Normal Distribution Dr. Anshul Singh Thapa

An Introduction The observed frequency distributions are based on observation and experimentation. The observed frequency distribution are obtained by grouping data. They help in understanding properly the nature of data. As distinguished from this type of distribution which is based on actual observation, it is possible to deduce mathematically what the frequency distribution of certain population should be. Such distributions as are expected on the basis of previous experience or theoretical considerations are known as ‘theoretical distribution’ or probability distribution.

Knowledge of expected behavior of a phenomenon or, in other words, the expected frequency distribution is of great help in a large number of problems in practical life. They serve as benchmarks against which to compare observed distributions and act as substitutes for actual distributions when the latter are costly to obtain or cannot be obtain at all. Amongst theoretical or expected frequency distributions, the following six are more popular: Binominal Distribution Multinomial distribution Negative binominal distribution Poisson Distribution Hyper geometric distribution Normal Distribution Among these the first five distributions are of discrete type and the last one of continuous type.

Probability The concept of probability is similar to the idea of percentage When we say that there is 50% chance of occurrence of an event, in probability we do not say it 50% chance but we can say that the probability is .5 Now the probability always in between 0 and 1. If an event has probability of 0 than tat event is impossible event. If an event have probability of 1 than it is a sure event. The sum of all the probability in any particular situation is always 1 which in percentage language is 100%. Example – flipping of a coin three times. Outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT Event: A: Getting exactly two Heads Probability = Number of possible outcomes Numbers of total outcomes

Outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT If we have three independent factor operating, the expression (p+q)n becomes for three coins (H + T)3. expanding this we get H3 + 3 H2T + 3 HT2 + T3, which may be written 1 H3 1 chance in 8 of 3 heads = 1/8 3 H2T 3 chances in 8 of 2 heads and 1 tail = 3/8 3 HT2 3 chances in 8 of 1 heads and 2 tail = 3/8 1 T3 1 chance in 8 of 3 tails = 1/8 8 (Total)

If we toss ten coins simultaneously, for instance, we have (p+q)n If we toss ten coins simultaneously, for instance, we have (p+q)n. this expression may be written (H+T)10. where H stand for probability of head, T stands for probability of nonhead (tail). When (H+T)10 is expanded H10 + 10H9T1 + 45H8T2 + 120H7T3 + 210H6T4 + 252H5T5 + 210T6H4 + 120T7H3 + 45T8H2 + 10T9H1 + T10

H10 - 1 chance in 1024 of all coins falling heads = 1/1024 10H9T1 - 10 chances in 1024 of 9 heads and I tail = 10/1024 45H8T2 - 45 chances in 1024 of 8 heads and 2 tails = 45/1024 120H7T3 - 120 chances in 1024 of 7 heads and 3 tails = 120/1024 210H6T4 - 210 chances in 1024 of 6 heads and 4 tails = 210/1024 252H5T5 - 252 chances in 1024 of 5 heads and 5 tails = 252/1024 210T6H4 210 chances in 1024 of 4 heads and 6 tails = 210/1024 120T7H3 - 120 chances in 1024 of 3 heads and 7 tails = 120/1024 45T8H2 - 45 chances in 1024 of 2 heads and 8 tails = 45/1024 10T9H1 - 10 chances in 1024 of 1 head and 9 tails = 10/1024 T10 - 1 chance in 1024 of all coins falling tails = 1/1024

250 200 150 100 50 10 T10 10 T9H1 45 T8H2 120 T7H3 210 T6H4 252 H5T5 210 H6T4 120 H7T3 45 H8T2 10 H9T1 H10

Normal Curve In Normal distribution the measures are concentrated closely around the centre and tapper off from this central high point or crest to the left and right. There are relatively few measures at the ‘low-score’ end of the scale; an increasing number up to a maximum at the middle position; and a progressive falling – off towards the ‘high-score’ end of the scale. If we divide the area under the curve by a line drawn perpendicularly through the central high point to the base line, the two part thus formed will be similar in shape and very nearly equal in are. This bell shaped figure is called normal probability curve or simply normal curve.

