Two sets each of 20 observations have the same standard derivation 5. The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.
Given two sets each of 20 observations, have the same standard derivation 5 . The first set has a mean 17 and the second a mean 22 .
Now we have to show that the standard deviation of the set obtained by
combining the given two sets
As per given criteria, for first set
Number of observations, $n_{1}=20$
Standard deviation, $\mathrm{s}_{1}=5$
And mean, $\bar{x}_{1}=17$
For second set, number of observations, $\mathrm{n}_{2}=20$
Standard deviation, $\mathrm{S}_{2}=5$
And mean, $\bar{x}_{2}=22$
We know the standard deviation for combined two series is
S. D $(\sigma)=\sqrt{\frac{\mathrm{n}_{1} \mathrm{~s}_{1}^{2}+\mathrm{n}_{2} \mathrm{~s}_{2}^{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}+\frac{\mathrm{n}_{1} \mathrm{n}_{2}\left(\overline{\mathrm{x}}_{1}-\overline{\mathrm{x}}_{2}\right)^{2}}{\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)^{2}}}$
Substituting the corresponding values, we get
S. D $(\sigma)=\sqrt{\frac{(20)(5)^{2}+(20)(5)^{2}}{20+20}+\frac{(20 \times 20)(17-22)^{2}}{(20+20)^{2}}}$
On simplifying we get
S. D $(\sigma)=\sqrt{\frac{(20) 25+(20) 25}{40}+\frac{(400)(-5)^{2}}{(40)^{2}}}$
S. $D(\sigma)=\sqrt{\frac{2(500)}{40}+\frac{(400)(25)}{1600}}$
S. D $(\sigma)=\sqrt{\frac{1000}{40}+\frac{10000}{1600}}$
S. D $(\sigma)=\sqrt{25+\frac{25}{4}}$
Taking LCM and simplifying,
S. D $(\sigma)=\sqrt{\frac{100+25}{4}}$
S. D $(\sigma)=\sqrt{\frac{125}{4}}$
$\mathrm{Or}, \sigma=5.59$
Hence the standard deviation of the set obtained by combining the given two sets is $5.59$
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All Study Material
- 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,