Confidence Interval Calculator
Your DataHow to Use the Confidence Interval Calculator
Your Data:
- Mean , Variance , and Sample Size: Input the arithmetic mean, variance, and sample size of your data.
- Raw Data: If you opt for raw data, the calculator will find the arithmetic mean, standard deviation, and sample size for you. Please note that each data point should be separated by a comma.
- Decimal Places: Numbers that are greater than 0 are rounded to a specific number of decimal places. If the number of digits is negative, the number is rounded to the left of the decimal point. For example, when 3 decimal places are chosen, 94.1234 will be rounded to 94.123 and 0.004321 will be rounded to 0.004.
Do you know the population standard deviation (𝝈)?
Choose Yes, if you know the population standard deviation. Then input it in the space on the right. The calculator will use the standard normal distribution, z-score, and population standard deviation (σ).
Choose No, if you don’t know the population standard deviation. Leave the space on the right blank. The calculator will use the t-distribution, t-statistic, and sample standard deviation (s).
Step-by-Step Confidence Interval Calculation
We need to perform the following steps to estimate a confidence interval:
Calculate the sample’s mean value:
Determine the sample’s standard deviation:
Alternatively, use the population standard deviation when it’s known.
Compute the sample mean’s standard error using the following formula when the population variance is:
known:
unknown:
Choose the confidence level.
Common confidence levels are 90%, 95%, and 99% (with α=10%, α=5%, and α=1%, respectively).Estimate the level of significance:
Obtain the z-score from the standard normal distribution table when the population variance is known for probability (p) = 1 - α/For instance, the z-score for a 95% confidence interval is 1.9You can find the standard normal distribution critical values and the corresponding (1-a) in the z-table .
Alternatively, calculate the t-statistic from the Student’s t table when the population variance is unknown for probability (p) = 1 - α/2 and n-1 degrees of freedom.
Estimate the margin of error by multiplying the standard error by the z-score (when the population variance is known) or the t-statistic (when the population variance is unknown):
Calculate the confidence interval by adding and subtracting the margin of error from the mean value:
When the population variance is known:
When the population variance is unknown:
Confidence intervals provide us with an estimation of where the parameters are located. However, when making a decision, you need a Yes-or-No answer. The correct approach, in this case, is to use hypothesis testing , which you can perform using the Hypothesis Testing Calculator.
Moreover, the Confidence Interval Calculator can be only used for one population mean. When you need to explore intervals by looking into two populations, you can use the Confidence Interval (Two Populations) Calculator.
Confidence Interval Calculator
A confidence interval is the range within which you expect the population parameter to be. Its estimation is based on the data you provide in the sample.
The Confidence Interval Calculator allows you to compute the confidence interval of a mean by providing the standard deviation, mean, and sample size. Alternatively, if you don’t know the aforementioned measures, the tool will estimate them for you. In addition, you can use it with any arbitrary confidence level.
There are two circumstances where we calculate the confidence intervals for a population—when the population variance is known and when it’s unknown. Depending on that, the calculator would use the normal distribution or the Student's t-distribution for the confidence interval of the mean.
The calculator provides the necessary formulas, as well as a step-by-step solution.
What Is a Confidence Interval?
A confidence interval is the range within which you expect the population parameter to be. When we use sample data, we aren’t certain of the exact value of the parameter we examine. Therefore, confidence intervals help us calculate whether the results we obtain will match those from a population. They are usually expressed at a designated confidence level — the degree of certainty that the estimated parameter’s true value will be in the confidence interval. The most widely used one is 95%, but others, such as 90% or 99%, are also common.
For example, a 95% confidence interval would imply that we are 95% confident that the true population mean (µ) falls within this interval, with a 5% chance that it does not contain the true population mean (µ).
We build the confidence interval around the point estimate (the estimate of the population parameter). In fact, the point estimate is located exactly in the middle of the interval. However, the latter provides much more information and is the preferred choice for making inferences.
![Confidence Interval. Interval start, end and the estimated point between them](https://365datascience.com/resources/assets/images/confidence-interval-01-desktop.png)
Which Confidence Level Should You Use?
The confidence level is equal to (1 – α). Alpha (α) denotes the level of significance or the probability of rejecting the null hypothesis when it‘s true—in other words, the probability of making a Type I Error.
When constructing a confidence interval (or using a calculator), you pick the value of α. Put simply, you decide what risk you are willing to accept based on the certainty you need. Typical values are 0.01, 0.05, and 0.1.
You want to be very precise, therefore, pick a low significance level such as 0.01. For example, to check whether a machine works properly, you expect the test to make little or no mistakes. In most cases, the context you operate in will determine α, but 0.05 is the most commonly used value.
Calculating the Confidence Interval (Formula)
The confidence interval’s lower bound equals the point estimate, minus the reliability factor, times the standard error. Meanwhile, the upper bound equals the point estimate, plus the reliability factor, times the standard error.
There are two circumstances where we calculate the confidence intervals for a population—when the population variance is known and when it’s unknown. Depending on that, you or the Confidence Interval Calculator would use a different method to estimate.
