*40*

AÂ **continuity correctionÂ **is applied when you want to use a continuous distribution to approximate a discrete distribution.Â 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 obtainingÂ *xÂ *successes inÂ *nÂ *trials, given the probability of success in a single trial isÂ *p*.Â To answer questions about probability with a binomial distribution we could simply use a Binomial Distribution Calculator, but we could alsoÂ *approximateÂ *the probability using a normal distribution with a continuity correction.

A continuity correction is the name given toÂ **adding or subtracting 0.5Â ****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 findÂ P(XÂ â‰¤ 45). To use the normal distribution to approximate the binomial distribution, we would instead findÂ P(XÂ â‰¤ 45.5).

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

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

X = 45 | 44.5 |

XÂ â‰¤ 45 | X |

X | X |

XÂ â‰¥ 45 | X > 44.5 |

X > 45 | X > 45.5 |

Note:ÂÂ

Itâ€™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.

Â

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

Â

n*p = 15 * 0.6 = 9

Â

n*(1-p) = 15 * (1 â€“ 0.6) = 15 * (0.4) = 6

Â

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 isÂ **0.09667**.

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

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

n*p = 100*0.5 = 50

n*(1-p) = 100*(1 â€“ 0.5) = 100*0.5 = 50

Both numbers are greater than or equal to 5, so weâ€™re good to proceed.

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

Referring to the table above, we see that weâ€™re supposed toÂ **addÂ ****0.5Â **when weâ€™re working with a probability in the form ofÂ XÂ â‰¤ 43. Thus, we will be finding P(X

**Step 3: Find the mean (Î¼) and standard deviation (Ïƒ) of the binomial distribution.**

**Î¼** = n*p = 100*0.5 = 50

**ÏƒÂ **=Â âˆšn*p*(1-p) =Â âˆš100*.5*(1-.5) =Â âˆš25 = 5

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

**z **=Â (x â€“Â Î¼) /Â Ïƒ = (43.5 â€“ 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 isÂ **0.0968**.

Thus, the exact probability we found using the binomial distribution wasÂ **0.09667Â **while the approximate probability we found using the continuity correction with the normal distribution wasÂ **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.Â 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, itâ€™s simply a topic discussed in statistics classes to illustrate the relationship between a binomial distribution and a normal distribution and to show that itâ€™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.