Mastering the Normal Approximation to the Binomial Distribution

Introduction to the Normal Approximation

In A-Level Statistics, you are familiar with the binomial distribution, $X \sim B(n, p)$, which models the number of successes in $n$ independent trials. While calculating probabilities for small $n$ is straightforward, the arithmetic becomes increasingly tedious as $n$ grows. For instance, calculating $P(X \leq 50)$ for $n=100$ requires summing 51 individual binomial probabilities, which is inefficient and prone to error.

Fortunately, the Central Limit Theorem provides a powerful shortcut. When $n$ is sufficiently large, the shape of the binomial distribution begins to resemble the symmetric, bell-shaped curve of the normal distribution. By using the normal approximation, we can transform complex discrete calculations into simple continuous ones using standard normal tables or calculators. Mastering this technique is essential for your exams, as it allows you to handle large-scale data sets with ease.

Conditions for a Valid Approximation

Before applying the normal approximation, you must verify that the binomial distribution is sufficiently 'normal' in shape. The rule of thumb used in A-Level specifications is that the approximation is valid if:

1. $np \geq 5$ (or sometimes $10$ depending on the textbook, but $5$ is the standard minimum).
2. $nq \geq 5$ (where $q = 1 - p$).

If these conditions are met, we can model $X$ using a normal distribution $Y hicksim N(\mu, \sigma^2)$, where:

• The mean $\mu = np$
• The variance $\sigma^2 = npq$
• The standard deviation $\sigma = \sqrt{npq}$

If $p$ is close to $0.5$, the distribution is symmetric and the approximation is excellent even for moderate $n$. If $p$ is close to $0$ or $1$, the distribution is skewed, and you will need a much larger $n$ for the approximation to be accurate.

The Continuity Correction

The most critical step in this process is the continuity correction. The binomial distribution is discrete (it only takes integer values), while the normal distribution is continuous (it can take any value). To bridge this gap, we represent each integer $k$ as an interval $[k - 0.5, k + 0.5]$.

When converting inequalities, follow these rules:

• $P(X = k)$ becomes $P(k - 0.5 < Y < k + 0.5)$
• $P(X \leq k)$ becomes $P(Y < k + 0.5)$
• $P(X < k)$ becomes $P(Y < k - 0.5)$
• $P(X \geq k)$ becomes $P(Y > k - 0.5)$
• $P(X > k)$ becomes $P(Y > k + 0.5)$

Always draw a quick sketch of the bars to ensure you are including or excluding the correct boundary.

Worked Example 1: Calculating a Probability

Suppose $X \sim B(100, 0.4)$. Find the probability $P(X \leq 45)$.

Step 1: Check conditions. $np = 100 \times 0.4 = 40 \geq 5$. $nq = 100 \times 0.6 = 60 \geq 5$. Conditions met.

Step 2: Find parameters. $\mu = np = 40$. $\sigma^2 = npq = 100 \times 0.4 \times 0.6 = 24$. $\sigma = \sqrt{24} \approx 4.899$.

Step 3: Apply continuity correction. $P(X \leq 45) \approx P(Y < 45.5)$.

Step 4: Standardise and calculate. $Z = \frac{45.5 - 40}{4.899} = \frac{5.5}{4.899} \approx 1.12$. Using tables, $P(Z < 1.12) \approx 0.8686$.

Worked Example 2: Range Probability

Suppose $X \sim B(200, 0.5)$. Find $P(90 \leq X \leq 110)$.

Step 1: Parameters. $\mu = 200 \times 0.5 = 100$. $\sigma^2 = 200 \times 0.5 \times 0.5 = 50$. $\sigma = \sqrt{50} \approx 7.071$.

Step 2: Continuity correction. We want the range from 90 to 110 inclusive. This covers the interval from $89.5$ to $110.5$. $P(89.5 < Y < 110.5)$.

Step 3: Standardise. $Z_1 = \frac{89.5 - 100}{7.071} = -1.485$. $Z_2 = \frac{110.5 - 100}{7.071} = 1.485$.

Step 4: Calculate. $P(-1.485 < Z < 1.485) = P(Z < 1.485) - P(Z < -1.485) = 0.9312 - 0.0688 = 0.8624$.

Common Mistakes

1. Forgetting the continuity correction: This is the most frequent error. If you calculate $P(X \leq 45)$ as $P(Y < 45)$ instead of $P(Y < 45.5)$, your answer will be inaccurate.
2. Using the wrong variance: Remember that the variance of a binomial distribution is $npq$, not $np$. Always square root the variance to get the standard deviation before standardising.
3. Incorrect inequality boundaries: Be careful with 'greater than' versus 'greater than or equal to'. For example, $P(X > 50)$ is $P(Y > 50.5)$, not $P(Y > 49.5)$.

Frequently Asked Questions

Q: When should I use the normal approximation instead of the binomial? A: Use it when $n$ is large (typically $n > 30$ or $50$) and calculating binomial probabilities directly is impractical.

Q: Does the continuity correction apply to all distributions? A: No, it is specifically used when approximating a discrete distribution (like binomial or Poisson) with a continuous one (like the normal).

Q: What if $np < 5$? A: The normal approximation is not appropriate. You should use the binomial distribution directly or, if $n$ is large and $p$ is very small, consider a Poisson approximation.

Conclusion

The normal approximation is a vital tool for simplifying complex statistical problems. By checking your conditions, applying the correct continuity correction, and standardising carefully, you can solve these problems with confidence. Ready to see these concepts in motion? Head over to MathInstructor AI to generate a free, narrated animated lesson on the normal approximation to the binomial distribution.