Normal Distribution - Essay Example

NORMAL DISTRIBUTION Table of contents 3 Introduction 3 Literaturereview 4 Objectives 5 Scope of study 5 Methodology 5 Discussion 18 Conclusion 18 Recommendations 18 References 19 Normal Distribution ABSTRACT Normal distribution curves are used to describe the trends of a given data to cluster around the mean. This paper seeks to identify whether the normal distribution is an accurate statistic method for analyzing a given system behavior. To determine this, the researcher first collects the data about the number of emails received per day for a period of 20 days. Then, the data collected is evaluated to find the mean, mode, median, variance and standard deviation. Using the collected data, frequency distribution graph and normal distribution curves are drawn. The normal distribution table is then subjected to the X 2 tests. From the results obtained a conclusion is drawn as to whether normal distribution curves are suitable for analysing any system behavior. INTRODUCTION A normal distribution curve is one of the most commonly used statistical tools. Many natural phenomena conform to a normal distribution with most elements clustering at the centre and a few extremes on the right and the left. Normal distribution is also called Gaussian distribution or the bell shaped distribution as the resulting graph has a bell shaped nature. Data tends to cluster near the mean or the average. The bell shaped curve is a probability density function with a peak at the mean. During the study of system behavior, the random variable is used to describe the unpredictable outcome. When a survey or an experiment is carried out, the data collected is also referred to as a variable. Variables can be classified as discrete random variable or continuous random variable. Discrete random variable consists of a set of data that takes discrete values that is, the values can be counted. These values are finite and denumerable. Continuous random variables are on the other hand are not denumerable (Cary 2008). Normal distribution has been used extensively in natural and social sciences and also in the evaluation of statistical data. It great use is derived from the fact that it's a simple model that represents complex data (Feller, 1968). LITERATURE REVIEW Abraham De Moivre introduced the normal distribution in the year 1733; this was printed in his book 'the doctrine of chances' in 1738. He used the normal distribution to evaluate large binomial distribution. The normal distribution theory was later extended by Laplace in 1812 to form the theorem of De Moivre-Laplace. Laplace used normal distribution to evaluate analytical errors during his experimental work. Gauss in 1809 used the method to analyse his astronomical work while Legendre used the method in 1805. The term bell shaped distribution function was first used by Esprit Jouffret in 1872 and later the graph was referred to as the normal distribution by Charles Pierce and Francis Galton. Normal distribution has been extensively used to study scientific and natural phenomena as well as analyse statistical data. The method is however not appropriate for the study of all phenomena, instead other distribution are preferred in some cases. The other popular distributions are the binomial and poison distribution. In light of this, the researcher seeks to subject a given collected data to normal distribution and determine its practicability and adaptability in the evaluation of a given system behavior. The researcher will collect data and from this data, draw the normal distribution curve and tests it overall suitability as a statistical tool. OBJECTIVES The main objective of this study is to determine whether the normal distribution curve is the most suitable method of analysing a system behavior. In this case, the number of emails sent per day. The specific objectives are; Collect the data. Calculate the mean, median, mode, variance and standard deviation. Draw the frequency distribution and the normal distribution curve. Carry out the X2 test. Determine the accuracy of using normal distribution curves to study a system behavior. SCOPE OF THE STUDY This study is limited to the determination of the effectiveness of the normal distribution curve