Standard 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.
Both sets have mean 10, but set B’s values sit farther from the mean, so its standard deviation is about three times larger. The shaded band is the mean plus or minus 1 standard deviation.
Formal definition and notation
For a population of N values with mean μ, the standard deviation is σ = √Σ (x − μ)2N. For a sample of n values with mean x, the sample standard deviation is s = √Σ (x − x)2n − 1. It is always 0 or positive, is 0 only when every value is equal, and has the same units as the data.
Standard deviation has the same units as the data. If heights are in inches, the standard deviation is in inches; the variance is in square inches.
Adding the same number to every value does not change the standard deviation. Multiplying every value by k multiplies it by |k|.
Like the mean, it is pulled up a lot by an outlier. For skewed data or data with outliers, the median and interquartile range describe the data better.
For data shaped like a bell (a normal distribution), about 68 percent of values lie within 1 standard deviation of the mean and about 95 percent within 2.
Dividing by n − 1 for a sample makes the sample standard deviation a little larger, which makes up for a sample usually being less spread out than the whole population.
Standard deviation formula
Standard deviation formula
σ = √Σ (x − μ)2Ns = √Σ (x − x)2n − 1
x
each data value
μ, x
the mean of the population or the sample
N, n
the number of values
x − μ
the deviation: how far one value is from the mean
Steps: (1) find the mean; (2) subtract the mean from each value; (3) square each deviation; (4) add the squares; (5) divide by N for a population or n − 1 for a sample; (6) take the square root.
Try it: measure the spread
Interactive
Edit the data, drag a dot, or press Add an outlier. The blue band covers one standard deviation on each side of the mean, and the readout shows each step of the calculation for a population and for a sample.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Standard Deviation examples, step by step
Example 1: Population standard deviation, step by step
Warm-up
Find the standard deviation of the population 2, 4, 4, 4, 5, 5, 7, 9.
Mean: 2 + 4 + 4 + 4 + 5 + 5 + 7 + 98 = 408 = 5.
Deviations from 5: −3, −1, −1, −1, 0, 0, 2, 4.
Squares: 9, 1, 1, 1, 0, 0, 4, 16. Their sum is 32.
Divide by N = 8: 328 = 4. This is the variance.
Square root: σ = √4 = 2.
Answer: The standard deviation is 2.
Example 2: Sample standard deviation
Core
A sample of 5 plants grew 6, 8, 10, 12 and 14 centimeters. Find the sample standard deviation.
Mean: 505 = 10 centimeters.
Squared deviations: 16, 4, 0, 4, 16. Sum: 40.
It is a sample, so divide by n − 1 = 4: 404 = 10.
Square root: s = √10 ≈ 3.16.
Answer: The sample standard deviation is √10, about 3.16 centimeters.
Example 3: What happens when every value changes?
Test-style
The quiz scores in a class have a mean of 72 and a standard deviation of 6. The teacher adds 5 points to every score. What are the new mean and standard deviation?
Adding 5 to every score moves the whole data set up by 5, so the mean becomes 72 + 5 = 77.
Every score moves together, so each distance from the mean stays the same.
Same distances give the same standard deviation: still 6.
Answer: The new mean is 77 and the standard deviation stays 6.
Common mistakes
✗ MistakeFor 2, 4, 4, 4, 5, 5, 7, 9: stopping at the variance and reporting 4.
✓ InsteadTake the square root: σ = √4 = 2.
The variance is in squared units. The square root brings the answer back to the units of the data, which is what makes the standard deviation easy to read.
✗ MistakeAdding the deviations: −3 − 1 − 1 − 1 + 0 + 0 + 2 + 4 = 0, so the spread is 0.
Variance is the average squared distance from the mean; standard deviation is its square root. For 2, 4, 4, 4, 5, 5, 7, 9 the variance is 4 and the standard deviation is 2.
The range uses only the largest and smallest values (9 − 2 = 7). Standard deviation uses every value’s distance from the mean (2).
Standard Deviationvs.Mean absolute deviation
Mean absolute deviation averages the distances without squaring: for 2, 4, 4, 4, 5, 5, 7, 9 it is 12 ÷ 8 = 1.5, smaller than the standard deviation of 2.
Quick check: 3 questions
Question 1 of 3
Which data set has the largest standard deviation?
