Enter the sample size, sample mean, standard deviation and confidence level. The calculator finds the confidence interval for the population mean (lower and upper bounds), the margin of error, and a table for common confidence levels.
For the standard deviation, enter the population standard deviation σ if you know it. If you do not, you can use the standard deviation of your sample instead, as long as the sample is large enough (30 or more as a rule of thumb). Enter the confidence level as a percentage (for example, 95 for 95%).
Result and graph
Enter numbers in the fields on the left and press "Calculate". The result and a graph will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formula and graph
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the sample size \(n\), the sample mean \(\bar{x}\), the standard deviation \(\sigma\) and the confidence level (%), and you get the confidence interval for the population mean (lower and upper bounds) on the spot
- It also calculates the margin of error \(E\) (how far off, plus or minus, the estimate may be), the standard error and the critical value z for the confidence level
- See the confidence intervals at common confidence levels such as 90%, 95% and 99% in one table, and check on a graph how the interval widens
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This calculator finds a confidence interval based on the normal distribution (the z method), for when the population standard deviation σ is known (or the sample is large enough to use the sample standard deviation instead). It does not handle confidence intervals based on the t-distribution, which are used when the sample is small and the population standard deviation is unknown.
What is this calculation used for?
What the "± points" in polls really tells you (news and politics)
The "margin of error" in a poll result such as "presidential approval at 45% (margin of error ±2.5 points)" is exactly the margin of error \(E\) on this page. For a poll with 45% approval and 1,600 respondents, the spread is \(\sigma = \sqrt{0.45 \times 0.55} \approx 0.497\), so at a 95% confidence level \(E = 1.96 \times 0.497 \div \sqrt{1600} \approx 0.024\) (about ±2.4 points). This assumes the share of answers can be approximated by a normal distribution. The "±2.5 points" in news reports matches an estimate made with 50% approval, where the spread is largest.
You will be able to judge for yourself whether news that "approval moved 1 point" is within the margin of error.
Quality control in a factory: backing up the "500 mL" label with numbers (manufacturing)
A factory cannot open and measure every bottle, so it checks a sample. If 50 bottles (the standard 500 mL, or 16.9 fl oz, size) average 500.2 mL with a standard deviation of 2 mL, the 95% confidence interval for the average content of everything produced is about 499.65 to 500.75 mL.
Checking statistically, from sample inspections alone, that products are not below the labeled amount is a calculation used every day in food, beverage and drug manufacturing.
Reading how sure we are that a new treatment works (medicine)
Results in medical papers are almost always reported with a confidence interval, as in "the new drug lowered blood pressure by 8.0 mmHg on average (95% confidence interval: 5.6 to 10.4)". In a 100-person study where the drops have a standard deviation of 12 mmHg, \(E = 1.96 \times 12 \div \sqrt{100} \approx 2.4\).
If the whole interval is on the plus side (it does not include 0), that supports "the drug had an effect". If the interval includes 0, read it as "it may be chance". This is one of the most important ways to read health information for yourself.
Running a national survey without asking everyone (sample surveys in education and government)
Testing every student in the country is a huge job, so national studies of fitness and achievement are often done with samples. For example, if a sample of 400 students has an average 50-meter dash time of 9.2 seconds with a standard deviation of 0.8 seconds, the 95% confidence interval for the average time of all students is about 9.12 to 9.28 seconds. Just 400 students give a fairly narrow range.
Government statistics, such as the Current Population Survey behind the monthly unemployment rate, are also based mainly on samples, and this calculation backs up how precise they are.
Telling real improvements from noise on a website (IT and marketing)
When the average time on your site goes up from 180 seconds to 184 seconds, is that the effect of your change, or just chance? With 2,500 visitors and a standard deviation of 90 seconds, the 95% confidence interval for the average time is about 176.5 to 183.5 seconds, so there is a range of about ±3.5 seconds.
If a change is about the same size as the width of the confidence interval, you can tell it "cannot yet be told apart from chance". This idea is the foundation for not misreading A/B test results.
Formula and graph
Margin of error \(E\) (half the width of the interval)
Graph
Standard notation
(the usual math form)
\(E\)
\(=\)
\(z\)
\(\times\)
\(\sigma\)
\(\div\)
\(\sqrt{n}\)
In words
(symbols replaced with words)
④
\(E\): margin of error
\(=\)
③
\(z\): critical value for the confidence level
\(\times\)
①
\(\sigma\): population standard deviation
\(\div\)
②
\(\sqrt{n}\): square root of the sample size \(n\)
The formula in words
①
Take the
\(\sigma\): population standard deviation
②
divide it by the
\(\sqrt{n}\): square root of the sample size \(n\)
to get the standard error (the spread of the sample mean)
③
multiply by the
\(z\): critical value for the confidence level
④
and you get the
\(E\): margin of error
Quick example
You check the contents of 25 bottles. With a standard deviation of 2 mL and a 95% confidence level (z = 1.96), the margin of error is
\(E\): margin of error
\(=\)
z (1.96)
\(\times\)
standard deviation (2)
\(\div\)
square root of 25 (5)
\(1.96 \times 2 \div 5 = 1.96 \times 0.4 = 0.784\)
Key idea
\(\sigma \div \sqrt{n}\) is called the standard error. It is the size of the spread of the sample mean itself. The key point is that you divide by \(\sqrt{n}\) (the square root), not by \(n\): making the sample 4 times larger only cuts the margin of error in half. This is why you need a lot of data to gain precision.
