UGC NET COMMERCE November 2017(Paper-II) Q18
0. Which one of the following is not the characteristics of normal distribution?
- Option : D
- Explanation : Characteristics of the Normal Probability Distribution: There is a family of normal distributions. Each normal distribution may have a different mean μ or standard deviation σ. A unique normal distribution may be defined by assigning specific values to the mean μ and standard deviation σ in the normal probability density function. Large value of σ reduce the height of the curve and increase the spread; small values of σ increase the height of the curve and reduce the spread. Figure 1 shows three normal distributions with different values of the mean μ and a fixed standard deviation σ, while in Figure 2 normal distributions are shown with different values of the standard deviation σ and a fixed mean μ.
From Figs. 1 and 2 the following characteristics of a normal distribution and its density function may be derived:
(i) For every pair of values of μ and σ, the curve of normal probability density function is bell shaped and symmetric.
(ii) The normal curve is symmetrical around a vertical line erected at the mean μ with respect to the area under it, that is, fifty per cent of the area of the curve lies on both sides of the mean and reflect the mirror image of the shape of the curve on both sides of the mean μ. This implies that the probability of any individual outcome above or below the mean will be same. Thus, for any normal random variable x,
P(x ≤ μ) = P(x ≥ μ) = 0.50
(iii) Since the normal curve is symmetric, the mean, median and mode for the normal distribution are equal because the highest value of the probability density function occurs when value of a random variable, x = μ.
(iv) The two tails of the normal curve extend to infinity in both directions and theoretically never touch the horizontal axis.
(v) The mean of the normal distribution may be negative, zero, or positive as shown in Fig. (1).
(vi) The mean μ determines the central location of the normal distribution, while standard deviation σ determines its spread. The larger the value of the standard deviation σ, the wider and flatter is the normal curve, thus showing more variability in the data, as shown in Fig. (2). Thus standard deviation σ determines the range of values that any random variable is likely to assume.
(vii) The area under the normal curve represents probabilities for the normal random variable, and therefore, the total area under the curve for the normal probability distribution is 1.
