NDA Maths · Statistics

Regression and Correlation

How two variables move together — correlation measures the strength of the link, regression draws the best-fit line.

Why this matters

27 PYQs across 2017–2026, with 6 HARD — the highest hard-rate of any Statistics subtopic. Almost every recent paper asks one of three shapes: properties of the correlation coefficient r under linear transformation, finding regression lines, identifying which equation is which, or computing the angle between them. Five tight concepts cover the entire surface.

Test yourself — a quick 12-question Statistics recall quiz, instant score.

Concept 1 of 5: Correlation Coefficient and Its Properties

Correlation coefficient

r

is a single number between

-1

and

+1

that summarises how strongly two variables move together.

r = +1

is perfect positive (one rises, the other rises by the same proportion);

r = -1

is perfect negative;

r = 0

means no linear relationship. Crucially,

r

is unaffected by shifts (change of origin) and unaffected in magnitude by positive scale changes — but a negative scale flips its sign.

Definition

For paired observations, $r = \dfrac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$ , bounded by $-1 \leq r \leq 1$ . If $U = aX + b$ and $V = cY + d$ , then $r_{UV} = \text{sign}(ac)\,r_{XY}$ — magnitude is preserved, sign flips when one of $a, c$ is negative.

Correlation Coefficient and Invariance Rule

r = \dfrac{\text{Cov}(X,Y)}{\sigma_X\,\sigma_Y} \qquad r_{(aX+b,\,cY+d)} = \text{sign}(ac)\,r_{XY}

$\text{Cov}(X,Y)$ covariance of X and Y
$\sigma_X,\sigma_Y$ standard deviations of X and Y
$\text{sign}(ac)$ $+1$ if $a, c$ have same sign, $-1$ otherwise

Visualization · slide r, watch the cloud tighten

r = 0.80

At r = ±1 the points fall exactly on a line; toward r = 0 the cloud loses any linear shape. Positive r slopes up, negative r slopes down. r is unitless and always lies in [−1, 1] — it captures tightness and direction, not how steep the line is.

Worked example

The correlation between

x

and

y

r = 0.6

. Find the correlation between

U = 2x + 5

and

V = -3y + 1

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Sep 2023 · Q111Easy

Example 1 · Statistics · Regression and Correlation

r

is the correlation coefficient between

x

and

y

, then what is the correlation coefficient between

(3x+4)

and

(-3y+3)

$r$ is bounded by $-1$ and $+1$ — always

If a calculation gives

|r| > 1

, the arithmetic is wrong. Use this as a sanity check at the end of any correlation computation.

Shift does not change $r$ ; only scale-with-negative-sign flips it

Adding constants to either variable is invisible to

r

. Multiplying by a positive constant is also invisible. Only a negative multiplier flips the sign — and even then, the magnitude is preserved.

Drill 8 more on correlation coefficient and its properties

Concept 2 of 5: Lines of Regression

For two variables there are TWO regression lines —

y

x

(used to predict

y

from

x

) and

x

y

(used to predict

x

from

y

). Both lines always pass through the mean point

(\bar{x}, \bar{y})

. If

r = \pm 1

the two lines coincide; otherwise they intersect at

(\bar{x}, \bar{y})

at a non-zero angle.

Definition

The regression line of $y$ on $x$ has slope $b_{yx} = r\,\dfrac{\sigma_y}{\sigma_x}$ and passes through $(\bar{x}, \bar{y})$ . The regression line of $x$ on $y$ has slope $b_{xy} = r\,\dfrac{\sigma_x}{\sigma_y}$ and also passes through $(\bar{x}, \bar{y})$ .

Lines of Regression (point-slope form)

y - \bar{y} = b_{yx}(x - \bar{x}) \qquad x - \bar{x} = b_{xy}(y - \bar{y})

$b_{yx}$ slope of $y$ on $x$ line $= r\,\sigma_y/\sigma_x$
$b_{xy}$ slope of $x$ on $y$ line $= r\,\sigma_x/\sigma_y$
$(\bar{x},\bar{y})$ the only point on BOTH regression lines

Visualization · drag the line, watch the error

Left endpoint y: 1.0Right endpoint y: 11.0

SSE = 2.97Best possible: 0.27

Red dashes are residuals (vertical distance from each point to the line). SSE is the sum of their squares. The least-squares regression line is the one that makes SSE as small as possible.

Worked example

Find the regression line of

y

x

passing through the only two data points

(1, 1)

and

(5, 9)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Apr 2023 · Q109Easy

Example 2 · Statistics · Regression and Correlation

A bivariate data set contains only two points

(-1,1)

and

(3,2)

. What will be the line of regression of

y

x

Both regression lines always pass through $(\bar{x}, \bar{y})$

If a PYQ gives you two regression lines and asks for the means, solve the two equations simultaneously — their intersection is exactly

(\bar{x}, \bar{y})

. No need to compute anything from raw data.