in the evaluating a given system behavior. The study is limited to subjecting the normal distribution curve to the X2 test only and does not compare this distribution to other distribution methods. METHODOLOGY In evaluation of normal distribution curve it is necessary to conduct a survey using a practical example. This involves collecting of data, analysing the data to find the mean, variance and the standard deviation and then drawing the normal distribution curves. After which, the chi square test is carried out. The variable used in this case study is discrete as the number of emails received per day is always a whole number. The most important parameters in drawing the normal distribution curves are the mean and standard deviation. The arithmetic mean is calculated by the summation of the variable divided by the number of the variables. This is mathematically represented as; MEAN = sum/ counts (1) The standard deviation is the square root of the variance; variance can be defined as the mean of the square of the deviation of data from the mean. The formula for calculation of variance is VAR = MSD = (2) The standard deviation is calculated by; SDEV = (3) Where VAR = variance SDEV= standard deviation DEV = deviation from the mean F = frequency MSD= mean square of deviation (Durham business school) (a) Data Collection The first step was for the researcher to collect data from the internet. The researcher studied and collected data about the number of emails received each day. The data was collected for a period of 20 days. To avoid errors in data collection, the numbers of emails received were collected at 0000 hrs. The table below shows this information. Table 1 showing the number of emails collected per day for a period of 20 days Day Emails 1 7 2 4 3 7 4 3 5 9 6 11 7 5 8 8 9 2 10 6 11 6 12 8 13 4 14 4 15 9 16 3 17 6 18 9 19 2 20 5 (b) Compiling the data Arranging the data in terms of the number of email received Table 2 showing the number of emails and the frequency Number of emails Frequency/occurrence 2 2 3 2 4 3 5 2 6 3 7 2 8 2 9 3 11 1 (c) Frequency Distribution Graph Plotting the frequency distribution graph (number of emails received against frequency) (Durham business school) Fig 1 showing the frequency distribution graph From the above data it can be seen that the modal number of emails are 4, 6 and 9. Computing the cumulative frequency and the relative cumulative frequency for this discrete variable the following table is obtained Table 3 showing tabulated data for determining the cumulative frequency and relative cumulative frequency. Number of emails Frequency Cumulative frequency Relative cumulative Frequency % 2 2 2 10 3 2 4 20 4 3 7 35 5 2 9 45 6 3 12 60 7 2 14 70 8 2 16 80 9 3 19 95 10 0 19 95 11 1 20 100 Using this data it is possible to draw the cumulative frequency distribution for this set of discrete data. The graph is as shown below. Fig 2 showing the cumulative frequency graph From the graph above the median value is the value intersected by the 50 % relative cumulative frequency line. The median value is therefore "6". Emails above 8 lie in the upper quartile mark while email below 4 lie in the lower quartile mark. An alternative method for the computation of the median is as shown below. The numbers are arranged from the smallest to the largest and the middle value is found. (Durham business school) 2,2,3,3,4,4,4,5,5,6,6,6,7,7,8,8,9,9,9,11. The median is (d)Computation of the mean. The mean was computed from the data collected Table 4 showing the computation of the arithmetic mean Day Emails 1 7 2 4 3 7 4 3 5 9 6 11 7 5 8 8 9 2 10 6 11 6 12 8 13 4 14 4 15 9 16 3 17 6 18 9 19 2 20 5 Where D is the number of days and E represent the total number of emails The arithmetic mean is calculated by Mean = sum / count (1) The mean is then equal to which equals 6 (4) (e)Computation of variance and standard deviation To help in drawing the normal distribution curve, the values of Variance and standard deviation were calculated. Table 5 showing tabulated data for computation of variance Number of emails Frequency(F) DEV=value-mean (DEV)2 F* (DEV)2 2 2 -3.9 15.21 30.42 3 2 -2.9 8.41 16.82 4 3 -1.9 3.61 10.83 5 2 -0.9 0.81 1.62 6 3 0.1 0.01 0.03 7 2 1.1 1.21 2.42 8 2 2.1 4.41 8.82 9 3 