Normal approximations

Introduction to Normal Approximations

• Normal approximations are commonly used in statistics when dealing with large data sets.
• They are based on the central limit theorem, which states that the sum or mean of a large number of independent and identically distributed variables will be approximately normal, regardless of the original distribution.

Using Normal Approximations

• When using normal approximations, the z-score is commonly used. It measures how far a data point is from the mean in terms of standard deviations.
• To perform a normal approximation, you will need the mean (μ) and the standard deviation (σ) of the population.

Converting to a Normal Distribution

• The z-score can be computed using the formula: z = (X - μ) / σ.
• Once you have computed the z-score, you can use a standard normal distribution table to find the corresponding percentile.

When to Use Normal Approximations

• Normal approximations can be used when the sample size is large (n > 30).
• They should be used with caution, as they are an approximation and there may be some error associated with this.

Binomial to Normal Approximation

• A binomial distribution can be approximated by a normal distribution when np > 5 and n(1-p) > 5.
• If these conditions are met, the binomial random variable X can be transformed into the standard normal variable Z.

Poisson to Normal Approximation

• Poisson distribution can also be approximated by a normal distribution when the mean λ is large (say greater than 20).