# Normal Distribution

One of the most important of the many distributions which occur in the study of statistics is the normal, or Gaussian distribution. The equation of the normal curve is 1 (x−μ)2 /2σ where f(x) is the frequency function. The total area under this curve, from x = −∞ to x = +∞, is one square unit. Therefore the area between any two points, say from x = a to x = b, is the proportion of cases which lie between the two points. A plot of Eq. 1 yields a bell-shaped curve having a maximum ordinate at x = μ and decreasing to zero at both +∞ and −∞ If the standard deviation σ is small, the curve is tall and thin; if σ is large, the curve is short and broad; see the Figure 1 below- f(x)=√ e 2π [1] Figure 1 : The Shape of the normal distribution curve (a) Small (b) Large In order to analyze different statistical cases, it is convenient to replace x in Eq. 1 with the standard variable z defined as z=x−μ [2] σ where now x is to be normally distributed with a mean of zero. Equation 1 then becomes a unit normal distribution and is written as 1 x2/2 f(x)=√ e [3] 2π The distribution is illustrated in the following figure, Figure 2. Figure 2 : The standard normal distribution

2/3 The ordinate at z is f(z), obtained using Eq 3. The areas Q(z) and P(z) under the curve can be found in statistical tables or approximated. Since the total area under the curve is unity P (z) = 1 − Q(z) [4] The term Q(z) is obtained by integrating the area under the curve. It can be approximated and then programmed for calculation using a polynomial expansion (Hewlett-Packard Applications Book, op. cit., p. 107). The results area found to be where Q(z) = f(z)(b1t + b2t2 ∗ b3t3 + b4t4 + b5t5) + θ(z) [5] t = 1 , p = 0.2316419, |θ| < 7.5 × 10−8 1+p|z| and b1 = 0.319381530, b2 = −0.356563782, b3 = 1.781477937, b4 = −1.821255978, b5 = 1.330274429. From the above equation we can calculate Q(z) for any distribution. In this problem you have to determine the standard deviation and mean of some given data. Then for some given value find the number of population falling in the region Q(z) and P(z). Input The input consists of several segments. The first line of input of each segment is an integer N (0 < N < 231) indicating the population. The next line contains N real numbers describing the population data (such as age of people, length of shaft or diameter of bearing etc.). The following line contains an integer Q (0 < Q < 1000) number of query. And the next line similarly contains Q number describing the query. Output For each set of input output the data set number followed by mean, standard deviation, then 3Q numbers answering the queries. Each answer to query describes P(z) then Q(z) and then percentage of population T falling in Q(z) area. Every number must be rounded to the nearest fourth digit after the decimal point. Sample Input 11 96 93 90 86 86 80 78 76 68 84 100 2 60 100 13 78 60 74 82 96 68 94 86 78 90 87 56 88 3 100 30 59 Sample Output Data Set #1 Mean = 85.1818 Standard Deviation = 9.3895 P(z) = 0.9963, Q(z) = 0.0037, T = 0.0403 P(z) = 0.9427, Q(z) = 0.0573, T = 0.6299

3/3 Data Set #2 Mean = 79.7692 Standard Deviation = 12.4777 P(z) = 0.9475, Q(z) = 0.0525, T = 0.6821 P(z) = 1.0000, Q(z) = 0.0000, T = 0.0004 P(z) = 0.9520, Q(z) = 0.0480, T = 0.6241