Question

Which of the following are correct regarding the normal distribution curve?
(i) Symmetrical about line $X = \mu $ (Mean)
(ii) Mean = Median = Mode
(iii) Unimodal
(iv) Points of inflexion are at $X = \mu \pm \sigma $
A) (i), (ii)
B) (ii), (iv)
C) (i), (ii), (iii)
D) All of these

Answer 1

Hint:
In the given question, we are asked to find what is correct regarding the normal distribution curve. By looking at the properties of the normal distributive, the given question can be solved easily. Find out which points are correct in regards to the normal distribution curve.
Thus, choose the correct option.

Complete step by step solution:
In the given question, we are asked to find which of the following is correct regarding the normal distribution curve.
Symmetrical about line $X = \mu $ (Mean)
Mean = Median = Mode
Unimodal
Points of inflexion are at $X = \mu \pm \sigma $
In a normal distribution, the values of mean, median and mode are equal. So, the point (ii) is correct in the context of a normal distribution curve.
Also, the normal distribution curve or bell curve is symmetric about the line $X = \mu $ . So, the point (i) is also correct in this context.

Now, for a normal curve to be Unimodal the first derivative must be 0 at $X = \mu $ and it must be less than 0 for $X > \mu $ and greater than 0 for $X < \mu $ , which is the exact case here. So, point (iii) is also correct.
For points of inflexion, the second derivative must be 0 at those points. Here, the second derivative is 0 at $X = \mu + \sigma $ and $X = \mu - \sigma $ . So, the points of inflexion are $X = \mu \pm \sigma $. Thus, point (iv) is also correct.

Hence, option (D) is the correct answer as all the given four points are correct in the context of a normal distribution curve.

Note:
Properties of a normal distribution curve:
i) The mean, mode and median are all equal.
ii) The curve is symmetric at the centre, i.e. around mean $\mu $.
iii) Exactly half of the values are to the left of centre and exactly half the values are to the right.
iv) The total area under the curve is 1.