Quizizz worksheet on quartiles of ungrouped data featuring multiple-choice questions on statistical concepts.

Let’s solve each question step by step.

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Question 1: The mean is equal to the 2nd quartile.

- The mean is the average of all numbers.
- The 2nd quartile (Q2) is the median — the middle value when data is ordered.
- These are only equal in perfectly symmetric distributions (like a normal bell curve), but not always.
- So, saying “the mean is equal to the 2nd quartile” as a general rule is false.

✔ Answer: A False

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Question 2: What is the value such that 25% of data is less than it and 75% is greater?

- That’s the definition of the lower quartile (Q1).
- Q1 = 25th percentile → 25% below, 75% above.

✔ Answer: D Lower Quartile

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Question 3: In computing upper quartile, if U falls halfway between two integers, round ______.

- This depends on the method taught. But commonly in school-level stats (especially with small datasets), if the position is halfway (e.g., 6.5), you round up to get the next integer position.
- Example: For n=9, Q3 position = 0.75*(9+1) = 7.5 → take average of 7th and 8th? Or sometimes just round up to 8th?
- But many curricula say: if it's halfway, round up for simplicity in ungrouped data.

Wait — let’s check standard practice:

Actually, there are multiple methods. But since this is likely following a specific textbook or curriculum (Quizizz lesson), and given common teaching at this level:

→ If the index is halfway (like 6.5), some systems say round up, others say interpolate.

But looking at Questions 4 and 5, we can infer what method they’re using.

Let’s hold off and come back after solving 4 and 5.

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Question 4 & 5: Same dataset: 25, 8, 37, 16, 45, 40, 29, 12, 42

First, sort the data:

Original: 25, 8, 37, 16, 45, 40, 29, 12, 42
Sorted: 8, 12, 16, 25, 29, 37, 40, 42, 45

Number of data points: n = 9

We need to find:

- Lower Quartile (Q1)
- Upper Quartile (Q3)

Common method for ungrouped data (used in many schools):

Use positions:

- Q1 position = (n + 1)/4 = (9 + 1)/4 = 10/4 = 2.5
- Q3 position = 3(n + 1)/4 = 3*10/4 = 30/4 = 7.5

So:

→ Q1 is halfway between 2nd and 3rd values:
2nd = 12, 3rd = 16 → average = (12 + 16)/2 = 14

Wait — but 14 is not among the answer choices! Choices are: 37, 16, 40, 29

Hmm… maybe they’re using a different method?

Alternative method (sometimes used): Use floor or ceiling without averaging?

Or perhaps they use:

Q1 = value at position ceil((n+1)/4) ? Let’s try another approach.

Another common method (especially in older textbooks):

Divide data into halves.

Sorted: [8, 12, 16, 25, 29, 37, 40, 42, 45]

Median (Q2) is 5th value = 29

Lower half: first 4 values: 8, 12, 16, 25 → median of these = (12 + 16)/2 = 14 → again not an option.

Upper half: last 4 values: 37, 40, 42, 45 → median = (40 + 42)/2 = 41 → not an option either.

Still doesn’t match.

Wait — look at the answer choices for Q4 (lower quartile): A 37, B 16, C 40, D 29

And Q5 (upper quartile): A 40, B 16, C 29, D 37

Maybe they’re using a simpler method: just pick the value at position rounded from formula?

Try:

Position for Q1 = (n * 0.25) = 9 * 0.25 = 2.25 → round to nearest whole number? 2 → 2nd value = 12 → not an option.

Or (n+1)*0.25 = 2.5 → if they round up, then 3rd value = 16 → which IS an option (B).

Similarly, Q3 = (n+1)*0.75 = 7.5 → if round up, 8th value = 42 → not an option.

Wait — 8th value is 42, but options for Q5 are 40, 16, 29, 37.

If they round down for Q3: 7.5 → 7th value = 40 → which IS option A for Q5.

Ah! So perhaps:

- For Q1: position 2.5 → round up → 3rd value = 16
- For Q3: position 7.5 → round down → 7th value = 40

That matches the options!

Check:

Sorted:
1:8, 2:12, 3:16, 4:25, 5:29, 6:37, 7:40, 8:42, 9:45

Q1 = 3rd value = 16 → matches option B for Q4
Q3 = 7th value = 40 → matches option A for Q5

Perfect.

So for Question 3: When position is halfway (like 2.5 or 7.5), how do they handle it?

In Q1: 2.5 → they took 3rd → so rounded up

In Q3: 7.5 → they took 7th → so rounded down? Wait, inconsistency.

Wait — no: for Q3, if they rounded down from 7.5 to 7, that’s rounding down.

But for Q1, 2.5 to 3 is rounding up.

That’s inconsistent.

Unless... maybe they use different rules for lower and upper? Unlikely.

Alternative idea: Perhaps they use the method where you don't average, but take the value at the position given by:

For Q1: position = (n+1)/4 = 2.5 → since it's .5, take the higher of the two? i.e., 3rd → 16

For Q3: position = 3(n+1)/4 = 7.5 → take the lower? 7th → 40

Still inconsistent.

Wait — let’s see what happens if we use the "inclusive" method or something else.

Another possibility: Maybe they are using the method where:

Q1 = median of lower half including median if odd? No.

Let me recalculate using the method that gives us the answers matching the options.

Given that:

- Q1 = 16 → which is 3rd value
- Q3 = 40 → which is 7th value

Positions: 3 and 7

How to get 3 from n=9?

