Correlation — Pearson Correlation Coefficient — Part D

Correlation — Pearson Correlation Coefficient — Part D. Practice questions to deepen understanding of correlation — Pearson correlation coefficient — Part D. Online statistics practice with full solutions and step-by-step explanations.

Pearson correlation Part D advanced practice — computation + in-depth interpretation, limitations of r, extreme points, biases, when Pearson is not appropriate.

Understanding + computations + in-depth interpretation (stage 4 — advanced).

30 questions

Question 1

3.33 pts

📈 Question 1 — An outlier changes the whole picture: In a scatter plot of X and Y there is a weak positive relationship (r≈0.20). After adding one outlier in the upper-right corner, r jumps to 0.85. What is the correct conclusion?

r is very sensitive to outliers, so one point can bias the relationship and create an illusion of a strong association.

The increase in r proves that the relationship is truly strong.

If a point is added, r must always increase.

An outlier does not affect r at all.

Explanation:

r is very sensitive to extreme points. One distant point can pull the trend line toward it, so r can increase sharply. The increase does not necessarily mean the true relationship became stronger; it may simply be distortion caused by the outlier.

Question 2

3.33 pts

📉 2 — No :
U: asX , Y . r?

r No — because .

r 1.

r -1.

r No .

Explanation:

— No . r , Yes U . No : " " → r . No Correct. r , No .

Question 3

3.33 pts

📊 3 — :
“X Y”. r = 0.70.

small , — because .

?

r ignores scale — it measures only how close points are to a straight line, not the spread itself

always more.

large r .

two different clusters cannot give the same r

Explanation:

r — No . between , r . : ” large” → r small. Correct — No .

Question 4

3.33 pts

🔄 4 — :
r between X Y?

r will remain completely identical.

r will increase.

r will decrease.

r will become 0.

Explanation:

: \[ r_{xy} = r_{yx} \] X Y No Yes No r. : X → Y Y → X. — equal.

Question 5

3.33 pts

📐 5 — , :
, find .

r?

r will drop slightly but will still remain very high.

r will remain exactly the same.

r will drop to near zero.

r will become negative.

Explanation:

r — , . No outlier . : “” correlation. .

Question 6

3.33 pts

🧮 6 — because :
X Y between 1–5. X Y between 100–500. : 100 .

r?

r .

r large .

r small .

r .

Explanation:

r calculate . No between Yes r . r “”.

Question 7

3.33 pts

📊 7 — negative correlation :
.

r Yes?

-0.98

-0.30

0.98

Explanation:

negative correlation r -1. -0.98 . : negative correlation “” — No.

Question 8

3.33 pts

🧠 8 — No :
Y X , . r ?

No — because because.

high positive

high negative

calculate r.

Explanation:

No (No always No always ). “asX Y /”. because → → r No. correlation “because ”.

Question 9

3.33 pts

🧮 9 — sample small :
3 . Correct r?

calculate r, No — .

calculate r 10 .

r must be 0

1 -1.

Explanation:

calculate r 3 — . . : sample small "" r — No Correct, .

Question 10

3.33 pts

📊 10 — :
, .

r?

r small — .

r .

r No .

r .

Explanation:

, → r . No : “ ”. — more "".

Question 11

3.33 pts

📈 Question 11 — A strong relationship with two opposite outliers: The graph shows an almost straight increasing line, but at the lower-left and upper-right edges there are two opposite outliers. What is expected to happen to r?

r will decrease significantly because the outliers act in opposite directions and damage the linearity.

r will increase

r will remain completely unchanged

r will become negative

Explanation:

When two opposite outliers pull the pattern in different directions, they seriously damage the straight-line structure. Therefore r decreases substantially. A common mistake is to think that one outlier cancels the other; in practice, both harm the correlation.

Question 12

3.33 pts

📉 Question 12 — Very different variance between X and Y: X has huge variance (values 1 to 1,000). Y has small variance (values 10 to 12). But the relationship structure is linear. What will happen to r?

r will be very high, because r standardizes deviations and different variances do not interfere.

r will be low because of asymmetry.

r will be negative.

r cannot be calculated.

Explanation:

r does not measure scale. It standardizes deviations in relative terms rather than absolute units. Therefore, even if X is widely spread and Y changes only slightly, a linear structure can still produce a high r.

Question 13

3.33 pts

🧠 13 — sample small + Outlier:
sample 4 , , Outlier .

Correct?

r — because sample small outliers.

r because .

r .

r No sample.

Explanation:

. known: r sample small . : ” ”. — outlier large more.

