A confidence interval is an estimate in **statistics** drawn from a **sample population** to evaluate the overall **population value**. This type of measurement is important when aiming to assess the **level of certainty or uncertainty** in statistics.

## Definition: Confidence interval

Confidence in statistics refers to probability.

The confidence interval refers to the average of your estimate in statistics, including the **negative or positive variations**. The desired confidence is usually one minus the alpha value applied in the **statistical test**: (1 − a).^{2}

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## When do you use a confidence interval?

You use confidence intervals for **diverse statistical estimates** like **proportions, population means**, and **variations** between population means and proportions. The confidence interval helps to communicate the difference surrounding the point estimate.

## Calculating a confidence interval

There are certain aspects a student needs to consider before calculating the confidence interval. These include:

- The point of estimate
- Critical values used in the statistic test
- The
**standard deviation**of the sample - The sample size

### Point estimate

The point estimate refers to the statistical estimate the student makes.

### Critical value

The critical value guides the students on the number of standard deviations they need to achieve their desired confidence level for the confidence interval.

To find the critical values, you need to follow these three steps:

**Choose the alpha value:**

The alpha value refers to the probability verge for statistical importance. The most used alpha value is *p=0.05*, but you can use *0.1, 0.01,* or *0.001.*

**Decide between one-tailed and two-tailed:**

In the case of the two-tailed interval, you should divide the alpha value by two to get the alpha values of the higher and lower tails.

**Find the corresponding critical value:**

For **normal distribution** or when the sample size is larger than thirty, you can use the **z-distribution** to find the critical values.

Here are some of the common values used for **z statistics:**

Confidence Level: |
90%, 95%, 99% |

Alpha for one-tailed CI: |
0.1, 0.05, 0.01 |

Alpha for two-tailed CI: |
0.05, 0.025, 0.005 |

Z statistic: |
1.64, 1.96, 2.57 |

A student should use a **t distribution** in the case of small datasets with **normal distribution**.

### Standard deviation

The student should find the data’s sample **variance** and then perform a square root to get the standard deviation.^{3}

**Find sample variance:**

You can find the sample variance by adding the squared differences from the average, also referred to as **mean-squared-error (MSE)**:

= Sample variance

= The value of the one observation

= The mean value of all observation

=The number of observation

To get the MSE, deduct the sample mean from each value in the data set, square it, and divide the result by the sample size: -1 (n-1).

Add all these values to find the total sample variance .

**Square root sample variance:**

In the example above, the variance in the Asian estimate is *100*, while the variance in the US estimate is *25*. The square roots are *10* and *5*, respectively.

### Sample size

The sample size refers to the total observations in a data set. In the above survey, the sample size is *100* Americans and *100* Asians.

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## Confidence interval in normal distribution

The confidence interval in this case is:

= The population mean

= The critical value of the z distribution

= The population standard deviation

= The square root of the sample size

In the case of a t distribution, use the same formula but replace *Z** with *t**.

## Confidence interval for proportions

You should use the same formula for proportions, but the SD, in this case, equates to the same proportion multiplied by one subtracting the proportion.

= Proportion of the sample

= Critical value of z distribution

= Sample size

## Confidence interval in non-normal distribution

There are two methods that you can use to calculate the confidence interval for data with non-normal distribution:

**Find a distribution that matches the shape of your data:**

Apply this distribution to get the confidence interval.

**Data transformation to make it fit a normal distribution:**

Perform a reverse transformation on data, then calculate the maximum and minimum bounds of the confidence interval.

## Reporting the confidence interval

When reporting the confidence interval in papers, include the higher and lower bounds of the confidence level.

They are used in graphs when demonstrating variations between groups, variations around estimates, and creating a linear regression.

## Confidence interval – Common misinterpration

A common misinterpretation is that the real value of one’s estimate is between the higher and lower range of the confidence interval.

This is false, as the CI is calculated using a sample rather than an entire population.

## FAQs

To determine how good an estimate is. The higher the CI, the more caution you should take.

The number of observations in a statistical sample.

The square root of sample variance.

## Sources

^{1} National Library of Medicine. “Confidence Intervals.” Accessed January 2, 2023. https://www.nlm.nih.gov/nichsr/stats_tutorial/section2/mod2_confidence.html.

^{2} Zhang, Jing, Bruce W. Hanik and Beth H. Chaney. “Confidence Intervals: Evaluating and Facilitating Their Use in Health Education Research.” The Health Educator 40, no. 1 (Spring, 2008): 29-36. https://files.eric.ed.gov/fulltext/EJ863507.pdf.

^{3} Sullivan, Lisa. “Confidence Intervals.” Boston University School of Public Health. Accessed January 2, 2023. https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_confidence_intervals/bs704_confidence_intervals_print.html.