From raw bivariate data, compute $b_{yx}$ via the Pearson form

When given

n

raw paired points (e.g. four

(x_i, y_i)

values), use the computational formula

b_{yx} = \dfrac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2}

with

\bar{x}, \bar{y}

read straight from the column sums. The regression line is then

y - \bar{y} = b_{yx}(x - \bar{x})

. Faster than the deviation-from-mean form because it works directly off the column totals.

Drill 7 more on lines of regression

Concept 3 of 5: Regression Coefficients and Their Link to r

The two regression slopes

b_{yx}

and

b_{xy}

carry the same information as

r

— their product equals

r^2

, and they share the same sign as

r

. This means once you know any two of

\{r, b_{yx}, b_{xy}\}

, the third is forced.

Definition

$b_{yx} \cdot b_{xy} = r^2$ , with $0 \leq r^2 \leq 1$ . Therefore $b_{yx} \cdot b_{xy} \leq 1$ always. Also $\text{sign}(b_{yx}) = \text{sign}(b_{xy}) = \text{sign}(r)$ — the two slopes can never have opposite signs. The coefficients themselves: $b_{yx} = r\,\dfrac{\sigma_y}{\sigma_x}$ and $b_{xy} = r\,\dfrac{\sigma_x}{\sigma_y}$ (multiply them and the SDs cancel to $r^2$ ). This is what a "given $\bar x, \bar y, \sigma_x, \sigma_y, r$ , write the regression equation" question wants: the line of $y$ on $x$ is $y - \bar y = b_{yx}(x - \bar x)$ , the line of $x$ on $y$ is $x - \bar x = b_{xy}(y - \bar y)$ , both through $(\bar x, \bar y)$ . Coefficient of determination: $r^2$ is the fraction of the variation in $y$ explained by the regression on $x$ ; $1 - r^2$ is the unexplained fraction. With $r = 0.8$ , 64% is explained, 36% is not.

Product Identity

b_{yx} \cdot b_{xy} = r^2, \qquad r = \pm\sqrt{b_{yx}\,b_{xy}}

Sign of $r$ same as the common sign of $b_{yx}$ and $b_{xy}$

Worked example

Two lines of regression are

3x - 5y + 10 = 0

and

5x - 3y - 6 = 0

. Find the correlation coefficient

r

between

x

and

y

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Apr 2026 · Q111Hard

Example 3 · Statistics · Regression and Correlation

Let two lines of regression be

x+y+11=0

and

2x+3y+4=0

for some data. What is the value of correlation coefficient between x and y?

$b_{yx} \cdot b_{xy} \leq 1$ is non-negotiable

If your computed product exceeds 1, you have assigned the wrong line to

y

x

. Swap the assignment and recompute — the inequality

b_{yx} b_{xy} = r^2 \leq 1

picks the correct pairing every time.

Both slopes share the sign of $r$

You cannot have

b_{yx} > 0

and

b_{xy} < 0

— if a problem seems to suggest this, the lines have been labelled wrong.

Drill 7 more on regression coefficients and their link to r

Concept 4 of 5: Identifying Which Regression Line is Which

When NDA gives you two equations and doesn't label them, you must figure out which is

y

x

and which is

x

y

. Use the inequality

b_{yx}\,b_{xy} \leq 1

as a sieve: there are only two possible pairings; one will satisfy the inequality, the other won't.

Definition

Two regression lines $L_1$ and $L_2$ can be paired in two ways. The correct pairing is the one for which the product of the slopes (interpreted as $b_{yx} \cdot b_{xy}$ ) is at most 1. The wrong pairing always gives a product greater than 1 (provided the lines are distinct).

Sieve Inequality

\text{Correct pairing satisfies } b_{yx}\,b_{xy} \leq 1; \text{ wrong pairing gives } > 1.

Diagram · which regression line is which

Both lines pass through the mean point (x̄, ȳ). The y-on-x line is the flatter one (it minimises vertical gaps); x-on-y is steeper. Their slopes are b_yx and 1/b_xy, with b_yx·b_xy = r².

Worked example

Two lines of regression are

2x - 3y + 1 = 0

and

4x - 5y + 3 = 0

. Identify which is

y

x

and find

b_{yx}

and

b_{xy}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Sep 2024 · Q101Hard

Example 4 · Statistics · Regression and Correlation

Let

x-3y+4=0

and

2x-7y+8=0

be two lines of regression computed from some bivariate data. If

b_{yx}

and

b_{xy}

are regression coefficients of lines of regression of

y

x

and

x

y

respectively, then what is the value of

b_{xy}+7b_{yx}

Try the inequality before doing anything else

Always compute both candidate pairings of

b_{yx} \cdot b_{xy}

. The pairing that satisfies the

\leq 1

condition is the correct one. Don't try to reason geometrically from the slopes — the inequality is mechanical and unambiguous.

Drill 4 more on identifying which regression line is which

Concept 5 of 5: Angle Between the Two Regression Lines

The two regression lines coincide when correlation is perfect and stand perpendicular when there is no correlation. In between, they make an acute angle whose tangent you can read straight off the line equations using the ordinary "angle between two lines" formula from coordinate geometry — no need to compute

r

\sigma_x

\sigma_y

first.

