NASEERA ASHARAF: NORMAL PROBABILITY CURVE

Normal Probability Curve

This bell shaped curve is called as the ‘Normal Probability Curve’. Thus the “graph of the probability density function of the normal distribution is a continuous bell shaped curve, symmetrical about the mean” is called normal probability curve.

In statistics it is important because:

(а) It is the distribution of many naturally occurring variables, such as intelligence of 8th grade students, height of the 10th grade students etc.

(b) The distribution of the means of samples drawn from most parent populations is normal or approximately so when the samples are sufficiently large.

Therefore normal curve has great significance in social sciences and behavioral sciences. In behavioral measurement most of the aspects approximates to the normal distribution. So that Normal Probability Curve or most popularly known as NPC is used as a reference curve. In order to understand the utility of the NPC we must have to understand the properties of the NPC.

Characteristics of Normal Probability Curve

Some of the major characteristics of normal probability curve are as follows:

· The curve is bilaterally symmetrical. The curve is symmetrical to its ordinate of the central point of the curve. It means the size, shape and slope of the curve on one side of the curve is identical to the other side of the curve. If the curve is bisected then its right hand side completely matches to the left hand side.

· The curve is asymptotic. The Normal Probability Curve approaches the horizontal axis and extends from-∞ to + ∞. Means the extreme ends of the curve tends to touch the base line but never touch it.

· The mean, median and mode. The mean, Median and mode fall at the middle point and they are numerically equal.

· The Points of inflection occur at ± 1 Standard deviation unit. The points of influx in a NPC occur at ± 1σ to unit above and below the mean. Thus at this point the curve changes from convex to concave in relation to the horizontal axis.

· The total area of npc is divided in to ± standard deviations. The total of NPC is divided into six standard deviation units. From the center it is divided in to three +ve’ standard deviation units and three —ve’ standard deviation units. Thus ± 3σ of NPC include different number of cases separately. Between ± 1σ lie the middle 2/3rd cases or 68.26%, between ± 2σ lie 95.44% cases and between ± 3σ lie 99.73% cases and beyond + 3σ only 0.37% cases fall.

· The y ordinate represents the height of the normal probability curve. The Y ordinate of the NPC represents the height of the curve. At the center the maximum ordinate occurs. The height of the curve at the mean or mid point is denoted as Y0. In order to determine the height of the curve at any point we use the following formula:

· It is unimodal. The curve is having only one peak point. Because the maximum frequency occurs only at one point.

· The height of the curve symmetrically declines. The height of the curve decline to both the direction symmetrically from the central point. Means the M + σ and M — σ are equal if the distance from the mean is equal.

· The mean of npc is µ and the standard deviation is σ. As the mean of the NPC represent the population mean so it is represented by the µ (Meu). The standard deviation of the curve is represented by the Greek Letter, σ.

· In normal probability curve the standard deviation is the 50% larger than the q. In NPC the Q is generally called the probable error or PE. The relationship between PE and a can be stated as following:

1 PE = .6745σ

1σ = 1.4826PE

The average deviation about the mean of npc is .798σ. There is a constant relationship between standard deviation and average deviation in a NPC.
The model ordinate varies increasingly to the standard deviation. In a Normal Probability curve the modal ordinate varies increasingly to the standard deviation. The standard deviation of the Normal Probability Curve increases, the modal ordinate decreases and vice-versa.