Essentials of Data Science – Probability and Statistical Inference – Normal Approximation

By / July 26, 2023

In this note series on Probability and Statistical Inference, we have already seen the importance of probability distributions and their associated probability functions for discrete random variables and continuous random variables. In addition, we have learned to resemble a natural random phenomenon with these probability distributions. These distributions were Degenerate distribution, Uniform distribution, Bernoulli distribution, Binomial distribution, Poisson distribution, Geometric distribution, and Normal distribution.

This note will cover how to approximate various probability distributions of discrete random variables using a continuous random variable probability distribution function. 

Table of Contents hide
1 Need for Normal Approximation
1.1 Normal Approximation to the Binomial Distribution
1.1.1 Continuity Correction
2 References
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Need for Normal Approximation

There are various cases where the data sample is huge, and a normal approximation could provide approximately similar results compared to finding results using the actual distribution function. However, there would be certain conditions which must be satisfied.

As a standard normal distribution, computation is tabular based (lookup table), which makes it less computing intensive compared to other distribution functions where we need to solve mathematical equations.

Normal Approximation to the Binomial Distribution

The normal distribution can be used to approximate binomial probabilities for cases where n is large it is because finding \binom{n}{k} is computational power intensive, so if accuracy is not the primary criterion, then it can be approximated by the normal distribution.

For example, consider a histogram, where each of the bars represents binomial probabilities. The area of bars can be approximated by the area under the normal density function as normal distribution is continuous. A continuous normal distribution is used to approximate a discrete binomial distribution, and a modification is needed, referred to as a continuity correction. Let us understand the concept behind continuity correction.

Continuity Correction

It is an adjustment that is made when a discrete distribution is approximated by a continuous distribution. In the below example, we try to create a smooth curve from the bars of the histogram. The steps are as follows:

  • Mark the midpoints of the bars of the histogram.
  • Join them at all the midpoints from a smooth curve.
Normal approximation to the binomial distribution

We can observe that some parts of the bars are excluded, and other parts are included when we draw a smooth curve using the midpoints over the bars of the histogram. 

If X is a binomial random variable with parameters n and p, where \mu = np and \sigma^2 = np(1-p), then

Z = \frac{X- np}{\sqrt{np(1-p)}}

is approximately a standard normal random variable. Note: It is a general formula for standardizing the random variable.

To approximate a binomial probability with a standard normal distribution, a continuity correction is applied as follows: 

P(X \leq x) = P(X \leq x + .5) = P \left ( Z \leq \frac{x + 0.5 - np}{\sqrt{np(1-p)}} \right) 

and 

P(X \geq x) = P(X \geq x - .5) = P \left ( Z \geq \frac{x - 0.5 - np}{\sqrt{np(1-p)}} \right)

The approximation is good when n is large relative to p and (mean) \mu > 5 and (variance) \sigma^2 > 5. However, if these parameters are smaller, the binomial distribution will be quite skewed, and the symmetric standard normal distribution will not be a good approximation.

References

  1. Essentials of Data Science With R Software – 1: Probability and Statistical Inference, By Prof. Shalabh, Dept. of Mathematics and Statistics, IIT Kanpur.

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