Normal Distribution Probability Calculator

Enter the mean, the standard deviation, and the lower and upper bounds of the range you want. The calculator finds the probability of falling in that range and a confidence interval table.

Mean μ

Standard deviation σ

Lower bound

Upper bound

If there is no lower bound, enter "-inf". If there is no upper bound, enter "inf" (they stand for minus infinity and plus infinity).

Result and graph

Enter the mean, standard deviation and range in the fields on the left and press "Calculate". The result will appear here.

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What you can do on this page
What is this calculation used for?
How to Use
Formula and graph
- Probability of falling in a range (using the cumulative distribution function \(\Phi\))
- Formula for the normal curve (probability density function)
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter the mean \(\mu\) and the spread (standard deviation \(\sigma\)), and you get the probability that a value falls in a given range on the spot
It also finds the probability of being at or above (or at or below) a value, such as "What percent of students scored 80 or higher?"
A table of the ranges often used in statistics (such as the range that holds 95% of the whole) is shown at the same time
A plain-language explanation of the normal distribution and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

Use it when the data is spread out close to a normal distribution (a symmetric bell shape around the mean). Many real-world data, such as test scores, heights and the weights of products, are known to come close to this shape.

What is this calculation used for?

Reading test scores and percentiles correctly (IQ and standardized tests)

IQ scores are set so that the mean is 100 and one standard deviation is 15 points. In a normal distribution, only about 16% of people are at \(+1\sigma\) or above (an IQ of 115 or more), and only about 2.3% are at \(+2\sigma\) or above (130 or more).
Once you see this link, percentile ranks on standardized tests stop being a mystery. You can read what "a 10-point jump" or "the 90th percentile" really stands for in numbers, not just by feel.

Reading the "reference range" on lab results calmly (health)

The "reference range" on blood tests and similar lab results is often set as the range that holds the middle 95% of measurements from healthy people (about the mean \(\pm 2\sigma\)).
So for any test item, about 5% of healthy people (1 in 20) fall outside the reference range. On a checkup with many items, it is actually common for a healthy person to be flagged on at least one. Knowing this, you understand that "slightly outside the range" is not automatically a sign that "something is wrong", and you can listen to your doctor's explanation calmly. It is especially useful if test results tend to worry you.

Quality control in a factory: how many products miss the spec?

The weights and sizes of products vary in a way that comes close to a normal distribution. For example, if bags labeled 16 oz are filled with an average of 16.2 oz and a standard deviation of 0.1 oz, the bags under 16 oz are those beyond \(-2\sigma\), about 2.3%.
Factories use this calculation to predict the defect rate and to decide where to set the average fill (how much extra to put in) to balance customer complaints against cost.

Deciding A/B tests on websites and apps (marketing)

When you compare whether design A or design B gets the higher click-through rate, you use the normal distribution (a confidence interval) to decide whether the difference is just chance or a real difference.
The "0.95" row in the table shown in the result is exactly the "95% confidence interval" used most often in practice. Behind decisions that move large sums of money, such as ad budgets, this calculation is used every day.

The math that makes insurance work (actuarial science)

Professionals called actuaries estimate the risk of paying out claims as probabilities and set insurance premiums from them.
Single accidents cannot be predicted, but the total over many policyholders comes close to a normal distribution (the central limit theorem), so the total payout can be predicted surprisingly accurately. This math is why we can get large coverage for an affordable premium.

Formula and graph

Probability of falling in a range (using the cumulative distribution function \(\Phi\))

Graph

Standard notation (the usual math form)

\(P(a \le X \le b)\) \(=\) \(\Phi\!\left(\dfrac{b-\mu}{\sigma}\right)\) \(-\) \(\Phi\!\left(\dfrac{a-\mu}{\sigma}\right)\)

In words (symbols replaced with words)

③ probability of falling between \(a\) and \(b\) \(=\) ① share up to the upper bound \(b\) \(-\) ② share up to the lower bound \(a\)

The formula in words

① Find the share up to the upper bound \(b\) with the cumulative distribution function \(\Phi\) (phi). Inside the parentheses of \(\Phi\) is the \(z\)-score: the distance from the mean \(\mu\) divided by the standard deviation \(\sigma\)