3.1 9.61 28.83 11 1 5.1 26.01 26.01 125.8 The mean squared deviation (MSD) is equal to variance (VAR). (MSD)= VAR = (5) The standard deviation is the square root of the variance. SDEV = (6) SDEV= (7) (f)The normal distribution curves The data for the computation of the normal distribution is shown below; the use of many variables enables the plotting of a smooth curve. The values were generated using the spreadsheets. (Durham business school) Table 6 showing the data used to draw the normal distribution curve x Y=x*SDEV+MEAN NORMDIST(B2,MEAN,SDEV,FALSE) -5 -6.6395 5.92815E-07 -4.9 -6.38871 9.72511E-07 -4.8 -6.13792 1.57953E-06 -4.7 -5.88713 2.5399E-06 -4.6 -5.63634 4.04356E-06 -4.5 -5.38555 6.37336E-06 -4.4 -5.13476 9.94556E-06 -4.3 -4.88397 1.53655E-05 -4.2 -4.63318 2.3503E-05 -4.1 -4.38239 3.55922E-05 -4 -4.1316 5.33635E-05 -3.9 -3.88081 7.92119E-05 -3.8 -3.63002 0.000116411 -3.7 -3.37923 0.000169377 -3.6 -3.12844 0.00024399 -3.5 -2.87765 0.000347973 -3.4 -2.62686 0.000491335 -3.3 -2.37607 0.000686857 -3.2 -2.12528 0.000950631 -3.1 -1.87449 0.001302611 -3 -1.6237 0.001767155 -2.9 -1.37291 0.002373513 -2.8 -1.12212 0.003156207 -2.7 -0.87133 0.004155243 -2.6 -0.62054 0.005416073 -2.5 -0.36975 0.006989234 -2.4 -0.11896 0.008929595 -2.3 0.13183 0.011295123 -2.2 0.38262 0.014145139 -2.1 0.63341 0.017538018 -2 0.8842 0.021528357 -1.9 1.13499 0.026163649 -1.8 1.38578 0.031480585 -1.7 1.63657 0.037501127 -1.6 1.88736 0.044228572 -1.5 2.13815 0.051643844 -1.4 2.38894 0.059702327 -1.3 2.63973 0.068331509 -1.2 2.89052 0.077429744 -1.1 3.14131 0.086866373 -1 3.3921 0.096483402 -0.9 3.64289 0.106098828 -0.8 3.89368 0.115511604 -0.7 4.14447 0.124508128 -0.6 4.39526 0.132869972 -0.5 4.64605 0.140382522 -0.4 4.89684 0.146844029 -0.3 5.14763 0.152074571 -0.2 5.39842 0.155924357 -0.1 5.64921 0.158280852 0 5.9 0.159074238 0.1 6.15079 0.158280852 0.2 6.40158 0.155924357 0.3 6.65237 0.152074571 0.4 6.90316 0.146844029 0.5 7.15395 0.140382522 0.6 7.40474 0.132869972 0.7 7.65553 0.124508128 0.8 7.90632 0.115511604 0.9 8.15711 0.106098828 1 8.4079 0.096483402 1.1 8.65869 0.086866373 1.2 8.90948 0.077429744 1.3 9.16027 0.068331509 1.4 9.41106 0.059702327 1.5 9.66185 0.051643844 1.6 9.91264 0.044228572 1.7 10.16343 0.037501127 1.8 10.41422 0.031480585 1.9 10.66501 0.026163649 2 10.9158 0.021528357 2.1 11.16659 0.017538018 2.2 11.41738 0.014145139 2.3 11.66817 0.011295123 2.4 11.91896 0.008929595 2.5 12.16975 0.006989234 2.6 12.42054 0.005416073 2.7 12.67133 0.004155243 2.8 12.92212 0.003156207 2.9 13.17291 0.002373513 3 13.4237 0.001767155 3.1 13.67449 0.001302611 3.2 13.92528 0.000950631 3.3 14.17607 0.000686857 3.4 14.42686 0.000491335 3.5 14.67765 0.000347973 3.6 14.92844 0.00024399 3.7 15.17923 0.000169377 3.8 15.43002 0.000116411 3.9 15.68081 7.92119E-05 4 15.9316 5.33635E-05 4.1 16.18239 3.55922E-05 4.2 16.43318 2.3503E-05 4.3 16.68397 1.53655E-05 4.4 16.93476 9.94556E-06 4.5 17.18555 6.37336E-06 4.6 17.43634 4.04356E-06 4.7 17.68713 2.5399E-06 4.8 17.93792 1.57953E-06 4.9 18.18871 9.72511E-07 5 18.4395 5.92815E-07 (g) The normal distribution curve The resulting curve is as shown below (Tushar, 2008) Fig 3 showing the normal distribution curve From figure 3 it can be seen that a smooth curve is obtained. The mean, which is at the center of the distribution, is 6. The curve is not skewed and does not touch the x axis( Cambridge University,1999) (h)The properties of a normal distribution curve. The properties of a normal distribution curve are; The normal distribution curve is bell shaped hence the distribution is referred to as the bell shaped distribution or the Gaussian distribution. The data is evenly distributed around the mean. In some cases the data on the right and left may not be even resulting to skewed distributions. The area under the normal distribution curve is equal to 1 The mean and the median are equal and located at the center of the curve. The normal distribution curve has one mode. The curve is continuous and does not touch the X axis (Wittwer, 2004). (i)Calculating the Z values The Z value indicates the distance in terms of standard deviation that a particular X value is from the mean. (Durham business school) The Z value is calculated by; (8) The value represents the number of emails received; the