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). It is based upon the law of probability discovered by French mathematician Abraham Demoiver (1667 – 1754).

The normal curve offers a convenient and reasonably accurate description of a great number of variables. It also describes the distribution of many statistics from samples. For example, if you drew 100 random samples from a population of teenagers and computed the mean weight of each sample, you would find that the distribution of the 100 means approximates the normal curve. In such situations, the fit of the normal curve is often very good indeed.

Normal curve in Psychology

Sir Francis Galton (cousin of Darwin) began the first serious investigation of “individual differences,' an important area of study today in education and psychology. In his research on how people differ from one another on various mental and physical traits, Galton found the normal curve to be a reasonably good description in many instances. He became greatly impressed with its applicability to natural phenomena. He referred to the normal curve as the Law of Frequency of Error.

Properties of the Normal Curve

Normal curve is a theoretical invention, a mathematical model, an idealized conception of the form a distribution might take under certain circumstances. No empirical distribution—one based on actual data—ever conforms perfectly to the normal curve. However, empirical distributions often offer a reasonable approximation of the normal curve. In these instances, it is quite acceptable to say that the data are “normally distributed.'

 The equation of the normal curve describes a family of distributions. Normal curves may differ from one another with regard to their means and standard deviations. However, they are all members of the normal curve family because they share several properties, as discussed below:

1. Symmetrical about the mean: Normal curves are symmetrical about its mean, i.e., the left half of the distribution is a mirror image of the right half. The two halves have identical size, shape and slope.
2. Unimodal: They are unimodal, i.e., there is a single peak and the highest point occurs at x=µ.
   It follows from these first two properties that the mean, median, and mode all have the same value. 
3. Bell-shaped: Normal curves have a bell-shaped form. Starting at the center of the curve and working outward, the height of the curve descends gradually at first, then faster, and finally more slowly. The maximum ordinate occurs at the center. And, the height of the curve declines symmetrically in both directions. This property alone illustrates why an empirical distribution can never be perfectly normal.
4. Points of inflection: It has inflection points at the points µ - σ and µ + σ. At this point the curve changes its shape from convex to concave.
5. The Empirical Rule: Approximately 68% of the area under the normal curve is between x=µ - σ and x= µ + σ.
   Approximately 95% of the area under the normal curve is between x=µ - 2σ and x= µ + 2σ.
   Approximately 99.7% of the area under the normal curve is between x=µ - 3σ and x= µ + 3σ.
6. Total area =1: The area under the curve is 1. The area under the curve is considered to be 100 percent probability.
7. Area of each side = ½: The area under the curve to the right of µ equals the area under the curve to the left of µ which equals ½. Therefore, the curve can also be said to be bilateral.
8. Asymptotic: As x increases without bound (gets larger and larger), the graph approaches, but never reaches, the horizontal axis. As x decreases without bound (gets larger and larger in the negative direction), the graph approaches, but never reaches, the horizontal axis. Thus, it can be said that the curve is asymptotic.
9. Equation: Equation of the normal curve reads:
   

Application of normal curve using z-scores

The relationship between the normal curve area and standard deviation units can be put to good use for answering certain questions that are fundamental to statistical reasoning. For example, the following type of question occurs frequently in statistical work: Given a normal distribution with a mean of 100 and a standard deviation of 15, what proportion of cases fall above the score 115.

A standard score expresses a score’s position in relation to the mean of the distribution, using the standard deviation as the unit of measurement. z score is one type of standard score, it states how many standard deviation units the original score lies above or below the mean of its distribution.

A z score is simply the deviation score divided by the standard deviation, as the following formula illustrates:


Following are some types of problems that can be solved:
1. Finding area of curve when score is known – helps identify probability of occurrence of a certain score
2. Finding scores when the area is known – helps to identify which score is expected to occur
3. Comparing scores from different distributions – By converting all scores to z scores, we can get the standard normal distribution. These distributions have a mean of 0 and standard deviation of 1 – regardless of the distributions original mean and standard deviation. These standard normal distributions can be used to compare different distributions.

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The normal curve offers a convenient and reasonably accurate description of a great number of variables. It also describes the distribution of many statistics from samples, and therefore, is very useful in social sciences such as psychology. The major properties of the normal curve are that it is symmetrical about the mean, unimodal, bell-shaped, points of inflection, the empirical rule, total area is 1 and area of each side is half, asymptotic, and it has a standard equation.

Sources:
http://www.dmaictools.com/dmaic-measure/normal-probability-curve
Statistics in Psychology and Education, Henry E. Garrett
Fundamentals of Statistical reasoning in Education, Theodore Coladarci, Casey D. Cobb, Edward W. Minium and Robert C. Clarke (Click for eBook)