In normal probability curve, the mean, median and mode all fall exactly at the midpoint of the distribution and are numerically equal. Since the normal curve is bilaterally symmetrical, all the measures of central tendency must coincide at the center of the distribution. Between the mean and the ± 1 SD lie the middle two – thirds (68.26%) of the cases in the normal distribution. Between the mean and ± 2 SD are found approximately 95% of the distribution; and between the mean and ± 3 SD are found 99.7% of the distribution. There are about 68% chances in 100 that a case will lie within ± 1 SD from the mean in the normal distribution. There are 95% chances in 100 that it will lie within ± 2 SD from the mean and 99.7% chances in 100 that it ill lie within ± 3 SD from the mean.

The normal distribution was first described by Abrham De Moivre The normal distribution was first described by Abrham De Moivre. Normal distribution was rediscovered by Gauss in 1809

Properties of the Normal Distribution The normal curve is bell shaped and symmetrical in its appearance. If the curves are folded along its vertical axis, the two halves would coincide. The number of cases below the mean in a normal distribution is equal to the number of cases above the mean. The height of the curve for a positive deviation of 3 units is the same as the height of the curve for negative deviation of 3 units. The height of the normal curve is at its maximum at the mean. In normal distribution mean, median and mode are all equal. The height of the curve declines as we go in either direction from the mean. The curve approaches nearer and nearer to the base but it never touches it, i.e., the curve is asymptotic to the base on either side. The area under the normal curve distributed as follows: Mean ± 1 SD covers 68.27% area; 34.13% area will lie on either side of the mean. Mean ± 2 SD covers 95.45% area. Mean ± 3 SD covers 99.73% area.

Distance from the mean ordinate Percentage of the total area The following table shows the area of the normal curve between mean ordinate and ordinates at various distances from mean as percentage. Distance from the mean ordinate Percentage of the total area 0.5 SD 19.146 0.0 SD 34.134 1.5 SD 43.319 1.96 SD 47.500 2.0 SD 47.725 2.5 SD 49.379 2.5758 SD 49.500 3.0 SD 49.865 Thus the two ordinates at distance 1.96 SD from the mean on either side would enclose 47.5 + 47.5 = 95% of the total area, and two ordinates at 2.5758 SD distance from the mean on either side would enclose 49.5 + 49.5 = 99% of the total area. The various hypothesis are generally tested either at 5% level or at 1% level (i.e., taking into account 95% and 99% of the total are of the normal curve).

Divergence in Normality (The Non – Normal Distribution) In frequency polygon or histogram of test scores, usually the first thing that strikes is the symmetry or lack of symmetry in the shape of curve. In the normal curve model, the mean, median and the mode all coincide and there is perfect balance between the right and the left halves of the curve. Generally two types of divergence occur in the normal curve: Skewness Kurtosis

Skewness A distribution is said to be “skewed” when the mean and median falls at different points in the distribution and the balance, i.e., centre of gravity is shifted to one side or the other to left or right. In normal distribution the mean equals, the median exactly and the Skewness is of course zero. The more nearly the distribution approaches the normal form, the closer together are the mean and the median and the less the skewness. There are two types of skewness which appears in the normal curve: Negative Skewness Positive Skewness

Negative Skewness Distribution said to be skewed negatively or to the left when scores are massed at the high end of the scale, i.e., the right side of the curve are spread out more gradually towards the low end i.e., the left side of the curve. In negatively skewed distribution the value of the median will be higher than that of the value of the mean.

Positive Skewness Distribution are skewed positively or to the right, when scores are massed at the low, i.e., the left end of the scale, and are spread out gradually towards the high or right end

Note that the mean is pulled more towards the skewed end of the distribution than in the median. In fact, the greater the gap between mean and median, the greater the skewness. Moreover, when skewness is negative, the mean lies to the left of the median; and when skewness is positive, the mean lies to the right of the median.

Kurtosis The term kurtosis refers to the ‘peakedness’ or flatness of a frequency distribution as compared with the normal. It is also refers to the divergence in the height of the curve. There are three types of divergence in the peakedness of the curve. Leptokurtic Mesokurtic (normal) Platykurtic

Leptokurtosis A frequency distribution more peaked than the normal distribution curve is said to be leptokurtic.

Platykurtosis When the frequency distribution is flatter than the normal distribution curve the distribution is said to be Platykurtic distribution