1. Normal Distribution
Dr. Anshul Singh Thapa

2. 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.

3. 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.

4. 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

5. 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)

6. 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
H H9T1 + 45H8T H7T H6T H5T T6H T7H3 + 45T8H2 + 10T9H1 + T10

7. H chance in 1024 of all coins falling heads = 1/ H9T chances in 1024 of 9 heads and I tail = 10/ H8T chances in 1024 of 8 heads and 2 tails = 45/ H7T chances in 1024 of 7 heads and 3 tails = 120/ H6T chances in 1024 of 6 heads and 4 tails = 210/ H5T chances in 1024 of 5 heads and 5 tails = 252/ T6H4 210 chances in 1024 of 4 heads and 6 tails = 210/ T7H chances in 1024 of 3 heads and 7 tails = 120/ T8H chances in 1024 of 2 heads and 8 tails = 45/ T9H chances in 1024 of 1 head and 9 tails = 10/1024 T chance in 1024 of all coins falling tails = 1/1024

8. 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

9. 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.

10. 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.

11. 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

13. 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.

14. 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 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 = 95% of the total area, and two ordinates at SD distance from the mean on either side would enclose = 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).

15. 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

16. 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

17. 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.

18. 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

19. 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.

20. 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

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

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