Normal Approximation of Binomial

When can I use a normal distribution to approximate a binomial distribution?

A binomial distribution $X ~ B (n, p)$ can be approximated by a normal distribution $X_{N} ~ N (μ, σ^{2})$ provided
- n is large
- p is close to 0.5
  - $n p > 5$
  - $n q > 5 w h e r e q = 1 - p$
The mean and variance of a binomial distribution can be calculated by:
- $μ = n p$
- $σ^{2} = n p (1 - p)$

4-4-2-normal-approximation-of-binomial-diagram-1

Why do we use approximations?

If there are a large number of values for a binomial distribution there could be a lot of calculations involved and it is inefficient to work with the binomial distribution
- These days calculators can calculate binomial probabilities so approximations are no longer necessary
- However it is easier to work with a normal distribution
  - You can calculate the probability of a range of values quickly
  - You can use the inverse normal distribution function (most calculators don't have an inverse binomial distribution function)
In your exam you must use the formula and not a calculator to find binomial probabilities so you are limited to small values of n

What are continuity corrections?

The binomial distribution is discrete and the normal distribution is continuous
A continuity correction takes this into account when using a normal approximation
The probability being found will need to be changed from a discrete variable, X, to a continuous variable, X_N
- For example, X = 4 for binomial can be thought of as $3.5 \leq X_{N} < 4.5$ for normal as every number within this interval rounds to 4
- Remember that for a normal distribution the probability of a single value is zero so $P (3.5 \leq X_{N} < 4.5) = P (3.5 < X_{N} < 4.5)$

How do I apply continuity corrections?

Think about what is largest/smallest integer that can be included in the inequality for the discrete distribution and then find its upper/lower bound
$P (X = k) \approx P (k - 0.5 < X_{N} < k + 0.5)$
$P (X \leq k) \approx P (X_{N} < k + 0.5)$
- You add 0.5 as you want to include k in the inequality
$P (X < k) \approx P (X_{N} < k - 0.5)$
- You subtract 0.5 as you don't want to include k in the inequality
$P (X \geq k) \approx P (X_{N} > k - 0.5)$
- You subtract 0.5 as you want to include k in the inequality
$P (X > k) \approx P (X_{N} > k + 0.5)$
- You add 0.5 as you don't want to include k in the inequality
For a closed inequality such as $P (a < X \leq b)$
- Think about each inequality separately and use above
- $P (X > a) \approx P (X_{N} > a + 0.5)$
- $P (X \leq b) \approx P (X_{N} < b + 0.5)$
- Combine to give
- $P (a + 0.5 < X_{N} < b + 0.5)$

How do I approximate a probability?

STEP 1: Find the mean and variance of the approximating distribution
- $μ = n p$
- $σ^{2} = n p (1 - p)$
STEP 2: Apply continuity corrections to the inequality
STEP 3: Find the probability of the new corrected inequality
- Find the standard normal probability and use the table of the normal distribution
The probability will not be exact as it is an approximate but provided n is large and p is close to 0.5 then it will be a close approximation
- To decide if n is large enough and if p is close enough to 0.5 check that:
  - $n p > 5$
  - $n p > 5$ where $q = 1 - p$

Worked example

The random variable $X ~ B (1250, 0.4) .$

Use a suitable approximating distribution to approximate $P (485 \leq X \leq 530)$ .

3-4-2-normal-approximation-of-binomial-we-solution-with-addition

Examiner Tip

In the exam, the question will often tell you to use a normal approximation but sometimes you will have to recognise that you should do so for yourself. Look for the conditions mentioned in this revision note, n is large, p is close to 0.5, np > 5 and nq > 5.

Normal Approximation of Binomial (CIE A Level Maths: Probability & Statistics 1)

Revision Note

When can I use a normal distribution to approximate a binomial distribution?

Why do we use approximations?

What are continuity corrections?

How do I apply continuity corrections?

How do I approximate a probability?

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1. Data Presentation & Interpretation

1.1 Statistical Measures

1.1.1 Basic Statistical Measures

1.1.2 Frequency Tables

1.1.3 Standard Deviation & Variance

1.1.4 Coding

1.2 Representation of Data

1.2.1 Data Presentation

1.2.2 Stem and Leaf Diagrams

1.2.3 Box Plots & Cumulative Frequency

1.2.4 Histograms

1.3 Working with Data

1.3.1 Interpreting Data

1.3.2 Skewness

2. Probability

2.1 Basic Probability

2.1.1 Calculating Probabilities & Events

2.1.2 Venn Diagrams

2.1.3 Tree Diagrams

2.2 Permutations & Combinations

2.2.1 Arrangements & Factorials

2.2.2 Permutations

2.2.3 Combinations

2.3 Further Probability

2.3.1 Set Notation & Conditional Probability

2.3.2 Further Tree Diagrams

2.3.3 Further Venn Diagrams

2.3.4 Probability Formulae

3. Statistical Distributions

3.1 Probability Distributions

3.1.1 Discrete Probability Distributions

3.1.2 E(X) & Var(X) (Discrete)

3.2 Binomial & Geometric Distribution

3.2.1 The Binomial Distribution

3.2.2 Calculating Binomial Probabilities

3.2.3 The Geometric Distribution

3.3 Normal Distribution

3.3.1 The Normal Distribution

3.3.2 Standard Normal Distribution

3.3.3 Normal Distribution - Calculations

3.3.4 Finding Sigma and Mu

3.4 Working with Distributions

3.4.1 Modelling with Distributions

3.4.2 Normal Approximation of Binomial

Author: Amber