Question 14

3.33 pts

📈 14 — , small:
between X Y Y = 3X , because small .

r?

r will be very close to 1 but not exactly 1

r 1 .

r because .

r .

Explanation:

+ small = r No 1. 1. : ” ” = r=1. small r — .

Question 15

3.33 pts

🔄 15 — because Y 1/Y:
because because No ( 1/Y), r?

because between — No .

r will always remain the same

r will automatically become negative

r will become 1

Explanation:

No ( 1/Y) . — . r linear relationship Yes . log — No .

Question 16

3.33 pts

📊 16 — (spurious correlation):
r between between .

Correct?

— Yes .

because .

always because .

r between No .

Explanation:

(population/ ). r “ ”. r ≠ . more statistic.

Question 17

3.33 pts

📉 17 — negative correlation :
. r?

r — because .

r .

r -1.

r No .

Explanation:

because — No . , . No → r No No large . r≈-0.9 “because ”. No .

Question 18

3.33 pts

🧮 18 — No :
X 2, Y 3. r?

r does not change at all — multiplying by constants does not change the relative pattern

r .

r small.

r .

Explanation:

, : r . Yes No . : ”Y more → r more”. No Correct.

Question 19

3.33 pts

📊 19 — :
equal between between ; between . r≈0.75.

?

The strength of the linear relationship is the same, even if the nature of the variables is completely different.

You cannot compare r between different fields.

A high r always implies causation.

r only measures positive correlation.

Explanation:

r — . Yes equal between . : . : r No .

Question 20

3.33 pts

🧠 20 — ?
(). r?

r — .

r will increase

r will decrease

r will become negative

Explanation:

( ), No Yes r No . : “more = more r”. No Correct — , No .

Question 21

3.33 pts

📉 21 — :
: , . — .

r?

r will be close to zero — the two opposing trends cancel each other out

r will be high and positive

r will be high and negative

r cannot be calculated in this case

Explanation:

because because — — r . : r≈0 . known statistic: “” .

Question 22

3.33 pts

🧮 22 — positive correlation common ratio (non-linear monotonic):
, ( ). r?

r 1 — because No .

r 1.

r 0.

r .

Explanation:

always () (No ), correlation No . “” , “” . Yes r below1.

Question 23

3.33 pts

📈 23 — ( correlation):
between X Y (r≈0.8). No Z . ?

between X Y — Z.

because X Y.

Z No .

r always .

Explanation:

Z X Y , r between X Y. (spurious correlation). : “ r — ”. No Correct. Yes correlation .

Question 24

3.33 pts

🧠 24 — :
X , Y . , r≈0.90. ?

the strength of the relationship does not depend on absolute variance, but on the relative deviation pattern

the small variance of Y should cause r to be low

the large variance of X should cause r to be negative

r is undefined here

Explanation:

r . , No . X “” Y “small”, — r . Yes large/ No because .

Question 25

3.33 pts

📊 25 — :
positive correlation (r≈0.20). — r≈0.75.

?

— because .

No — r mean .

always correlation.

r must be 0

Explanation:

: , between — Yes r . correlation . more statistic.

Question 26

3.33 pts

📉 26 — linear relationship X because :
sample X=5 , Y . r?

r — because X No .

r 0.

r .

calculate r X .

Explanation:

X . r — , r . : “ because — ”. No Correct.

Question 27

3.33 pts

📈 27 — :
, .

r?

moderate positive (around 0.40–0.60) — trend exists but noise weakens linearity

perfect positive

strongly negative

0 because .

Explanation:

= large mean → No → r . , . : “ = ”. No Correct. , No .

Question 28

3.33 pts

🔄 Question 28 — Sign reversal in only part of the sample: In the first half of the sample, the relationship is positive. In the second half, the relationship is negative. What is the expected overall r?

close to zero — opposite directions cancel each other out

high and positive

high and negative

impossible to calculate

Explanation:

When two parts of the data pull in opposite directions, many products \((x-\bar{x})(y-\bar{y})\) cancel each other out. The result is r≈0. This does not mean there is no relationship; it means there is no single linear relationship.

Question 29

3.33 pts

📊 29 — r ?
Y ( 100). r?

does not change at all — a change of units is a constant multiplication

r .

r small.

r .

Explanation:

r — because . Yes No r. .

Question 30

3.33 pts

📉 30 — :
X small — Y large. X — Y small. X large — Y large.

r ?

close to zero — non-linear (inverted U-shaped) relationship

high and positive

high and negative

-1.

Explanation:

∩ — 😅 because No No . , → → r≈0. : ” ” → r . No — r No between .