② find the share up to the lower bound \(a\) in the same way, and subtract it

③ to get the probability of falling between \(a\) and \(b\)

Quick example

On a test with a mean of 62 points and a standard deviation of 14 points, the share of scores between 50 and 80 is

probability of 50 to 80 points \(=\) share up to 80 points \(\Phi((80-62)/14)\) \(-\) share up to 50 points \(\Phi((50-62)/14)\)

\(\Phi(1.29) - \Phi(-0.86) \approx 0.90 - 0.20 = 0.70\ \ (70\%)\)

Key idea

Dividing the distance from the mean \(\mu\) by the standard deviation \(\sigma\) is called standardization (turning a value into a \(z\)-score). Whatever the mean and standard deviation of a normal distribution, once you rewrite values as \(z = (x - \mu) \div \sigma\), you can find probabilities with a single standard normal distribution with mean 0 and standard deviation 1 (a common ruler). This calculator also does this conversion internally before calculating. IQ scores are built on the same idea: IQ \(= 100 + 15 \times z\).

Formula for the normal curve (probability density function)

Graph

Standard notation (the usual math form)

\(f(x)\) \(=\) \(\dfrac{1}{\sigma\sqrt{2\pi}}\) \(\times\) \(e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)

In words (symbols replaced with words)

③ height of the curve at \(x\), \(f(x)\) \(=\) ② scaling factor (makes the total area 1) \(\times\) ① part that shrinks away from the mean

The formula in words

① The part that shrinks away from the mean drops quickly as \(x\) moves away from the mean \(\mu\), which makes the bell shape

② the scaling factor (makes the total area 1) makes the area under the curve exactly 1

③ and together they give the height of the curve at \(x\), \(f(x)\) (the area between the curve and the horizontal axis is the probability)

Quick example

You will rarely calculate this formula by hand. Usually you convert to a \(z\)-score as in the first formula, or use the calculator on this page or Excel.

Key idea

In a normal distribution, about 68.3% of the data falls within the mean \(\pm 1\sigma\) (one standard deviation), about 95.4% within \(\pm 2\sigma\), and about 99.7% within \(\pm 3\sigma\) (the 68–95–99.7 rule, also called the empirical rule). For example, with a mean of 62 points and a standard deviation of 14 points, about 68% of students score between 48 and 76. Keep this rule of thumb in mind and you get a feel for how the shape of the curve matches the probabilities.

The key to normal distribution probabilities: whatever the mean and standard deviation, once you rewrite a value as "how many standard deviations it is from the mean" (the \(z\)-score), you can find the probability with one common ruler.

Symbols and terms

Symbols

\(\mu\)	mu	The mean. It is where the center of the peak of the distribution sits. A Greek letter that corresponds to the English m.
\(\sigma\)	sigma	The standard deviation. It is the size of the spread of the data and sets the width of the peak. The larger it is, the lower and wider the peak. A Greek letter that corresponds to the English s.
\(X\)	X (random variable)	A letter for a quantity whose value can change each time you measure it, such as a test score or a height.
\(\Phi\)	capital phi	The cumulative distribution function of the standard normal distribution (mean 0, standard deviation 1). It returns the probability of getting that value or less.
\(z\)	z (z-score)	A number that tells how many standard deviations a value is from the mean. It is found with \(z = (x - \mu) \div \sigma\).
\(e\)	e (Euler's number)	A constant, about 2.71828…, and the base of the natural logarithm. It is a special number at the heart of exponential functions.
\(\pi\)	pi	A constant, about 3.14159…, the ratio of a circle's circumference to its diameter. In the normal distribution formula it appears to make the total area under the curve exactly 1.

