Normal curveThe bell-shaped curve of a normal distribution. It is symmetric: the left half mirrors the right half. It is often called a bell curve.
MeanThe mean is the center of the curve, under its peak. The median and the mode of a normal distribution are at the same place.
Standard deviationOne step along the axis from the mean. The standard deviation sets how spread out the curve is.
Area under the curveThe whole region between the curve and the axis. Its total is 1, which stands for all of the data.
The 68-95-99.7 rule 3
About 68%The share one standard deviation or less from the mean. It is the middle band of the curve.
About 95%The share two standard deviations or less from the mean.
About 99.7%The share three standard deviations or less from the mean. That is nearly all of the data.
Same mean, different spread 2
Small standard deviationThe tall, narrow curve. Its values sit close to the mean.
Large standard deviationThe low, wide curve. Its values are spread far from the mean. Both curves have the same mean and the same total area.
How this list was compiled: Every name and share was read on 2026-10-03 in at least two of these. (1) NIST/SEMATECH e-Handbook of Statistical Methods, a United States government work: 1.3.6.6.1 Normal Distribution, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm (the mean, the median and the mode are all the location parameter; the standard deviation is the scale parameter; the standard normal distribution has mean 0 and standard deviation 1); 6.5.1 What do we mean by Normal data?, https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc51.htm (symmetric and unimodal, bell-shaped; the area under the curve equals 1; 68.27%, 95.45% and 99.73% of the population fall within 1, 2 and 3 standard deviations of the mean; z is the value less the mean, divided by the standard deviation); 1.3.6.4 Location and Scale Parameters, https://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm (a larger scale parameter stretches the graph out and lowers its maximum; a smaller one squeezes it). (2) Wikipedia, English edition: Normal distribution, https://en.wikipedia.org/wiki/Normal_distribution (informally called a bell curve; the mean is also the mode and the median; the area between the curve and the axis is one; two inflection points one standard deviation from the mean); 68–95–99.7 rule, https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule (about 68%, 95% and 99.7% of values fall within 1, 2 and 3 standard deviations of the mean; also called the empirical rule); Standard score, https://en.wikipedia.org/wiki/Standard_score (the z-score counts how many standard deviations a value is above or below the mean); Standard deviation, https://en.wikipedia.org/wiki/Standard_deviation (a low standard deviation means values close to the average, a high one values spread over a wider range). (3) Yale University, Department of Statistics, The Normal Distribution, http://www.stat.yale.edu/Courses/1997-98/101/normal.htm (a bell-shaped density curve set by its mean and standard deviation, symmetrical and centered about its mean, its spread determined by its standard deviation; about 68%, 95% and 99.7% within one, two and three standard deviations). (4) California Department of Education, California Common Core State Standards: Mathematics, https://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf (a data set's mean and standard deviation fitted to a normal distribution; areas under the normal curve). (5) Douglas Shafer and Zhiyi Zhang, Introductory Statistics, an open-license text, section 2.5 The Empirical Rule and Chebyshev's Theorem, https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Introductory_Statistics_(Shafer_and_Zhang)/02%3A_Descriptive_Statistics/2.05%3A_The_Empirical_Rule_and_Chebyshev's_Theorem (the Empirical Rule: about 68%, 95% and 99.7% of a bell-shaped data set is within 1, 2 and 3 standard deviations of the mean). Hints, detail lines and the order are our own. Licence: own work
The drawing: Drawn for this site by script from the formula of the normal density, 2026-10-03. Own work. Licence: own work
How it works
This normal distribution labeled chart marks the bell curve, the mean at its center, one standard deviation and the area under the curve. Two more charts show the 68-95-99.7 rule and how the standard deviation changes the spread.
Read the full guide
The curve
Four parts are on the first chart. The dashed line through the peak is the mean. The bracket under the axis is one standard deviation, and the numbers under the axis count standard deviations from the mean. The filled region is the area under the curve.
The rule and the spread
On the second chart each bracket spans one, two or three standard deviations to both sides of the mean: name the share of the data inside it. The third chart has two curves with one mean. The narrow curve has the small standard deviation.
The curve never touches the axis. It gets closer and closer on both sides.
Questions and answers
What is the 68-95-99.7 rule?
In a normal distribution about 68% of the values sit within a standard deviation of the mean, about 95% within two and about 99.7% within three. It is also called the empirical rule.
Where are the mean, median and mode on a normal curve?
All three are at the same place: the center of the curve, under its peak. The curve is symmetric, so half of its area lies on each side of that line.
What does the standard deviation do to a bell curve?
It sets the spread. A small standard deviation gives a tall, narrow curve with values close to the mean. A large one gives a low, wide curve. The total area under the curve stays the same.
What is a z-score on a normal curve?
The number of standard deviations a value lies above or below the mean: the value minus the mean, divided by the standard deviation. The numbers under the first chart, from −3 to 3, are z-scores.