Confidence interval (lower bound \(L\) and upper bound \(U\))
Graph
Standard notation
(the usual math form)
\(L,\ U\)
\(=\)
\(\bar{x}\)
\(\pm\)
\(E\)
In words
(symbols replaced with words)
③
\(L\), \(U\): lower and upper bounds of the confidence interval
\(=\)
①
\(\bar{x}\): sample mean
\(\pm\)
②
\(E\): margin of error
The formula in words
①
From the
\(\bar{x}\): sample mean
②
subtract and add the
\(E\): margin of error
(the symbol ± stands for both "minus" and "plus")
③
and you get the
\(L\), \(U\): lower and upper bounds of the confidence interval
Quick example
If the average content of the 25 bottles is 500.2 mL and the margin of error is 0.784 mL, the 95% confidence interval for the population mean is
\(L\), \(U\): lower and upper bounds of the confidence interval
\(=\)
sample mean (500.2)
\(\pm\)
margin of error (0.784)
\(L = 500.2 - 0.784 = 499.416\)
\(U = 500.2 + 0.784 = 500.984\)
Key idea
Put the first two formulas together and you get the familiar textbook formula for a confidence interval, \(\bar{x} \pm z \dfrac{\sigma}{\sqrt{n}}\).
Be careful how you read "95% confidence interval". It does not say "the population mean is inside this interval with 95% probability". It says "if you repeated the process of taking a sample and building an interval the same way 100 times, about 95 of those intervals would contain the population mean". The population mean is one fixed value; it is the interval that moves from sample to sample.
Finding the critical value z from the confidence level
Graph
Standard notation
(the usual math form)
\(z\)
\(=\)
\(\Phi^{-1}\!\left(\dfrac{1+C}{2}\right)\)
In words
(symbols replaced with words)
②
\(z\): critical value for the confidence level
\(=\)
①
the share counted from the left, (1 + confidence level \(C\)) ÷ 2, turned into a position on the z axis
The formula in words
①
Find
the share counted from the left, (1 + confidence level \(C\)) ÷ 2, turned into a position on the z axis
(\(\Phi^{-1}\), "phi inverse", is the function that turns "a share counted from the left" into "a position on the horizontal axis". Write the confidence level \(C\) as a decimal, such as 0.95 for 95%)
②
and that is the
\(z\): critical value for the confidence level
Quick example
The critical value z for a 95% confidence level (C = 0.95) is
critical value \(z\)
\(=\)
position 97.5% from the left, \(\Phi^{-1}(0.975)\)
\(\Phi^{-1}\!\left(\dfrac{1+0.95}{2}\right) = \Phi^{-1}(0.975) \approx 1.96\)
Key idea
The critical value z is the boundary on the standard normal distribution that holds exactly the confidence level (95% for 95%) in the middle. With 95% in the middle, 2.5% is left in each tail, so z is the position 97.5% from the left: \(\Phi^{-1}(0.975) \approx 1.96\).
The common values are fixed: 90% → 1.6449, 95% → 1.9600, 99% → 2.5758, 99.9% → 3.2905. This calculator also works out less common confidence levels that are not in the table (for example, 97%).
A confidence interval is "sample mean ± margin of error". The margin of error is \(z \times \sigma \div \sqrt{n}\), so all together it is \(\bar{x} \pm z \dfrac{\sigma}{\sqrt{n}}\). The larger the sample (the larger \(\sqrt{n}\)), the narrower the interval; the higher the confidence level (the larger z), the wider the interval.
