# How to Calculate Confidence Intervals

## What is a Confidence Interval?

Confidence intervals are an essential part of inferential statistics, upon which most market research is based.

Put simply, in statistics, a confidence interval is a type of interval estimate that is computed from the data that has been observed from a sample. In other words, a confidence interval is a range of values that researchers can be fairly certain their true value of interest lies in.

As we’ve previously discussed, the purpose of taking a random sample from a population of interest and computing a statistic, such as the mean, from the sample data, is to determine an approximate estimate of the mean of the greater population.

However, researchers must always be weary of how well the sample statistic actually estimates the underlying population value. Confidence intervals address this concern, as they provide a range of values which is likely to contain the population parameter of interest.

## What is a Confidence Level?

All confidence intervals are constructed at a confidence level — for instance, 95 percent. The confidence level is selected by the researchers conducting the statistical analysis.

But, what does a 95 percent confidence level mean?

If a confidence level is 95 percent, it means that if the same population were to be sampled on multiple occasions, and estimates of a parameter were made on each occasion, the resulting intervals would include the true population parameter in approximately 95 percent of the cases.

## 2 Types of Confidence Intervals

There are two types of confidence intervals: one-sided and two-sided.

### One-Sided Confidence Intervals vs. Two-Sided Confidence Intervals

The concept of one-sided and two-sided confidence intervals is fairly straightforward.

A **two-sided confidence interval** brackets the population parameter of interest from above and below.

A **one-sided confidence interval** brackets the population parameter of interest from either above or below, which establishes an upper or lower window in which the parameter exists.

## How to Calculate a Confidence Interval

To demonstrate how to calculate a confidence interval, let’s imagine a group of researchers that are interested in determining whether or not the oranges grown on a particular farm are large enough to be sold to a prospective grocery chain.

### Step #1: Find the number of samples (n).

The researchers randomly select 46 oranges from trees on the farm.

Therefore, **n = 46**.

### Step #2: Calculate the mean (x) of the the samples.

The researchers then calculate of a mean weight of 86 grams from their sample.

Therefore, **x = 86**.

### Step #3: Calculate the standard deviation (s).

It’s best to use the standard deviation of the entire population, however, in many cases researchers will not have access to this information. If this is the case, the researchers should use the standard deviation of the sample that they have established.

For our example, let’s say that the researchers have resorted to calculating the standard deviation from their sample. They receive a standard deviation of 6.2 grams.

Therefore, **s = 6.2**.

### Step #4: Decide the confidence interval that will be used.

95 percent and 99 percent confidence intervals are the most common choices in typical market research studies.

In our example, let’s say the researchers have elected to use a confidence interval of **95 percent**.

### Step #5: Find the Z value for the selected confidence interval.

The researchers would then utilize the following table to determine their Z value:

Confidence Interval | Z |

80% | 1.282 |

85% | 1.440 |

90% | 1.645 |

95% | 1.960 |

99% | 2.576 |

99.5% | 2.807 |

99.9% | 3.291 |

Since they have decided to use a 95 percent confidence interval, the researchers determine that **Z = 1.960**.

### Step #6: Calculate the following formula.

Next, the researchers would need to plug their known values into the formula.

Continuing with our example, this formula would appear as follows:

**86 ± 1.960 (6.2/6.782)**

When calculated, this formula gives the researchers the result of **86 ± 1.79** as their confidence interval.

### Step #7: Draw a conclusion.

The researchers have now determined that the true mean of the greater population of oranges is likely (with 95 percent confidence) between 84.21 grams and 87.79 grams.

## Conclusion

You now have the tools necessary to calculate confidence intervals and contextualize your research.

The next time you are working with a fairly straightforward and comprehensible data set, try playing with the confidence interval that you’ve selected.

How does choosing a 99 percent confidence interval over a 95 percent confidence interval affect your findings?