NORMAL PROBABILITY CURVE

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NORMAL PROBABILITY CURVE

A normal probability curve shows the theoretical shape of a normally distributed histogram. The shape of the normal probability curve is based on two parameters: mean (average) and standard deviation (sigma).
Properties of Normal Probability Curve
It is bell-shaped.
Mean, median and mode coincide.
It is uni-model
The highest point occurs in the Middle.
It is symmetric about the mean, m. One half of the curve is a mirror image of the other half, i.e., the area under the curve to the right of middle is equal to the area under the curve to the left of middle equals liz.
It has inflection points at -sigma and +sigma
The curve is asymptotic to the horizontal axis at the extremes.
The total area under the curve equals one. Empirical Rule:
Approximatelv 68% of the area under the curve is between -sigma and +sigma
Approximatelv 95% of the area under the curve is between -2sigma and +2 sigma 
Approximatelv 99.7% of the area under the curve is between -3 sima and +3 sigma


Uses of Normal Curve
   It is used to know the percentage of cases that lie within the givenlimits of scores.
    It is used to find the limits between the given percentage of cases fall
    It is used to convert raw scores into Z scores and T Scores.
    It  is used to divide a group in to a given number of sub groups based on a trait
which follows normal distribution.
   It is used to know the relative difficulties of test items.
  It is used to compare two frequency distributions
  It is used to test the hypotheses and levels of significance.

DEVIATIONS FROM THE NORMALITY
  1. Skewness
  2. Kurtosis
1. SKEWNESS 
Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point. The term skewness refers to "lack of symmetry" 
Two types 
a) positive skewness 
b) Negative skewness.




2. KURTOSIS
Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case.
Three types; 
a) Leptokurtic (more peak) 
b) Platykurtic (flatter) 
c) Mesokurtic (Normal curve)

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