The confidence interval for a sample with known population variance is equal to the following:
The confidence interval’s lower bound equals the sample mean minus
Whereas, we calculate the confidence interval for a sample with unknown population variance as follows:
There are two key differences between the formulas for known and unknown variances:
- Instead of the z-score, we use the t-statistic.
- Instead of the population standard deviation, we use the sample standard deviation.
The underlying logic in both cases is the same. The only two inputs that change are the standard deviation and the statistic at hand. When the population variance is known, we use the population standard deviation and the z-score. On the other hand, when the population variance is unknown, we use the sample standard deviation and the t-statistic.
![Known and unknown standard deviations graph.](https://365datascience.com/resources/assets/images/confidence-interval-02-desktop.png)
![Known standard deviations graph.](https://365datascience.com/resources/assets/images/confidence-interval-03-desktop.png)
In the table, this will match the value of 1– 0.025, or 0.9750. The corresponding
Standard Normal Table
![Standard Normal Distribution Table.](https://365datascience.com/resources/assets/images/confidence-interval-04-desktop.webp)
A commonly used term for is critical value. In our case, the critical value for the 95% confidence interval is 1.9 + 0.06, or 1.96.
When population variance is unknown, we need to use the t-statistic from the Student’s t table , which gives us the right tail area.
![Unknown standard deviations graph.](https://365datascience.com/resources/assets/images/confidence-interval-05-desktop.png)
One important characteristic of Student’s t-statistic is that it depends on the degrees of freedom. The latter is equal to the sample size (n) - 1. Put simply, for a sample of n, we have n-1 degrees. Following this logic, a sample of 20 observations would have 19 degrees of freedom.
The rows in the Student’s t table indicate different degrees of freedom, abbreviated as d.f., while the columns show common
Student’s t-Distribution Table
Area in Upper Tail
![Student\'s T-Distribution Table.](https://365datascience.com/resources/assets/images/confidence-interval-06-desktop.webp)
Note that after 30 degrees of freedom, the t-statistic table becomes almost the same as the z-score. As the degrees depend on the sample we observe that, in essence, the bigger the sample, the closer we get to the actual numbers. A common rule of thumb for a sample containing more than 50 observations is to use the z-table instead of the t-table.
In summary, the Confidence Interval Calculator involves determining three parameters: the sample mean value (μ), the population standard deviation (σ) when the variance is known, and the sample standard deviation (s) when the variance is unknown, as well as the sample size (n).
How to Calculate a Confidence Interval with Precision
The confidence intervals could be summarized as follows:
The true population mean (µ) falls within the interval defined by the sample mean +/- the margin of error (МЕ).
Since we want a better prediction, it is in our interest to have the narrowest confidence interval possible. If the range is too wide, we can’t be certain that the true population mean (µ) falls within the interval.
Getting a smaller margin of error means that the confidence interval would be narrower. As it happens, we can actually control the ME. It consists of three parts:
- Z-score and t-statistic
- Standard deviation
- Sample size
The statistic and the standard deviation are in the numerator, so smaller values for those will reduce the margin of error.
All else equals a higher level of confidence, which results in a higher margin of error. This leads to a wider confidence interval. Therefore, if we keep the standard deviation and the sample size at a constant lower confidence level will result in a narrower interval. In other words, when our confidence is lower, the confidence interval itself is smaller
What about the standard deviation? A lower value means that the dataset is more concentrated around the mean, so we have a better chance to get it right.
Lastly, we have the sample size in the denominator. Higher sizes will decrease the margin of error as the more observations you have in your sample, the more certain you are in the prediction. We can conclude, therefore, that the more observations in the sample, the higher the chances of getting a good idea about the entire population’s true mean.
![Variables effect on the confidence level width.](https://365datascience.com/resources/assets/images/confidence-interval-07-desktop.png)
To answer the question, we first need to define what normal distribution is. Otherwise known as Gaussian distribution, it’s a symmetrical distribution with equal mean, median and mode. Many statisticians and mathematicians also refer to it as the Bell Curve because of its bell-like shape.
The normal distribution is perfectly centered around its mean and it can be described by two values:
- The mean
- The standard deviation
However, the two measures can take on any value. So, we need a way to compare different normal distributions that may have different means and standard deviations. And this is where standardization comes into play—this is the process of transforming a variable to one with a mean of 0 and a standard deviation of 1.
Logically, a normal distribution can also be standardized; the result is called a standard normal distribution. The standardized variable is called a z-score and is equal to the original variable, minus its mean, and divided by its standard deviation.
By standardizing the normal distribution, you can easily calculate the probability of certain values occurring. In addition, you can compare datasets with different means and standard deviations. This is useful in hypothesis testing and confidence intervals.
A 95% confidence interval implies we are 95% confident that the true population mean falls within this interval. Logically, the other 5% denotes the chance that true population mean is not located there.
To calculate a 95% confidence interval, we use a point estimate of the population parameter (such as the sample mean or proportion), then add and subtract the appropriate critical value from the t-distribution (or z-distribution) times the standard error. The resulting range represents the plausible values for the population mean with 95% confidence.
You can use our Confidence Interval Calculator to obtain values, including the 95% confidence interval.