Symbols and terms
Symbols
| \(n\) | en | The sample size: how many items or people you actually measured. (Example: measure 50 bottles and \(n = 50\)) |
| \(\bar{x}\) | x-bar | The sample mean, the average of the sample you measured. The bar on top is the sign for an average. |
| \(\sigma\) | sigma | The population standard deviation, the size of the spread in the whole population. A Greek letter that corresponds to the English s. If the sample is large enough, the standard deviation of the sample can be used instead. |
| \(\sqrt{n}\) | square root of n | The square root of the sample size (example: \(\sqrt{25} = 5\)). It appears in the denominator of the standard error, and it is the reason 4 times the data only cuts the error in half. |
| \(z\) | zee (critical value) | The multiplier set by the confidence level. It is the boundary on the standard normal distribution that holds exactly the confidence level in the middle. For 95%, it is about 1.96. It is also written z*. |
| \(E\) | E | The margin of error. It is half the width of the confidence interval and a guide to how far, plus or minus, the sample mean may be from the population mean. E stands for "error". |
| \(L,\ U\) | L, U | The lower and upper bounds of the confidence interval, found with \(L = \bar{x} - E\) and \(U = \bar{x} + E\). |
| \(\Phi^{-1}\) | phi inverse (inverse of Φ) | The function that works the other way from \(\Phi\), the cumulative distribution function of the standard normal distribution. It takes a probability (area) and returns the matching z. (Example: \(\Phi^{-1}(0.975) \approx 1.96\)) |
| \(C\) | C | The confidence level written as a decimal. For 95%, \(C = 0.95\). |
Terms
| confidence interval | A range estimated from sample data where the population mean is likely to be. It is built by extending the sample mean up and down by the margin of error. |
| confidence level | How confident you can be in the interval. It is the share of intervals that would contain the population mean if you built intervals the same way over and over. 95% is used most often. It is also called the confidence coefficient. |
| population | The whole group you would really like to study. (Examples - every bottle a factory makes, or every voter in the country) |
| sample | The part of the population that you actually pick out and study. Using a sample to infer what the population looks like is what statistical estimation is about. |
| population mean | The mean of the whole population. You cannot measure everything, so its exact value is unknown. A confidence interval estimates where this unknown value is. It is usually written \(\mu\) (mu). |
| sample mean | The mean of the sample, \(\bar{x}\). It is the single most likely estimate of the population mean (a point estimate), but it changes a little every time depending on which sample you happen to take. |
| standard error | The size of the spread of the sample mean itself, found with \(\sigma \div \sqrt{n}\). It is easy to confuse with the spread of the individual data values (the standard deviation), so be careful. |
| interval estimate | A way to estimate an unknown value such as the population mean with a range instead of a single number. Compared with a point estimate, its strength is that it also shows how far off the estimate may be. |
| t-distribution | The distribution used instead of the normal distribution (z) when the sample is small and the population standard deviation is unknown. The calculator on this page supports only the z method. As the sample gets larger, the t-distribution gets closer to the normal distribution and the results become almost the same. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| The mean (Grade 6) |
|
| Square roots (Grade 8) |
|
| Random sampling (Grade 7) |
|
| Data analysis and standard deviation (high school statistics) |
|
| The normal distribution and interval estimates (high school statistics / AP Statistics) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the critical value z for a confidence level
| Confidence (%) | 95 |
| z | =NORM.S.INV((1+B1/100)/2) |
Table to find the margin of error E
| z | 1.96 |
| Standard dev. σ | 2 |
| Sample size n | 25 |
| Margin of error E | =B1*B2/SQRT(B3) |
Table to find the confidence interval (lower and upper bounds)
| Sample size n | 50 |
| Sample mean x̄ | 500.2 |
| Standard dev. σ | 2 |
| Confidence (%) | 95 |
| z | =NORM.S.INV((1+B4/100)/2) |
| Margin of error E | =B5*B3/SQRT(B1) |
| Lower bound | =B2-B6 |
| Upper bound | =B2+B6 |
"NORM.S.INV(probability)" is the function that finds z from a probability (the inverse of the cumulative distribution function, \(\Phi^{-1}\)). "SQRT" is the square root.
The first table finds z from the confidence level (B1) alone; enter 95 and B2 is about 1.959964. The second table is the margin of error; with the example values, B4 is 0.784.
The third table finds the confidence interval from all the inputs at once. With the example values (n = 50, x̄ = 500.2, σ = 2, 95%), the lower bound (B7) is about 499.6456 and the upper bound (B8) is about 500.7544. Just replace B1 to B4 with your own numbers.
The first table finds z from the confidence level (B1) alone; enter 95 and B2 is about 1.959964. The second table is the margin of error; with the example values, B4 is 0.784.