Show answer and explanation
Answer: D. All four sets have mean 10. The values 1 and 19 are farthest from the mean, so that set is most spread out. The first set has standard deviation 0 because every value is the same.
Question 2 of 3
The variance of a population is 49 square meters. What is its standard deviation?
Show answer and explanation
Answer: A. Standard deviation is the square root of the variance: √49 = 7 meters. 24.5 halves instead of taking a root, 49 forgets the root and 2401 squares instead.
Question 3 of 3
Every value in a data set is multiplied by 3. The old standard deviation was 4. What is the new one?
Show answer and explanation
Answer: C. Multiplying every value by 3 stretches every distance from the mean by 3, so the standard deviation becomes 3 × 4 = 12. 4 is the rule for adding a constant, 7 adds 3, and 36 multiplies by 9, which is how the variance changes, not the standard deviation.
Standard Deviation: frequently asked questions
What does a standard deviation tell you?
It tells you how much the values typically differ from the mean. If test scores have a mean of 80 and a standard deviation of 3, most scores are close to 80, roughly from 77 to 83. With a standard deviation of 15, the scores are much more spread out. A standard deviation of 0 means every value is identical.
When do you divide by n − 1 instead of N?
Divide by N when your data are the whole population, such as every student in one class when you only care about that class. Divide by n − 1 when your data are a sample used to estimate a larger population, such as 50 voters chosen from a city. Calculators show both: σ (population) and s (sample).
Can standard deviation be negative?
No. Each squared deviation is 0 or positive, so their average is 0 or positive, and the square root of that is never negative. The smallest possible standard deviation is 0, which happens only when every value in the data set is the same.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSS-ID.A.2, CCSS HSS-ID.A.4
Cite this entry
Effortless Math Team. “Standard Deviation.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/standard-deviation/
Flashcard
TermStandard Deviationσ = √Σ (x − μ)2N
DefinitionHow spread out data are from the mean. 2, 4, 4, 4, 5, 5, 7, 9 has mean 5 and standard deviation 2.
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 7PopulationA population in statistics is the whole group of people, animals or objects you want to learn about, such as every student in a school or every bulb a factory makes. Measuring all of them is often too costly, so you study a sample, a part of the population, and use it to draw conclusions about the whole group.Statistics · Grade 7, Statistics, Algebra 1SampleA sample in statistics is a part of a population that is actually observed or measured, such as 60 students surveyed out of a school of 900. You use the sample to estimate facts about the whole population. A random sample, where every member has the same chance to be chosen, tends to represent the population fairly.Statistics · Grade 7, Statistics, Algebra 1DataData 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 6VarianceVariance 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, StatisticsSquare RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1HeightHeight 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 6InchAn inch is a US customary unit of length equal to one twelfth of a foot, about the width of an adult thumb. Its abbreviation is in. Rulers split each inch into halves, quarters, eighths and sixteenths. One inch is exactly 2.54 centimeters, and 12 inches make 1 foot.Measurement · Grade 2, Grade 3, Grade 4SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3NumberA 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 2OutlierAn 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 8MedianThe 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 7Interquartile RangeThe interquartile range (IQR) is the spread of the middle half of a data set: the third quartile minus the first quartile, IQR = Q3 − Q1. Because it ignores the lowest and highest quarters of the data, outliers barely change it. In a box plot it is the length of the box.Statistics · Grade 6, Grade 7, Algebra 1Normal DistributionA 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.Statistics · Algebra 2, StatisticsCentimeterA centimeter is a metric unit of length equal to one hundredth of a meter, about the width of a fingernail. Its symbol is cm. There are 100 centimeters in 1 meter and 10 millimeters in 1 centimeter, and 1 inch equals 2.54 centimeters.Measurement · Grade 3, Grade 4, Grade 5RangeThe range of a data set is the difference between the greatest and least values: maximum minus minimum. For 2, 5, 9, the range is 9 − 2 = 7. Range is a simple measure of spread: it tells how far apart the extreme values are, though one outlier can make it misleading about the rest of the data.Statistics · Grade 5, Grade 6, Grade 7Box 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 8StatisticsStatistics is the branch of math that collects, organizes, analyzes and interprets data to answer questions. Descriptive statistics summarize data with graphs and numbers such as the mean and median. Inferential statistics use a sample to draw conclusions about a whole population, with a stated amount of uncertainty.Statistics · Grade 6, Grade 7, Grade 8