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
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?

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)

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
- Use the "Normal Cumulative Distribution" function on your calculator
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

Worked example

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

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

4-4-2-normal-approximation-of-binomial-we-solution

Examiner Tip

In the exam, only use a normal approximation if the question tells you to. Otherwise use the binomial distribution.

Normal Approximation of Binomial (Edexcel A Level Maths: Statistics)

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. Statistical Sampling

1.1 Sampling & Data Collection

1.1.1 Sampling & Data Collection

2. Data Presentation & Interpretation

2.1 Statistical Measures

2.1.1 Basic Statistical Measures

2.1.2 Frequency Tables

2.1.3 Standard Deviation & Variance

2.1.4 Coding

2.2 Data Presentation

2.2.1 Data Presentation

2.2.2 Box Plots & Cumulative Frequency

2.2.3 Histograms

2.3 Working with Data

2.3.1 Outliers & Cleaning Data

2.3.2 Intrepreting Data

2.4 Correlation & Regression

2.4.1 Correlation & Regression

2.5 Further Correlation & Regression (A Level only)

2.5.1 PMCC & Non-linear Regression

2.5.2 Hypothesis Testing for Correlation

3. Probability

3.1 Basic Probability

3.1.1 Calculating Probabilities & Events

3.1.2 Venn Diagrams

3.1.3 Tree Diagrams

3.2 Further Probability (A Level only)

3.2.1 Set Notation & Conditional Probability

3.2.2 Further Venn Diagrams

3.2.3 Further Tree Diagrams

3.2.4 Probability Formulae

4. Statistical Distributions

4.1 Probability Distributions

4.1.1 Discrete Probability Distributions

4.2 Binomial Distribution

4.2.1 The Binomial Distribution

4.2.2 Calculating Binomial Probabilities

4.3 Normal Distribution (A Level only)

4.3.1 The Normal Distribution

4.3.2 Normal Distribution - Calculations

4.3.3 Standard Normal Distribution

4.4 Choosing Distributions (A Level only)

4.4.1 Modelling with Distributions

4.4.2 Normal Approximation of Binomial

5. Hypothesis Testing

5.1 Hypothesis Testing

5.1.1 Hypothesis Testing

5.2 Hypothesis Testing (Binomial Distribution)

5.2.1 Binomial Hypothesis Testing

5.3 Hypothesis Testing (Normal Distribution) (A Level only)

5.3.1 Sample Mean Distribution

5.3.2 Normal Hypothesis Testing

6. Large Data Set

6.1 Large Data Set

6.1 Large Data Set

Author: Dan