Terms

normal distribution	A distribution spread symmetrically around the mean in a bell shape. It is used as a model for data where values near the mean are the most common and values get rarer the farther they are from the mean, such as heights, test scores and the weights of products.
standard deviation	One number that shows how spread out the data is. Two data sets can have the same mean but different spreads. The larger the standard deviation, the lower and wider the peak of the distribution.
standardization (z-score)	Rewriting a value as "how many standard deviations it is from the mean" (the \(z\)-score). Any normal distribution, once standardized, can be handled with the single standard normal distribution with mean 0 and standard deviation 1. IQ scores use the same idea: IQ \(= 100 + 15 \times z\).
cumulative distribution function	A function that returns the probability of getting that value or less. For the standard normal distribution it is written \(\Phi\) (phi). The probability of a range is "\(\Phi\) up to the upper bound" minus "\(\Phi\) up to the lower bound".
probability density function	The formula for the curve of the distribution itself. The height of the curve is not a probability but a probability density. The area between the curve and the horizontal axis over a range is the probability. For the normal distribution, it is a bell-shaped curve.
confidence interval	The table in the result lists, for each percentage, the range centered on the mean that holds that share of the whole distribution (\(\mu \pm z \times \sigma\)) and its multiplier \(z\). Strictly speaking, a "95% confidence interval" in statistics is a range for the true value (such as the population mean) estimated from sample data. But when a normal distribution is assumed, it uses the same \(z\) (1.96 for 95%), so the \(z\) values in this table apply directly. It is often used to find the "margin of error" of surveys and test results.
68–95–99.7 rule	A handy rule of thumb, also called the empirical rule: in a normal distribution, about 68.3% of the data falls within the mean \(\pm 1\sigma\), about 95.4% within \(\pm 2\sigma\), and about 99.7% within \(\pm 3\sigma\).
central limit theorem	The mathematical fact that a quantity built up from many small random effects (a sum or an average) gets close to a normal distribution, whatever the original distribution. It is the reason so much real-world data comes close to a normal distribution.

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)	Knowing that you find the mean by adding up all the data and dividing by how many values there are Knowing that the mean is a typical value that marks the middle of the distribution
Data analysis and standard deviation (high school statistics)	Knowing that two data sets with the same mean can be more spread out or less spread out Knowing that the standard deviation is one number that shows the size of that spread
Reading shares as areas (Grades 6–7)	Being able to see the size of an area on a graph as a share of the whole Being able to read the shape of a distribution from a histogram (a bar graph of the data)
Standardization and z-scores (high school statistics / AP Statistics)	Knowing that "how many standard deviations from the mean" lets you compare data measured on different scales Knowing that IQ scores are built on this idea (IQ \(= 100 + 15 \times z\))

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 probability of a range

Mean μ	62
Standard dev. σ	14
Lower bound	50
Upper bound	80
Probability between the bounds	=NORM.DIST(B4,B1,B2,TRUE)-NORM.DIST(B3,B1,B2,TRUE)

Table to find the probability of a value or more (top what percent)

Mean μ	62
Standard dev. σ	14
Value (find the share at or above it)	80
Probability of that value or more	=1-NORM.DIST(B3,B1,B2,TRUE)

Table to find the height of the curve at a point (probability density)

Mean μ	62
Standard dev. σ	14
Point x	80
Height of the curve at that point	=NORM.DIST(B3,B1,B2,FALSE)

NORM.DIST is the normal distribution function. NORM.DIST(x, mean, standard_dev, TRUE) returns the probability of getting x or less.
The probability of a range is "the probability of the upper bound or less" minus "the probability of the lower bound or less".
If you set the last argument to FALSE, you get the height of the curve at that point (the value of the probability density function) instead of a probability.
After pasting, column B holds your inputs and the formula in the last row calculates automatically. The first table shows about 0.70 (about 70%), and the second shows about 0.10 (about 10%, that is, the top 10% or so).

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 probability of a range

Mean μ	62
Standard dev. σ	14
Lower bound	50
Upper bound	80
Probability between the bounds	=NORMDIST(B4,B1,B2,TRUE)-NORMDIST(B3,B1,B2,TRUE)

Table to find the probability of a value or more (top what percent)

Mean μ	62
Standard dev. σ	14
Value (find the share at or above it)	80
Probability of that value or more	=1-NORMDIST(B3,B1,B2,TRUE)

Table to find the height of the curve at a point (probability density)

Mean μ	62
Standard dev. σ	14
Point x	80
Height of the curve at that point	=NORMDIST(B3,B1,B2,FALSE)

In Google Sheets, use the NORMDIST function (it works the same as NORM.DIST in Excel).
NORMDIST(x, mean, standard_deviation, TRUE) returns the probability of getting x or less, so the upper bound's value minus the lower bound's value is the probability of the range.
If you set the last argument to FALSE, you get the height of the curve at that point (the probability density) instead of a probability.

