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Bell Shaped Curve

Statistics and probability theory are two important and inter-connected branches of mathematics. Statistics deals with collection, organization, calculation of numerical data; while probability theory is basically concerned about the chances of happening of the events.

In probability and statistics, we study a concept of "probability distribution". It is the process of assigning the probabilities to all the measurable subsets of the possibly expected outcomes of a certain random statistical experiment or survey or some statistical procedure.

The probability distributions are mainly of two types: Continuous probability distribution and discrete probability distribution.Normal distribution is one of the most common continuous probability distribution. The common pattern of the curve of a normal distribution resembles the shape of a bell. Therefore, it may be called as the bell–shaped curve or simply bell curve. In this type of distribution, the points are likely to occur on either side of mean position in almost the same way. Due to this, the curve of a normal distribution takes the shape of a bell. However, many other distributions also look similar to normal distribution. But, only statistical calculations may be able to prove a normal distribution.

In this article below, we are going to go ahead and understand about the bell-shaped curve of the normal probability distribution in the reference of probability and statistics.

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Definition

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In probability theory and statistics, the bell-shaped curves are very commonly seen. A bell-shaped curve is a curve that is used to explain a statistical concept that is known as normal distribution, which is also referred to Gaussian distribution. The curve of normal distribution is called so, since it looks similar to the shape of a bell. When the boundary of the probability distribution resembles the image of a bell, it is called a bell-shaped curve or a bell curve. It is referred to a shape that is formed when a curve is plotted with the use of statistical data which meets the criteria of normal probability distribution.

A bell-shaped curve is demonstrated in the following figure:

Bell Shaped Curve
In this curve, there is a highest peak that contains the central part. The center will have the highest value of the data. This point is known as the mean or average of the distribution. In the more simple terms, this point possesses the greatest number of occurrences of the elements.

As the curve progresses away from the central position, it scatters towards the both sides almost in the similar manner. So, we can say that the bell curve has most of the concentration in the center and it decreases on the either side. Bell shaped curve is more significant when the data has less tendency to give extreme values known as outliers. This curve also signifies about the data that it is symmetrical. This makes a researcher enable to create reasonable expectations and anticipation of an outcome.

Normal Distribution

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The normal distribution also known as Gaussian distribution. It is a very commonly used continuous probability distribution. The curve of normal or Gaussian distribution is sometimes informally called as the bell curve. However in statistics, there are various other distributions that are bell-shaped, for example Student's, Cauchy's and logistic.

But usually, the term bell curve is used for normal distribution curve. In probability and statistics, normal distribution is extremely useful. It is also utilized in social science and natural science very often.

Normal distribution is very important due to the central limit theorem. In a normal curve, mean median and mode lie on a same point. The normal distribution function is given by the following equation:
f ($\sigma$, $\mu$, x) = $\frac{1}{2 \pi \sigma^{2}}$ $e^{\frac{-(x-\mu)^{2}}{2 \sigma^{2}}}$
Where, $\mu$ = Mean
$\sigma$ = Standard deviation

The few important features of normal distributions are described in the following points:
1) Normal distribution is symmetric about its mean.

2) In this distribution, mean, median and mode are coincident, i.e they fall on a same point.

3) Normal distribution is concentrated in center and less dense at the tails.

4) The area below a bell or normal curve is also equal to 1.

5) The 68% of total area of this distribution comes under the standard deviation of the mean. Also 95% (approximately) of the total area of a normal distribution come under two standard deviations of the mean.

6) There are two parameters in a normal distributions - one is mean ($\mu$) and other is standard deviation ($\sigma$).

Standard Deviation

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The standard deviation of the data plays an important role in the bell curve of normal distribution. The greater the standard deviation the more wide the graph on X axis. The smaller the standard deviation, the more shrunk the graph towards mean position. Basically, the shape of the bell curve is based upon the standard deviation of the given data.

The spread of the data in the form percentages of the total area are shown in the following graph:
Bell Shaped Curve Examples
If we read chart carefully, we get to know that approximately 19.1% of the area of normally distributed data is situated between mean and 0.5 standard deviations to the either side of the mean. Let us have a look at following facts based on percentages.

These are known as "empirical rules" which are as follows:
1) The 68% of normal distribution is located within 1 standard deviation around the mean.

2) The 95% of normal distribution is situated within 2 standard deviations around the mean.

3) Approximately 99.7% of the normal distribution lies within 3 standard deviations of the mean.

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