A **continuity correction **is applied when you want to use a continuous distribution to approximate a discrete distribution. Typically it is used when you want to use a normal distribution to approximate a binomial distribution.

Recall that the binomial distribution tells us the probability of obtaining *x *successes in *n *trials, given the probability of success in a single trial is *p*. To answer questions about probability with a binomial distribution we could simply use a Binomial Distribution Calculator, but we could also *approximate *the probability using a normal distribution with a continuity correction.

A continuity correction is the name given to **adding or subtracting 0.5 ****to a discrete x-value**.

For example, suppose we would like to find the probability that a coin lands on heads less than or equal to 45 times during 100 flips. That is, we want to find P(X ≤ 45). To use the normal distribution to approximate the binomial distribution, we would instead find P(X ≤ 45.5).

The following table shows when you should add or subtract 0.5, based on the type of probability you’re trying to find:

Using Binomial Distribution |
Using Normal Distribution with Continuity Correction |
---|---|

X = 45 | 44.5 < X < 45.5 |

X ≤ 45 | X < 45.5 |

X < 45 | X < 44.5 |

X ≥ 45 | X > 44.5 |

X > 45 | X > 45.5 |

Note:

It’s only appropriate to apply a continuity correction to the normal distribution to approximate the binomial distribution when n*p and n*(1-p) are both at least 5.

For example, suppose n = 15 and p = 0.6. In this case:

n*p = 15 * 0.6 = 9

n*(1-p) = 15 * (1 – 0.6) = 15 * (0.4) = 6

Since both of these numbers are greater than or equal to 5, it would be okay to apply a continuity correction in this scenario.

The following example illustrates how to apply a continuity correction to the normal distribution to approximate the binomial distribution.

**Example of Applying a Continuity Correction**

Suppose we want to know the probability that a coin lands on heads less than or equal to 43 times during 100 flips. In this case:

n = number of trials = 100

X = number of successes = 43

p = probability of success in a given trial = 0.50

We can plug these numbers into the Binomial Distribution Calculator to see that the probability of the coin landing on heads less than or equal to 43 times is **0.09667**.

To approximate the binomial distribution by applying a continuity correction to the normal distribution, we can use the following steps:

**Step 1: Verify that n*p and n*(1-p) are both at least 5**.

n*p = 100*0.5 = 50

n*(1-p) = 100*(1 – 0.5) = 100*0.5 = 50

Both numbers are greater than or equal to 5, so we’re good to proceed.

**Step 2: Determine if you should add or subtract 0.5**

Referring to the table above, we see that we’re supposed to **add ****0.5 **when we’re working with a probability in the form of X ≤ 43. Thus, we will be finding P(X< 43.5).

**Step 3: Find the mean (μ) and standard deviation (σ) of the binomial distribution.**

**μ** = n*p = 100*0.5 = 50

**σ **= √n*p*(1-p) = √100*.5*(1-.5) = √25 = 5

**Step 4: Find the z-score using the mean and standard deviation found in the previous step.**

**z **= (x – μ) / σ = (43.5 – 50) / 5 = -6.5 / 5 = -1.3.

**Step 5: Use the Z table to find the probability associated with the z-score.**

According to the Z table, the probability associated with z = -1.3 is **0.0968**.

Thus, the exact probability we found using the binomial distribution was **0.09667 **while the approximate probability we found using the continuity correction with the normal distribution was **0.0968**. These two values are pretty close.

**When to Use a Continuity Correction**

Before modern statistical software existed and calculations had to be done manually, continuity corrections were often used to find probabilities involving discrete distributions. Today, continuity corrections play less of a role in computing probabilities since we can typically rely on software or calculators to calculate probabilities for us.

Instead, it’s simply a topic discussed in statistics classes to illustrate the relationship between a binomial distribution and a normal distribution and to show that it’s possible for a normal distribution to approximate a binomial distribution by applying a continuity correction.

**Continuity Correction Calculator**

Use the Continuity Correction Calculator to automatically apply a continuity correction to a normal distribution to approximate binomial probabilities.