There are 60 students in a class. The following is the frequency distribution of the marks obtained by the students in a test:
Where x is a positive integer. Determine the mean and standard deviation of the marks.
Given there are 60 students in a class. The frequency distribution of the marks obtained by the students in a test is also given.
Now we have to find the mean and standard deviation of the marks.
It is given there are 60 students in the class, so
∑fi=60
⇒ (x – 2) + x + x2 + (x + 1)2 + 2x + x + 1 = 60
⇒ 5x – 1 + x2 + x2 + 2x + 1 = 60
⇒ 2x2 +7x = 60
⇒ 2x2 +7x – 60 = 0
Splitting the middle term, we get
⇒ 2x2 + 15x – 8x – 60 = 0
⇒ x (2x + 15) – 4(2x + 15) = 0
⇒ (2x + 15) (x – 4) = 0
⇒ 2x + 15 = 0 or x – 4 = 0
⇒ 2x = -15 or x = 4
Given x is a positive number, so x can take 4 as the only value.
And let assumed mean, a=3.
Now put x = 4 and a = 3 in the frequency distribution table and add other columns after calculations, we get
And we know standard deviation is
$\sigma=\sqrt{\frac{\sum f_{i} d_{i}^{2}}{n}-\left(\frac{\sum f_{i} d_{i}}{n}\right)^{2}}$
Substituting values from above table, we get
$\sigma=\sqrt{\frac{78}{60}-\left(\frac{-12}{60}\right)^{2}}$
Substituting values from above table, we get
$\sigma=\sqrt{\frac{78}{60}-\left(\frac{-12}{60}\right)^{2}}$
$\sigma=\sqrt{1.3-(0.2)^{2}}$
$\sigma=\sqrt{1.3-0.04}$
$\Rightarrow \sigma=1.12$
Hence the standard deviation is $1.12$
Now mean is
$\overline{\mathrm{x}}=\mathrm{A}+\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{d}_{\mathrm{i}}}{\mathrm{N}}$
$=3+\left(-\frac{12}{60}\right)$
$=3-\frac{1}{5}=\frac{14}{5}$
$=2.8$
Hence the mean and standard deviation of the marks are $2.8$ and $1.12$ respectively.
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- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,