A normal distribution is a symmetric, bell-shaped distribution of data centered at the mean, where the mean, median and mode are equal. Its spread is set by the standard deviation σ. About 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ. Heights and test scores are often close to normal.
In Spanish
distribución normal
Also called
bell curve, Gaussian distribution, normal curve, 68-95-99.7 rule
Grade level
Statistics
Algebra 2StatisticsCCSS HSS-ID.A.4SATACTAP
Normal Distribution in a picture
The empirical rule: 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three.
Formal definition and notation
A continuous random variable X is normally distributed with mean μ and standard deviationσ > 0, written X ∼ N(μ, σ), when its density curve is f(x) = 1σ√2πe−(x − μ)22σ2. The curve is symmetric about μ, has inflection points at μ ± σ, and the total area under it is 1, so areas give probabilities. The standard normal distribution has μ = 0 and σ = 1; any value converts to it with the z-scorez = x − μσ.
Symbol
Read it as
Meaning
N(μ, σ)
normal with mean mu and standard deviation sigma
names a normal distribution; some books write the varianceσ2 instead
μ
mu
the mean, at the center and peak of the curve
σ
sigma
the standard deviation, which sets the width
z
z
a z-score: how many standard deviations a value is from the mean
Key facts
The curve is symmetric, so exactly half the data are above the mean and half below.
Changing μ slides the curve left or right; a bigger σ makes it lower and wider, a smaller σ taller and narrower.
Because 95% lie within 2σ, the other 5% are split between the tails: 2.5% above μ + 2σ and 2.5% below μ − 2σ.
Values more than 3σ from the mean are rare (about 0.3% in total), so they are often flagged as unusual.
Not all data are normal. Incomes and wait times are usually skewed; check a histogram before using the normal model.
z-score and the empirical rule
z-score and the empirical rule
z = x − μσ
x
a data value
empirical rule
about 68% of data have |z| ≤ 1, 95% have |z| ≤ 2, 99.7% have |z| ≤ 3
The 68-95-99.7 percentages are rounded. For other z-values, use a z-table, a calculator or a spreadsheet (NORM.DIST).
Normal Distribution examples, step by step
Example 1: Use the 68% rule
Warm-up
Test scores are normally distributed with mean 70 and standard deviation 8. About what percent of scores are between 62 and 78?
70 − 8 = 62 and 70 + 8 = 78.
So 62 to 78 is the range within 1 standard deviation of the mean.
The standard deviation is one number that measures spread; the normal distribution is a whole shape that uses the mean and standard deviation as its two settings.
Normal Distributionvs.Uniform distribution
In a uniform distribution every value in a range is equally likely, so its graph is flat; a normal graph peaks at the mean and tapers off.
Normal Distributionvs.Skewed distribution
A skewed distribution has one long tail and its mean is pulled away from its median; in a normal distribution the mean and median are equal.
Quick check: 3 questions
Question 1 of 3
In a normal distribution, about what percent of the data lie within 1 standard deviation of the mean?
Show answer and explanation
Answer: B. The empirical rule gives about 68% within 1 standard deviation. 95% is for 2, 99.7% is for 3, and 50% is the share above the mean.
Question 2 of 3
IQ scores have mean 100 and standard deviation 15. What is the z-score of 130?
Show answer and explanation
Answer: C.z = 130 − 10015 = 2. 30 is the raw distance, 0.5 divides the wrong way, and 1.5 divides by 20.
Question 3 of 3
In a normal distribution, how does the median compare with the mean?
Show answer and explanation
Answer: A. The curve is symmetric about its center, so the mean, median and mode are all the same value. The median differs from the mean only in skewed distributions.
Normal Distribution: frequently asked questions
What is the 68-95-99.7 rule?
It is a shortcut for normal distributions, also called the empirical rule. About 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. With mean 50 and standard deviation 5, about 95% of values are between 40 and 60.
Why is it called a bell curve?
