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NORMAL PROBABILITY CURVE

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 Normal Probability Curve

A normal curve is a bell-shaped curve which shows the probability distribution of a continuous random variable.  Moreover, the normal curve represents a normal distribution.  The total area under the normal curve logically represents the sum of all probabilities for a random variable.   Hence, the area under the normal curve is one. Also, the standard normal curve represents a normal curve with mean 0 and standard deviation 1.  Thus, the parameters involved in a normal distribution is mean ( μ ) and standard deviation ( σ ). In statistics, the theoretical curve that shows how often an experiment will produce a particular result.  The curve is symmetrical and bell shaped, showing that trials will usually give a result near the average, but will occasionally deviate by large amounts.  The width of the “bell” indicates how much confidence one can have in the result of an experiment — the narrower the bell, the higher the confidence.  This curve is also called the Gaussian curve, after the nineteenth-century German mathematician Karl Friedrich Gauss.  ( See statistical significance.) The Normal Distribution Curve (NDC) is a bell shaped line (also called the Bell curve or Gaussian curve) in a graph that has values on the x-axis and frequency on the y-axis.  It is formed by a rather complex formula that includes pi and the natural logarithem, two ‘natural’ values or mathematical phenomena that happen to be everywhere around us.  The result (the Bell curve or NDC) is so common that it has become the standard to compare most observed data distributions with.  What does it represent? Below the Bell curve are all the data points (100% of the frequency). Because of the form you can identify that values on the left and right are less frequently observed or expected than values in the center.  

Characteristics of normal probably curve

some of the major characteristics of NPC are given below

1. The normal curve is symmetrical.  The Normal Probability Curve (N.P.C.) is symmetrical about the ordinate of the central point of the curve.  It implies that the size, shape and slope of the curve on one side of the curve is identical to that of the other.  That is, the normal curve has a bilateral symmetry.  If the figure is to be folded along its vertical axis, the two halves would coincide.  In other words the left and right values to the middle central point are mirror images.   

2. The normal curve is unimodal.  Since there is only one point in the curve which has maximum frequency, the normal probability curve is unimodal, i.e. it has only one mode.  

3. Mean, median and mode coincide.  The mean, median and mode of the normal distribution are the same and they lie at the centre.  They are represented by 0 (zero) along the base line. [Mean = Median = Mode]

4. The maximum ordinate occurs at the centre.  The maximum height of the ordinate always occurs at the central point of the curve that is, at the mid-point.  The ordinate at the mean is the highest ordinate and it is denoted by Y₀. (Y₀ is the height of the curve at the mean or mid-point of the base line).   

5. The normal curve is asymptotic to the x-axis.   The Normal Probability Curve approaches the horizontal axis asymptotically i.e., the curve continues to decrease in height on both ends away from the middle point (the maximum ordinate point); but it never touches the horizontal axis.  It extends infinitely in both directions i.e.  from minus infinity (-∞) to plus infinity (+∞) as shown in Figure below.  As the distance from the mean increases the curve approaches to the base line more and more closely.  

6. The height of the curve declines symmetrically.  In the normal probability curve the height declines symmetrically in either direction from the maximum point.  Hence the ordinates for values of X = µ ± K, where K is a real number, are equal.   

7. The points of influx occur at point ± 1 standard deviation (± 1 a).  The normal curve changes its direction from convex to concave at a point recognized as point of influx.  If we draw the perpendiculars from these two points of influx of the curve on horizontal axis, these two will touch the axis at a distance one Standard Deviation unit above and below the mean (± 1 σ).  

8. The total percentage of area of the normal curve within two points of influxation is fixed.  Approximately 68.26% area of the curve falls within the limits of ±1 standard deviation unit from the mean as shown in figure below.  

9. Normal curve is a smooth curve.  The normal curve is a smooth curve, not a histogram.  It is moderately peaked. The kurtosis of the normal curve is 263.  

10. The normal curve is bilateral.  The 50% area of the curve lies to the left side of the maximum central ordinate and 50% lies to the right side.  Hence the curve is bilateral.  

11. The normal curve is a mathematical model in behavioural sciences.  The curve is used as a measurement scale.  The measurement unit of this scale is ± σ (the unit standard deviation). 

12. Greater percentage of cases at the middle of the distribution.   There is a greater percentage of cases at the middle of the distribution. In between -1σ and + 1σ, 68.26% (34.13 + 34.13), nearly 2/3 of eases lie. To the right side of +1σ, 15.87% (13.59 + 2.14 + .14), and to the left of-1σ, 15.87% (13.59 + 2.14 + .14) of cases lie.  Beyond +2σ. 2.28% of eases lie and beyond -2σ also 2.28% of cases lie.  Thus, majority of eases lie at the middle of the distribution and gradually number of cases on either side decreases with certain proportions.   

FIGURE1 PROBABILITY CURVE

TABLE 1 shows the MERITS AND DEMERITS OF NPC

MEERITS	DEMERITS
ACCQURATE FOR LARGE CLASS	NOT GOOD TO USE FOR SMALL CALSS
ABOVE 200 STUDENTS	LESSTHAN 200 STUDENTS
HEALTHY	UNHEALTHY

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