted longer, so they 'weigh' more in the final answer.

Q: How do I know when to use weighted averages?

Use weighted averages when you have rates or values that apply for different amounts of time. Look for phrases like 'for X days' or 'over Y periods' in the problem.

Q: Can I solve this problem a different way?

Yes! You could list out all 16 individual days: 4,4,4,7,7,0,18,18,18... then find the simple average. But the weighted average method is much faster!

Q: What if the answer isn't a nice whole number?

That's normal! $ 12.875 $ means on average there are about 12-13 defective items per day. Real-world averages are often decimals.

Weighted Averages with Multi-Day Data

Below are the data from a factory quality control:

Over 3 days they found 4 defective items each day.

Over 2 days they found 7 defective items each day.

Over 1 day there were no defective items at all.

Over 10 days there were 18 defective items each day.

How many defective items are there on average each day?

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Understand the problem

Step-by-step solution

To solve this problem, we'll calculate the average number of defective items per day using weighted averages:

Step 1: Compute the total number of defective items.
- 3 days × 4 defective items/day = 12 defective items
- 2 days × 7 defective items/day = 14 defective items
- 1 day × 0 defective items/day = 0 defective items
- 10 days × 18 defective items/day = 180 defective items
Step 2: Sum all defective items calculated: $12 + 14 + 0 + 180 = 206 \text{ defective items}$
Step 3: Compute the total number of days: $3 + 2 + 1 + 10 = 16 \text{ days}$
Step 4: Calculate the average number of defective items per day using: $\frac{206}{16} = 12.875$

Therefore, the average number of defective items per day is $12.875$ .

Final Answer

$12.875$

Key Points to Remember

Essential concepts to master this topic

Formula: Total items divided by total days gives weighted average
Technique: Multiply days × daily count: 3×4 = 12, 2×7 = 14
Check: Sum all products (12+14+0+180=206) then divide by total days (16) ✓

Common Mistakes

Avoid these frequent errors

Adding daily averages instead of using weighted average
Don't add 4+7+0+18 and divide by 4 = 7.25! This treats each period equally when they have different durations. Always multiply each daily rate by its number of days first, then divide total items by total days.

Practice Quiz

Test your knowledge with interactive questions

Norbert buys some new clothes.

When he gets home, he decides to work out how much each outfit cost him on average.

What answer should he come up with?

FAQ

Everything you need to know about this question

Why can't I just average the four daily rates (4, 7, 0, 18)?

Because each rate applies to a different number of days! The 18 defective items happened for 10 days, so it has much more impact than the 0 defective items that only happened for 1 day.

What's the difference between a simple average and weighted average?

A simple average treats all values equally. A weighted average considers how often each value occurs. Here, some daily rates lasted longer, so they 'weigh' more in the final answer.

How do I know when to use weighted averages?

Use weighted averages when you have rates or values that apply for different amounts of time. Look for phrases like 'for X days' or 'over Y periods' in the problem.

Can I solve this problem a different way?

Yes! You could list out all 16 individual days: 4,4,4,7,7,0,18,18,18... then find the simple average. But the weighted average method is much faster!

What if the answer isn't a nice whole number?

That's normal! $12.875$ means on average there are about 12-13 defective items per day. Real-world averages are often decimals.

How can I check if 12.875 is reasonable?

Look at your data: most days had 18 defective items, some had 4 or 7, and one had 0. Since 18 happened for 10 out of 16 days, the average should be closer to 18 than to the smaller numbers. $12.875$ makes sense!

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