Compare Averages: Determining if 4,4,4,4 Equals 1,4,5,7

Q: Do I need to calculate both averages even if one group has identical numbers?

Yes, always calculate both! Even when one group has identical numbers like 4,4,4,4, you still need to find its average (which equals that repeated number) to make a proper comparison.

Q: What if the groups have different numbers of values?

The average formula automatically accounts for different group sizes! Just divide each group's sum by its own count. That's why we use averages instead of just comparing sums.

Q: How do I know which inequality symbol to use?

Calculate both averages first, then compare the numerical results. If Average A is smaller than Average B, use <. If larger, use >. If equal, use =.

Q: Can I use decimals or do I need fractions?

Both work perfectly! $ \frac{17}{4} = 4.25 $ and 4.25 are the same value. Use whichever form makes the comparison easier for you.

Average Comparison with Same Sample Sizes

Calculate the average of each group and choose the appropriate sign (?):

$4,4,4,4\stackrel{?}{=}1,4,5,7$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the average of each group and choose the appropriate sign:

00:05 To calculate the average, we need to divide the sum by the number of occurrences

00:32 This is the average, now let's use the same method for the second group

01:02 Let's compare and match the sign

01:09 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Calculate the average of each group and choose the appropriate sign (?):

$4,4,4,4\stackrel{?}{=}1,4,5,7$

Step-by-step solution

To solve this problem, we'll calculate the average of each group and compare them.

Step 1: Calculate the average of the first group $4, 4, 4, 4$ :
$\text{Sum of Group A} = 4 + 4 + 4 + 4 = 16$
$\text{Number of elements in Group A} = 4$
$\text{Average of Group A} = \frac{16}{4} = 4$
Step 2: Calculate the average of the second group $1, 4, 5, 7$ :
$\text{Sum of Group B} = 1 + 4 + 5 + 7 = 17$
$\text{Number of elements in Group B} = 4$
$\text{Average of Group B} = \frac{17}{4} = 4.25$
Step 3: Compare the two averages:
$\text{Average of Group A} = 4$
$\text{Average of Group B} = 4.25$
Since $4 < 4.25$ , the correct inequality is $4 < 4.25$ .

Therefore, the appropriate sign to use is $<$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Average equals sum of values divided by count
Calculation: Group A: (4+4+4+4)÷4 = 4, Group B: (1+4+5+7)÷4 = 4.25
Verification: Check sums: 16÷4 = 4 and 17÷4 = 4.25, so 4 < 4.25 ✓

Common Mistakes

Avoid these frequent errors

Comparing individual numbers instead of averages
Don't compare 4 vs 7 or look at highest values = wrong comparison! This ignores the overall group performance. Always calculate the complete average for each group before comparing.

Practice Quiz

Test your knowledge with interactive questions

Calculate the average of $$ 11 $$ and $$ 7 $$ .

FAQ

Everything you need to know about this question

Do I need to calculate both averages even if one group has identical numbers?

Yes, always calculate both! Even when one group has identical numbers like 4,4,4,4, you still need to find its average (which equals that repeated number) to make a proper comparison.

What if the groups have different numbers of values?

The average formula automatically accounts for different group sizes! Just divide each group's sum by its own count. That's why we use averages instead of just comparing sums.

How do I know which inequality symbol to use?

Calculate both averages first, then compare the numerical results. If Average A is smaller than Average B, use <. If larger, use >. If equal, use =.

Can I use decimals or do I need fractions?

Both work perfectly! $\frac{17}{4} = 4.25$ and 4.25 are the same value. Use whichever form makes the comparison easier for you.

What if my sums are really close together?

Even tiny differences matter! If one sum is 16 and another is 17, their averages will be different. Always complete the division to get the exact averages before comparing.

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