Statistics is a very rich subject that contains a lot of concepts that are incredibly helpful in data analysis. Whether we talk about standard deviation or central tendency, there are many useful statistical measures used to obtain desired results from data. Confidence interval is also an essential term that is utilized to estimate the reliability of data.
Experts use this concept to determine how close their findings are to the true values. If you are a data analyst or a student of statistics, it is important for you to understand how it is calculated. Learning to compute confidence intervals will help you throughout your professional career, as it is an integral part of many problems where data is involved. In this article, we will walk you through some practical examples of finding confidence intervals. However, before that, let’s start with some basics.
Understanding Confidence Intervals
A confidence interval is simply a range of numbers in which the true average of a population falls. It is not a single number but rather a combination of an upper and lower limit that shows where the true value is likely to be.
When it comes to calculating it, you have to rely on a statistical formula that is mentioned below:
Confidence Interval = X̄ ± Z× σ/√n
Here is the breakdown of elements of this expression:
- X̄ indicates the mean of the whole sample.
- Z is simply the z-score for the chosen confidence level.
- n tells about the size of the sample.
- σ refers to the standard deviation in the data.
Here, an important thing that needs to be remembered is that the portion after the ± signs is usually known as the margin of error. Another important thing is that the value of Z depends on the confidence level. It varies as you choose the more or less level of confidence.
Process Of Calculating Confidence Interval
Consider that you have sample data that has size n. You need to follow the steps below to calculate the confidence interval.
- First, set the base for your calculations by finding the average of the given sample data.
- Then, estimate the variability of your data by calculating the standard deviation.
- After that, find the appropriate value of the z-score for the chosen confidence level.
- Following that, calculate the margin of error using the formula mentioned above.
- Lastly, put all determined values in the formula of confidence intervals.
- As you do so, you will get the upper and lower limits of confidence intervals.
Remember, if you are already given the values of the standard deviation and mean, you just need to input them into the confidence intervals’ formula.
Calculation Of Real Life Examples Of Confidence Interval
Here is the calculation of some simple real-life examples of confidence intervals:
Example 1: Imagine a botanist studies the average height of a plant species by measuring 50 plants. He finds the mean height of 30 cm with a standard deviation of 5 cm. Now, he wants to calculate the confidence interval with a 95% confidence level. The calculation process will be as follows:
Solution:
X̄ = 30 cm
n = 50
Z-score for 95% confidence level = 1.96
By putting these values in the formula of confidence interval, we get:
CI = X̄ ± Z× σ/√n
CI = 30 ± 1.96 × 5 /√50
CI = 30 ± 1.386
CI = (30 − 1.386) to (30 + 1.386)
CIl = (28.614) to (31.386)
Hence, the true average height of plants lies between the interval 18.614—31.386.
Example 2: A brand conducts a survey to know the satisfaction rate of customers. They collect data from 25 customers with an average satisfaction score of 8.6 out of 10. If the standard deviation is 1.5, what will be the confidence interval with a 90% confidence level?
Solution:
X̄ = 8.6
n = 25
Z-score for 90% confidence level = 1.645
If we use these values in the mathematical expression, it will result as follows:
CI = X̄ ± Z× σ/√n
CI = 8.6 ± 1.645 × 1.5/√25
CI = 8.6 ± 0.493
CI = (8.6 − 0.493) to (8.6 + 0.493)
CI = (8.107) to (9.093)
It means the true value of ratting is more likely to be between the interval 8.107—9.093.
Easy Method To Calculate Confidence Intervals
What if we say that calculating confidence intervals is easier than you read above? Well, there is also a quick and easiest method to perform this calculation. Yes, you heard that right. What makes it possible is an efficient online confidence interval calculator. It is an automated tool that eliminates the need to be involved in frustrating manual calculations.
All you have to do is provide the calculator with the size of your sample data, standard deviation, mean value, and confidence level. It automatically chooses the accurate z-score value according to the confidence level and applies the formula using advanced algorithms. Within seconds, the range of confidence intervals appears on your screen.
To Sum Up
After going through this guide, we hope that you have learned how you can solve real-life examples of confidence intervals. So, from now on, this calculation doesn’t have to be difficult for you.
