Features of Normal Distribution
| * Normal distribution becomes counterpart of binomial distribution when the variables in stead of being discrete, become continuous. |
| * The assumption is that events are random, mutually independent, similar, equal in strength and the probability of happening or not happening is the same. For example, in the tossing of a coin, the probability of head or tail is same (at 0.5) random and mutually independent if several coins are tossed. |
| * In Normal Distribution function, y is the height of the curve i.e frequency of a given x-value where x is the probability of the event happening, N-> no. of cases; σ--> standard deviation of the distribution; π ->3.1416; e--> 2.7183. |
| * Maximum height of the curve represents mean. Again in normal distribution, mean=mode=median. |
| * Total area under the curve between maximum height and the heights corresponding to a particular x-value i.e ± x, represent the % of total cases happening. |
| * Distance from the mean to a point in x-axis is given by z = x / σ i.e. x is represented in the units of standard deviation. We may put the x-axis in terms of Z in stead of x. |
| * Though the normal curve does not meet the base line until at infinity to the right or left, for all practical purpose, it is treated to be ending at ±3σ as 99.73% of the entire distribution lie within - 3σ to + 3σ . For large samples and extremely precise results, one may consider cases outside ±3σ. |
| * We introduce a quantity, probable error (PE or Q ) where Q=0.6745 σ or σ = 1.4826Q. |
| * The normal performance is the middle 50% of the curve +25% & -25% from the mean. |
| * The curve is perfectly symmetrical around the vertical line to the mean. |
| * The curve is skewed ( designated as sk )when mean and
median do not coincide.
>Skewed negatively when the peak shifts to the right because results accumulate at high end of the scores. >Skewed positively when the peak shifts to the left because results accumulate at the low end of the score. |
| * sk = 3(mean -median)/ σ . In terms of percentile sk= (P90 +P10)/2 - P50 ; |
| * Kurtosis or peakedness -- (ku)- ku= Q / (P90 - P10 ) . |
| * Leptokurtic-->When the distribution is more peaked than normal. |
| * Platykurtic ---> When the distribution is flatter than the normal. |
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* Cause of Skewedness / Kurtosis - Scores made by small, homogeneous groups are more likely to be narrow, leptokurtic . Scores for large, heterogeneous groups are likely to be broad & platykurtic. When tests are too easy, scores are skewed to the left as scores pile up at the high end of the score and for tests that are too difficult, curve is skewed to the right as scores pile up at the low end. Asymmetry is due to various perturbations on normal distribution. |