The third table finds the confidence interval from all the inputs at once. With the example values (n = 50, x̄ = 500.2, σ = 2, 95%), the lower bound (B7) is about 499.6456 and the upper bound (B8) is about 500.7544. Just replace B1 to B4 with your own numbers.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the critical value z for a confidence level
| Confidence (%) | 95 |
| z | =NORMSINV((1+B1/100)/2) |
Table to find the margin of error E
| z | 1.96 |
| Standard dev. σ | 2 |
| Sample size n | 25 |
| Margin of error E | =B1*B2/SQRT(B3) |
Table to find the confidence interval (lower and upper bounds)
| Sample size n | 50 |
| Sample mean x̄ | 500.2 |
| Standard dev. σ | 2 |
| Confidence (%) | 95 |
| z | =NORMSINV((1+B4/100)/2) |
| Margin of error E | =B5*B3/SQRT(B1) |
| Lower bound | =B2-B6 |
| Upper bound | =B2+B6 |
In Google Sheets, the function that finds z from a probability is written "NORMSINV(probability)" (it works the same as "NORM.S.INV" in Excel). All the other formulas are the same as in Excel.
Copy the whole table, paste it into cell A1, and replace the numbers in column B with your own. The results are the same as in the Excel version: in the third table, the lower bound is about 499.6456 and the upper bound is about 500.7544.
Copy the whole table, paste it into cell A1, and replace the numbers in column B with your own. The results are the same as in the Excel version: in the third table, the lower bound is about 499.6456 and the upper bound is about 500.7544.
How to calculate it in Python
import math
from statistics import NormalDist
sample_size = 50 # sample size n
sample_mean = 500.2 # sample mean x̄
sd = 2 # standard deviation σ
confidence_level = 95 # confidence level (%)
z = NormalDist().inv_cdf((1 + confidence_level / 100) / 2) # critical value z for the confidence level
standard_error = sd / math.sqrt(sample_size) # standard error σ ÷ √n
margin_of_error = z * standard_error # margin of error E
lower = sample_mean - margin_of_error # lower bound of the confidence interval
upper = sample_mean + margin_of_error # upper bound of the confidence interval
print(f"z: {z}")
print(f"Margin of error E: {margin_of_error}")
print(f"{confidence_level}% confidence interval: {lower} to {upper}")
Runs with the standard library only. NormalDist().inv_cdf() is the function that finds z from a probability (\(\Phi^{-1}\)). Replace the first four values with your own numbers and run it. With the example values, it shows about 499.65 to 500.75.
How to write it in LaTeX and other math languages (copy and paste)
Margin of error \(E\) (half the width of the interval)
E = z × σ ÷ √n
E = z \times \dfrac{\sigma}{\sqrt{n}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>E</mi>
<mo>=</mo>
<mi>z</mi>
<mo>×</mo>
<mfrac>
<mi>σ</mi>
<msqrt><mi>n</mi></msqrt>
</mfrac>
</mrow>
</math>
E = z sigma / sqrt(n)
z*sigma/Sqrt[n]
E := z*sigma/sqrt(n);
E = z*sigma/sqrt(n);
E = zσ/√n
Confidence interval (lower bound \(L\) and upper bound \(U\))
L = x̄ − E, U = x̄ + E
L = \bar{x} - E,\quad U = \bar{x} + E
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>L</mi>
<mo>=</mo>
<mover><mi>x</mi><mo>¯</mo></mover>
<mo>−</mo>
<mi>E</mi>
<mo>,</mo>
<mspace width="1em"/>
<mi>U</mi>
<mo>=</mo>
<mover><mi>x</mi><mo>¯</mo></mover>
<mo>+</mo>
<mi>E</mi>
</mrow>
</math>
L = bar x - E, U = bar x + E
{xbar - e, xbar + e}
L := xbar - E; U := xbar + E;
L = xbar - E; U = xbar + E;
L = x̄ - E, U = x̄ + E
Finding the critical value z from the confidence level
z = Φ⁻¹((1 + C) ÷ 2)
z = \Phi^{-1}\!\left(\dfrac{1+C}{2}\right)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>z</mi>
<mo>=</mo>
<msup><mi>Φ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup>
<mrow>
<mo>(</mo>
<mfrac>
<mrow><mn>1</mn><mo>+</mo><mi>C</mi></mrow>
<mn>2</mn>
</mfrac>
<mo>)</mo>
</mrow>
</mrow>
</math>
z = Phi^-1((1 + C)/2)
InverseCDF[NormalDistribution[0, 1], (1 + c)/2]
with(Statistics): Quantile(RandomVariable(Normal(0, 1)), (1 + c)/2);
z = norminv((1 + c)/2);
z = Φ^(-1)((1+C)/2)
How to have ChatGPT do the calculation
You are a statistics calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). 50 bottles were taken off a factory line and their contents were measured. The mean was 500.2 mL. The spread of the fill amount (the population standard deviation) is known to be 2 mL. 1. Find the critical value z for a 95% confidence level (use the inverse of the cumulative distribution function of the standard normal distribution; Python's standard library statistics.NormalDist().inv_cdf can be used). 2. Find the margin of error E = z × σ ÷ √n. 3. Find the 95% confidence interval (lower and upper bounds) for the population mean (the true average content of all the bottles). Show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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