Because the graph of a normal distribution looks like the outline of a bell: highest in the middle at the mean and falling off smoothly and evenly on both sides. The name describes the shape only. Many bell-shaped data sets are close to normal, but not every bell-looking graph is exactly normal.
What is a standard normal distribution?
It is the normal distribution with mean 0 and standard deviation 1. Any normal value x can be moved onto it with the z-score z = x − μσ, so one table of areas, the z-table, works for every normal distribution. A z-score of 1.5 means 1.5 standard deviations above the mean.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSS-ID.A.4
Cite this entry
Effortless Math Team. “Normal Distribution.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/normal-distribution/
Flashcard
TermNormal Distributionthe bell curve
DefinitionA symmetric bell-shaped distribution set by its mean and standard deviation. About 68%, 95% and 99.7% of data lie within 1, 2 and 3 SDs.
MeanThe mean is the sum of the values divided by the number of values. It is the arithmetic average: add every value and divide by how many there are. For 3, 7, 5 and 9, the mean is 24 ÷ 4 = 6. The mean is the balance point of the data, so one outlier can pull it far.Statistics · Grade 5, Grade 6, Grade 7Standard DeviationStandard deviation is a number that measures how spread out data values are from their mean: roughly, the typical distance of a value from the mean. A small standard deviation means the values cluster near the mean; a large one means they are spread out. It is the square root of the variance.Statistics · Algebra 1, StatisticsDensityDensity is the amount of mass packed into each unit of volume, found by dividing mass by volume: D = mV. Its units combine the two, such as grams per cubic centimeter. A 40 g block with a volume of 16 cm³ has D = 40 ÷ 16 = 2.5 g/cm³. Population density works the same way: people per unit of area.Problem Solving · Grade 8, Algebra 1, GeometryZ-ScoreA z-score tells how many standard deviations a value is above or below the mean. Find it with z = x − μσ: subtract the mean, then divide by the standard deviation. A positive z-score is above the mean, a negative one is below, and z = 0 is exactly at the mean.Statistics · Algebra 2, StatisticsVarianceVariance is a measure of spread: the average of the squared distances of the data values from their mean. Find each value’s distance from the mean, square it, add the squares, and divide by n for a population (or n − 1 for a sample). Larger variance means more spread. Its square root is the standard deviation.Statistics · Algebra 1, StatisticsDataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6HistogramA histogram uses bars to show how numerical data are grouped into intervals. The data are split into equal-width intervals, called bins, and each bar’s height is the count of values in that interval. The bars touch, because the intervals are continuous, which is what distinguishes a histogram from a bar graph.Statistics · Grade 6, Grade 7, Grade 8Empirical RuleThe empirical rule says that in a bell-shaped (normal) distribution about 68% of the data fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. It is also called the 68-95-99.7 rule, and it only works for data that are roughly symmetric and bell-shaped.Statistics · Algebra 2, StatisticsHeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2MedianThe median is the middle value when data are listed in order. If there is an even number of values, the median is the average of the two middle values. Half the data lie at or below the median and half at or above it, and outliers barely move it.Statistics · Grade 5, Grade 6, Grade 7SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2ProbabilityProbability measures how likely an event is, from 0 for impossible to 1 for certain. For equally likely outcomes it is the number of favorable outcomes divided by the total number of outcomes: for a fair coin, the probability of heads is 12. Probabilities can be written as fractions, decimals or percents.Probability · Grade 6, Grade 7, Grade 8OutlierAn outlier is a data value that is far away from most of the other values in a set. In 4, 5, 6, 7, 50, the value 50 is an outlier. A common test flags any value more than 1.5 times the interquartile range below the first quartile or above the third. Outliers pull the mean strongly.Statistics · Grade 6, Grade 7, Grade 8Box PlotA box plot displays a data set through five numbers: minimum, first quartile, median, third quartile, and maximum. The box holds the middle half of the data, a line inside it marks the median, and whiskers reach out to the smallest and largest values. It makes spread easy to compare between sets.Statistics · Grade 6, Grade 7, Grade 8