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NORMAL DISTRIBUTION — Statistics STD 12 Commerce — Question

Gujarat BoardEnglish MediumSTD 12 CommerceStatisticsNORMAL DISTRIBUTION3 Marks
Question
State the properties of normal distribution.
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Answer

The normal distribution possesses the following properties:
$(1)$ The normal distribution is a probability distribution of continuous random variable.
$(2)\ \mu$ and $\sigma$ are its parameters.
$(3)$ The graph of density function of this distribution is continuous graph of bell- shape.
$(4)$ The normal curve is symmetric abou $x=\mu$
$(5)$ The total area under normal curve is and the area of the region of normal curve o both sides of $X=\mu$ is equal and $0.5$.
$(6)$ The mean, median and mode of the distribution are equal, i.e., $\mu=M=M_{0}$
$(7)$ The skewness of this distribution is zero.
$(8)$ The first quartile $Q_{1}$ and third quartile $Q_{3}$ are equidistant from the second quartile $Q_{2}$ in this distribution, i.e., $Q_{3}-Q_{2}=Q_{2}-Q_{1}$ and $M =\frac{Q_{3}+Q_{1}}{2}$.
$(9)$ In the form of $\mu$ and $\sigma$ the estimate values of $Q_{1}$ and $Q_{3}$ are obtained as follows: $Q_{1}=\mu-0.675 \sigma$ and $Q_{3}=\mu+0.675 \sigma$
$(10)$ Two tails of the curve of normal distribution are asymptotic to $X$-axis, i.e., they never touch $X$-axis.
$(11)$ In this distribution, $(i)$ Quartile deviation $=\frac{2}{3} \sigma$ $(ii)$ Mean deviation $=\frac{2}{3} \sigma$
$(12)$ The important areas under normal curve are as follows: $(i)$ The area under the normal curve between $\mu \pm \sigma=0.6826$ $(ii)$ The area under the normal curve between $\mu \pm 2 \sigma=0.9545$ $(iii)$ The area under the normal curve between $\mu \pm 3 \sigma=0.9973$ $(iv)$ The area under the normal curve between $\mu \pm 1.96 \sigma=0.95$ $(v)$ The area under the normal curve between $\mu \pm 2.575 \sigma=0.99$

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