Question:
If \(\mu_1,\mu_2\) and \(\mu_3\) are the mean, median and mode respectively of a normal distribution, then
\[
3(\text{Median})-2(\text{Mode})
=
?
\]
If \(\mu_1,\mu_2\) and \(\mu_3\) are the mean, median and mode respectively of a normal distribution, then
\[
3(\text{Median})-2(\text{Mode})
=
?
\]
Show Hint
Remember the standard empirical relation:
\[
\text{Mode}=3(\text{Median})-2(\text{Mean}).
\]
It is one of the most important formulas in descriptive statistics.
Updated On: Jun 25, 2026
- \(\mu_1\)
- \(\mu_2\)
- \(\mu_3\)
- \(0\)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept:
For a moderately skewed distribution, the empirical relationship among mean, median and mode is
\[
\text{Mode}
=
3(\text{Median})
-
2(\text{Mean}).
\]
This relation is one of the most frequently used results in statistics.
For a normal distribution specifically,
\[
\text{Mean}=\text{Median}=\text{Mode},
\]
and the empirical relation is also satisfied.
Step 1: Write the empirical relation.
\[ \text{Mode} = 3(\text{Median}) - 2(\text{Mean}). \] Using the notation of the question, \[ \mu_3 = 3\mu_2 - 2\mu_1. \]
Step 2: Rearrange the expression.
Transposing terms, \[ 2\mu_1 = 3\mu_2-\mu_3. \] Therefore, \[ \mu_1 = \frac{3\mu_2-\mu_3}{2}. \] For a normal distribution, \[ \mu_1=\mu_2=\mu_3. \] Hence, \[ 3\mu_2-2\mu_3 = \mu_1. \]
Step 3: Substitute the required expression.
The quantity asked is \[ 3(\text{Median})-2(\text{Mode}) = 3\mu_2-2\mu_3. \] Using the relation above, \[ 3\mu_2-2\mu_3 = \mu_1. \] Therefore, \[ \boxed{\mu_1} \]
Step 1: Write the empirical relation.
\[ \text{Mode} = 3(\text{Median}) - 2(\text{Mean}). \] Using the notation of the question, \[ \mu_3 = 3\mu_2 - 2\mu_1. \]
Step 2: Rearrange the expression.
Transposing terms, \[ 2\mu_1 = 3\mu_2-\mu_3. \] Therefore, \[ \mu_1 = \frac{3\mu_2-\mu_3}{2}. \] For a normal distribution, \[ \mu_1=\mu_2=\mu_3. \] Hence, \[ 3\mu_2-2\mu_3 = \mu_1. \]
Step 3: Substitute the required expression.
The quantity asked is \[ 3(\text{Median})-2(\text{Mode}) = 3\mu_2-2\mu_3. \] Using the relation above, \[ 3\mu_2-2\mu_3 = \mu_1. \] Therefore, \[ \boxed{\mu_1} \]
Was this answer helpful?
0
0
Top TS PGECET Mathematics Questions
- If \[ \{(2,-3,5),\ (1,\alpha,7),\ (3,\beta,3)\} \] is not a basis of the vector space \(\mathbb{R}^3\), then \((\alpha,\beta)\) lies on the locus:
Consider the linear system of equations \[ \begin{bmatrix} 3 & -1 & 4 \\ 6 & 3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}. \] In this system of equations, if \(x\) is always a fixed constant, then the system has:& nbsp;
- Evaluate \[ \iint_D x^2y\,dx\,dy, \] where \[ D:\;x^2+y^2\le16. \]
- If \[ \overline{V}=x^{2}y\,\overline{i}-yz^{2}\,\overline{j}+xyz\,\overline{k} \] is the linear velocity of a particle in circular motion at the point \((1,1,1)\), then its angular velocity is
- Evaluate \[ \oint_C \frac{z+2}{z^2+2z+2}\,dz, \] where \(C\) is the circle \[ |z-\alpha|=1, \qquad \alpha=-1+\frac{3}{2}i. \]
View More Questions
Top TS PGECET Statistics Questions
Top TS PGECET Questions
- If \( P = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix} \) is the modal matrix of \( A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{pmatrix} \) then the sum of all the elements of \( P^{-1}AP \) is
- If the system of equations \( kx + y + z = k - 1 \), \( x + ky + z = k - 1 \), and \( x + y + kz = k - 1 \) has infinite solutions, then \( k \) is
- The value of the integral \( \int_C (2xy - x^2) \, dx + (x^2 + y^2) \, dy \) where \( C \) is the boundary of the region enclosed by \( y = x^2 \) and \( y^2 = x \), described in the positive sense, is
- Let \( f(x) = \log x \). The number \( C \) strictly between \( e^2 \) and \( e^3 \) such that its reciprocal is equal to \( \frac{f(e^3) - f(e^2)}{e^3 - e^2} \) is
- The solution of the differential equation \( (D^2 + 2)y = x^2 \) is
View More Questions