mean is 5.9 while the standard deviation is 2.5079. For X= 2 Z = (2-5.9)/ 2.5079= -1.555 The probability = 0.061 (from normal probability tables) The percentage is given by 0.061* 100= 6.1 % For X= 3 Z = (3-5.9)/2.5079 = -1.156 The probability = 0.125 The percentage = 12.5 % Table 7 showing the computation of the Z value Number of emails Area of tail Percentage 2 1.555 0.061 6.1 3 1.156 0.125 12.5 4 0.757 0.227 22.7 5 0.358 0.363 36.3 6 0.039 0.488 48.8 7 0.438 0.334 33.4 8 0.837 0.203 20.3 9 1.236 0.109 10.9 10 1.634 0.052 5.2 11 2.033 0.021 2.1 From the data in the table above, it can be seen that most of the variables fall between the 0.68 %. This means that the collected data in this system can be accurately surveyed using the normal distribution curve. From the sample data there are no values that are unusually high or unusually low (Papoulis, 1984). (j)Conducting the X2 test Calculating the degrees of freedom The degree of freedom = (r-1)*(c-1) r = number of rows c = number of columns From the data provided there are 20 rows and 2 columns Degree of freedom = 19* 1 = 19 It is expected that the mean number of emails is 6 Hypothesis The calculated value of X2 should be less than the critical value given by the tables at a 5 % significant level (NIST SEMATECH, 2008). The value of X2 can be calculated by taking into consideration the following. Table 8 showing values used in the computation of X2 Number of emails(observed emails )(o) Expected Emails (e) Observed - expected (o-e) (o-e)2 (o-e)2/ e 2 6 -4 16 2.67 3 6 -3 9 1.5 4 6 -2 4 0.67 5 6 -1 1 0.167 6 6 0 0 0 7 6 1 1 0.167 8 6 2 4 0.67 9 6 3 9 1.5 10 6 4 16 2.67 11 6 5 25 4.167 85 (9) The calculated value of X2 is equal to 14.18 Assuming a tail probability of 5% and a degree of freedom of 19 the tables give the critical value of X2 to be equal to 30.14. Since the critical value of X2 from the tables is greater than the calculated X2, it shows that all the data fall within the normal distribution. The results indicate that the normal distribution can be used to accurately study the behavior of the system under study (Greenwood & Nikulin, 1996). Calculated X2< critical X2.Based on the assumption taken in the hypothesis, the results are accepted. DISCUSSION From the results obtained above, it can be seen that collected data assumed a normal distribution. The data was clustered around the mean with few variables slightly away from the mean. By calculating the values of Z from the normal distribution table it can be shown that the is no value with a percentage greater than or less than 0.68% this shows that the distribution lies within a single mean. The resulting curve is perfectly uniform. By conducting the X2 test, it can further be seen that, the calculated value of X 2 is less than the critical value of X 2 given in the tables. This clearly shows that the normal distribution curve can be used to accurately study a system behavior. CONCLUSION The normal distribution curve can be used to effectively analyse statistical data. The results obtained are accurate and within the critical range. RECOMMENDATIONS The researcher consequently recommends the use of normal distribution in analysing different system behavior as well as natural and scientific phenomena. References Cambridge University., 1999, SMP advanced tables. Cambridge University Press, Cambridge. Cary, W., 2008, Standard normal distribution table to 7.5 SD, Viewed 28 May 2009 < http://www.adamssixsigma.com/Newsletters/standard_normal_table.htm > Durham Business School. Managing information. Sn. Feller, W., 1968. An Introduction to Probability Theory and Its Applications, Wiley, New York. Greenwood, P. & Nikulin, M. 1996. A guide to chi-squared testing, Wiley, New York. NIST SEMATECH, 2008, Chi-Square Goodness-Of-Fit Test. Viewed 29 May 2009 < http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm > Papoulis, A.1984, Probability, Random Variables, and Stochastic Processes, 2nd ed, McGraw-Hill, New York. Tushar M., 2008, Drawing a normal curve, Viewed 29 May 2009 < http://www.tushar-mehta.com/excel/charts/normal_distribution/ > Wittwer, J., 2004. "Graphing a Normal Distribution in Excel" Viewed 28 May 2009 < http://vertex42.com/ExcelArticles/mc/NormalDistribution-Excel.html > Read More