(9+1)/4 = 2.5 → if we always round up for both, then Q3 would be 8th = 42, not 40.

If we always round down, Q1 would be 2nd = 12, not 16.

So neither works for both.

Unless... for Q1, they round up; for Q3, they round down? That seems arbitrary.

Wait — perhaps they use:

Q1 position = floor((n+1)/4) + 1? Not sure.

Let’s think differently.

Maybe they are using the method where:

The lower quartile is the median of the first half, excluding the overall median.

Data: 8,12,16,25,29,37,40,42,45

Median is 29 (5th)

Lower half: 8,12,16,25 → median of these is (12+16)/2 = 14 → not an option.

Upper half: 37,40,42,45 → median = (40+42)/2 = 41 → not an option.

Not working.

Perhaps they include the median in both halves? Then lower half: 8,12,16,25,29 → median = 16 → yes! That’s option B.

Upper half: 29,37,40,42,45 → median = 40 → yes! Option A.

Oh! So they are using the method where:

- Split the data at the median, and include the median in both halves when finding Q1 and Q3.

So:

Full sorted: 8,12,16,25,29,37,40,42,45

Lower half including median: 8,12,16,25,29 → 5 values → median is 3rd = 16

Upper half including median: 29,37,40,42,45 → 5 values → median is 3rd = 40

Perfect! Matches the options.

So for Question 4: lower quartile = 16 → B

Question 5: upper quartile = 40 → A

Now, back to Question 3: “In computing for the upper quartile, if U falls halfway between two integers, round ______.”

In this case, for the upper half: 29,37,40,42,45 — we didn’t have a halfway situation because it was odd number of values.

But in general, if when finding the median of a subset, the position is halfway, what do we do?

In our calculation above, for the upper half of 5 values, position = (5+1)/2 = 3 → exact, no issue.

But suppose we had even number.

Example: if upper half was 4 values: say 37,40,42,45 — then median position = (4+1)/2 = 2.5 → halfway between 2nd and 3rd.

What did we do earlier? We averaged them: (40+42)/2=41.

But in the actual problem, they avoided that by including the median, making halves odd-sized.

However, the question is general: “if U falls halfway between two integers, round ___”

In many school contexts, especially if they want a single value (not average), they might say round up.

But in our successful method for Q4 and Q5, we didn’t encounter halfway, so we can’t directly infer.

Looking back at the options for Q3: A Up, B Down

And since in the position calculation for Q1 and Q3 using (n+1)/4 etc., we had 2.5 and 7.5, and to get 3 and 7, we effectively:

- For 2.5 → went to 3 (up)
- For 7.5 → went to 7 (down)

Inconsistent.

But wait — in the method we used (including median in both halves), we never used those formulas; we just took medians of subsets.

So perhaps for Question 3, it’s referring to a different method.

Maybe the "U" refers to the position calculated as 0.75*n or something.

Let’s assume the most common instruction in such quizzes: when the position is halfway, round up.

I recall that in some curricula, for quartiles, if the index is not integer, you round up to the next whole number.

For example, in Excel or some software, but for education, often simplified.

Given that for Q1, 2.5 became 3 (up), and if we applied same to Q3, 7.5 should become 8, but we got 40 which is 7th, so contradiction.

Unless for Q3 they used a different formula.

Perhaps they use:

Q1 position = (n+1)/4 = 2.5 → round up to 3

Q3 position = 3n/4 = 3*9/4 = 6.75 → round up to 7 → 7th value = 40

Yes! That could be it.

Some methods use:

- Q1: (n+1)/4
- Q3: 3n/4 or 3(n+1)/4

Let’s calculate:

If Q3 position = 3n/4 = 27/4 = 6.75 → round up to 7 → 7th value = 40

Q1 position = (n+1)/4 = 10/4 = 2.5 → round up to 3 → 3rd value = 16

Perfect! And consistent: always round up when fractional.

So for Question 3: if U (position) falls halfway or any fraction, round up.

Even though 6.75 is not halfway, but 2.5 is halfway, and they rounded up.

The question says "if U falls halfway", implying specifically when it's .5, but probably means when it's not integer.

And in this case, they round up.

So answer should be A Up

Now Question 6: Interpretation of upper quartile value from Q5, which is 40.

Upper quartile (Q3) = 40 means that 75% of the data is less than or equal to 40, and 25% is greater.

Standard interpretation: Q3 is the value below which 75% of the data falls.

So: "seventy-five percent of the scores are less than or equal to 40"

Look at options:

A: seventy-five percent greater than or equal to 37 → no

B: seventy-five percent less than or equal to 40 → yes

C: seventy-five percent greater than or equal to 40 → no, that would be top 25%

D: seventy-five percent less than or equal to 37 → no

So B is correct.

But let’s read carefully:

Option B: "It means that seventy-five percent of the scores are less than or equal to 40."

Yes, that’s standard.

Sometimes people say "less than", but technically, since 40 is included, and in our data, 40 is present, and it's the 7th out of 9, so 7/9 ≈ 77.8% are ≤40, which is more than 75%, but quartile is defined as the value where at least 75% are below or equal.

In this case, since we have discrete data, it's approximate, but the interpretation is still that Q3 represents the 75th percentile.

So B is correct.

Now summarize all answers:

1. A False
2. D Lower Quartile
3. A Up
4. B 16
5. A 40
6. B It means that seventy-five percent of the scores are less than or equal to 40.

Final Answer:

1. A
2. D
3. A
4. B
5. A
6. B