Definition

Treat the two regression lines as ordinary straight lines in the $(x,y)$ plane with slopes $m_1$ and $m_2$ (read directly from each equation after solving for $y$ ). The acute angle $\theta$ between them satisfies the standard formula below. When $r = \pm 1$ the slopes coincide and $\tan\theta = 0$ ; when $r = 0$ , $1 + m_1 m_2 = 0$ and the lines are perpendicular. In terms of the statistics, $m_1 = b_{yx} = r\sigma_y/\sigma_x$ and $m_2 = 1/b_{xy} = \sigma_y/(r\sigma_x)$ , which gives the textbook form $\tan\theta = \dfrac{1-r^2}{|r|}\cdot\dfrac{\sigma_x\sigma_y}{\sigma_x^2+\sigma_y^2}$ . If a question hands you the angle as a multiple ("the lines make an angle $3\theta$ "), the triple-angle formula $\tan 3\theta = \dfrac{3t - t^3}{1 - 3t^2}$ with $t = \tan\theta$ converts it.

Angle between two lines (applied to regression)

\tan\theta = \left|\dfrac{m_1 - m_2}{1 + m_1\,m_2}\right|

$m_1, m_2$ slopes of the two regression lines in the $(x,y)$ plane
$\theta$ acute angle between the lines

Diagram · angle between the regression lines

The lines meet at (x̄, ȳ) at angle θ, where tan θ = |(m₂ − m₁) / (1 + m₁m₂)|. As correlation strengthens (r → ±1) the two lines rotate together and θ → 0 — they coincide at perfect correlation. As r → 0 they splay apart, signalling no linear relationship.

Worked example

Two lines of regression are

x + 3y + 2 = 0

and

2x + 5y + 1 = 0

. Find the tangent of the acute angle between them.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Apr 2024 · Q104Hard

Example 5 · Statistics · Regression and Correlation

Let

x+2y+1=0

and

2x+3y+4=0

be two lines of regression computed from some bivariate data. If

\theta

is the acute angle between them, then what is the value of

488\tan 3\theta

Slope of the $x$ -on- $y$ line is NOT $b_{xy}$ in the $(x,y)$ plane

The slope of the

y

-on-

x

line in the

(x,y)

plane is

b_{yx}

. But the slope of the

x

-on-

y

line (written

x = a + b_{xy} y

) in the

(x,y)

plane is

1/b_{xy}

, NOT

b_{xy}

. When reading slopes off the line equation directly (solve for

y

, take the coefficient of

x

), you sidestep this trap.

Acute angle only — take absolute value

A negative tangent would correspond to the obtuse supplement. The formula's absolute value guarantees

\theta \leq 90^\circ

. PYQs always ask the acute angle.

Drill 1 more on angle between the two regression lines

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (5)

Correlation Coefficient and Its Properties
Correlation Coefficient and Invariance Rule
$r = \dfrac{\text{Cov}(X,Y)}{\sigma_X\,\sigma_Y} \qquad r_{(aX+b,\,cY+d)} = \text{sign}(ac)\,r_{XY}$
Lines of Regression
Lines of Regression (point-slope form)
$y - \bar{y} = b_{yx}(x - \bar{x}) \qquad x - \bar{x} = b_{xy}(y - \bar{y})$
Regression Coefficients and Their Link to r
Product Identity
$b_{yx} \cdot b_{xy} = r^2, \qquad r = \pm\sqrt{b_{yx}\,b_{xy}}$
Identifying Which Regression Line is Which
Sieve Inequality
$\text{Correct pairing satisfies } b_{yx}\,b_{xy} \leq 1; \text{ wrong pairing gives } > 1.$
Angle Between the Two Regression Lines
Angle between two lines (applied to regression)
$\tan\theta = \left|\dfrac{m_1 - m_2}{1 + m_1\,m_2}\right|$

Watch out for (9)

$r$ is bounded by $-1$ and $+1$ — always
→ Correlation Coefficient and Its Properties
Shift does not change $r$ ; only scale-with-negative-sign flips it
→ Correlation Coefficient and Its Properties
Both regression lines always pass through $(\bar{x}, \bar{y})$
→ Lines of Regression
From raw bivariate data, compute $b_{yx}$ via the Pearson form
→ Lines of Regression
$b_{yx} \cdot b_{xy} \leq 1$ is non-negotiable
→ Regression Coefficients and Their Link to r
Both slopes share the sign of $r$
→ Regression Coefficients and Their Link to r
Try the inequality before doing anything else
→ Identifying Which Regression Line is Which
Slope of the $x$ -on- $y$ line is NOT $b_{xy}$ in the $(x,y)$ plane
→ Angle Between the Two Regression Lines
Acute angle only — take absolute value
→ Angle Between the Two Regression Lines

Test yourself on Statistics

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Related notes

Measures of Central Tendency — both regression lines pass through $(\bar{x}, \bar{y})$ →