# 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 n
*p > 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**).