How to calculate it in Python

from statistics import NormalDist

mean = 62               # mean
standard_deviation = 14 # standard deviation
lower_bound = 50        # lower bound of the range
upper_bound = 80        # upper bound of the range

distribution = NormalDist(mean, standard_deviation)
p_between = distribution.cdf(upper_bound) - distribution.cdf(lower_bound)

print(f"Probability between {lower_bound} and {upper_bound}: {p_between}")

With Python 3.8 or later, the standard library's statistics module is all you need (nothing extra to install). cdf(x) returns the probability of getting x or less, so the code subtracts the lower bound's value from the upper bound's value.

How to write it in LaTeX and other math languages (copy and paste)

Probability of falling in a range (using the cumulative distribution function \(\Phi\))

P(a ≤ X ≤ b) = Φ((b − μ)/σ) − Φ((a − μ)/σ)

P(a \le X \le b) = \Phi\!\left(\frac{b-\mu}{\sigma}\right) - \Phi\!\left(\frac{a-\mu}{\sigma}\right)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>P</mi><mo>(</mo><mi>a</mi><mo>&#x2264;</mo><mi>X</mi><mo>&#x2264;</mo><mi>b</mi><mo>)</mo>
    <mo>=</mo>
    <mi>&#x3A6;</mi>
    <mrow>
      <mo>(</mo>
      <mfrac><mrow><mi>b</mi><mo>&#x2212;</mo><mi>&#x3BC;</mi></mrow><mi>&#x3C3;</mi></mfrac>
      <mo>)</mo>
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      <mo>)</mo>
    </mrow>
  </mrow>
</math>

P(a <= X <= b) = Phi((b - mu)/sigma) - Phi((a - mu)/sigma)

CDF[NormalDistribution[mu, sigma], b] - CDF[NormalDistribution[mu, sigma], a]

with(Statistics): CDF(RandomVariable(Normal(mu, sigma)), b) - CDF(RandomVariable(Normal(mu, sigma)), a);

p = normcdf(b, mu, sigma) - normcdf(a, mu, sigma);

P(a ≤ X ≤ b) = Φ((b-μ)/σ) - Φ((a-μ)/σ)

Formula for the normal curve (probability density function)

f(x) = 1/(σ√(2π)) · e^(−(x − μ)²/(2σ²))

f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^2}{2\sigma^2}}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo>
    <mo>=</mo>
    <mfrac>
      <mn>1</mn>
      <mrow><mi>&#x3C3;</mi><msqrt><mrow><mn>2</mn><mi>&#x3C0;</mi></mrow></msqrt></mrow>
    </mfrac>
    <msup>
      <mi>e</mi>
      <mrow>
        <mo>&#x2212;</mo>
        <mfrac>
          <msup><mrow><mo>(</mo><mi>x</mi><mo>&#x2212;</mo><mi>&#x3BC;</mi><mo>)</mo></mrow><mn>2</mn></msup>
          <mrow><mn>2</mn><msup><mi>&#x3C3;</mi><mn>2</mn></msup></mrow>
        </mfrac>
      </mrow>
    </msup>
  </mrow>
</math>

f(x) = 1/(sigma sqrt(2 pi)) e^(-(x - mu)^2 / (2 sigma^2))

PDF[NormalDistribution[mu, sigma], x]

with(Statistics): PDF(RandomVariable(Normal(mu, sigma)), x);

y = normpdf(x, mu, sigma);

f(x) = 1/(σ√(2π)) e^(-(x-μ)^2/(2σ^2))

How to have ChatGPT do the calculation

You are a probability 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).

The scores on a test follow a normal distribution with a mean of 62 points and a standard deviation of 14 points.
Find each of the following:
1. The probability that a score is between 50 and 80 points
2. The probability that a score is 80 points or more (the top what percent)
3. The probability that a score is 50 points or less

Use Python's statistics.NormalDist, and show the code you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula