Sigma notation is a compact way to write a long sum using the Greek capital letter Σ (sigma). In Σ4k=1k2, the index k starts at the number below sigma, stops at the number above it, and each value is put into the rule k2: 1 + 4 + 9 + 16 = 30.
In Spanish
notación sigma
Also called
summation notation, sum notation, capital sigma, Σ notation
Grade level
Algebra 2
Algebra 2PrecalculusCalculusStatisticsSATACTAP
Sigma Notation in a picture
Sigma notation packs a whole sum into one line. Plug k = 1, 2, 3, 4, 5 into 2k + 1 and add: 35.
Formal definition and notation
For integers m ≤ n and a rule ak, Σnk=mak = am + am + 1 + ⋯ + an. The variable k is the index of summation, m is the lower limit and n is the upper limit; the sum has n − m + 1 terms. The index is a dummy variable: any letter can be used without changing the value.
Symbol
Read it as
Meaning
Σ
sigma
add up the terms
Σnk=1ak
the sum of a sub k, from k = 1 to n
a1 + a2 + ⋯ + an
k (or i, j, n)
the index
the counter that steps by 1 from the lower limit to the upper limit
Capital Σ means “sum”. Lowercase σ is the standard deviation of a population. Both are the Greek letter sigma.
Sigma Notationvs.Pi product notation
Capital Π works like Σ but multiplies the terms: Π4k=1k = 24, while Σ4k=1k = 10.
Sigma Notationvs.Series
A series is the sum itself; sigma notation is one way to write it down.
Quick check: 3 questions
Question 1 of 3
What is Σ5n=2n?
Show answer and explanation
Answer: B.2 + 3 + 4 + 5 = 14. 15 starts at 1 instead of 2, 9 stops at 4, and 10 is 1 + 2 + 3 + 4.
Question 2 of 3
How many terms are in Σ10k=3 (k2 − 1)?
Show answer and explanation
Answer: D. Count k = 3, 4, …, 10: 10 − 3 + 1 = 8 terms. 7 forgets to include both ends, and 10 counts from 1.
Question 3 of 3
What is Σ20k=1 3?
Show answer and explanation
Answer: A. The constant 3 is added once for each of the 20 values of k: 3 × 20 = 60. 3 counts it once, and 630 is Σ 3k.
Sigma Notation: frequently asked questions
How do you read sigma notation?
Read Σ6k=1 (k + 2) as “the sum of k + 2 for k from 1 to 6″. The number under Σ is where the index starts, the number on top is where it stops, and the expression to the right is the rule for each term. Here the terms are 3, 4, 5, 6, 7 and 8, so the sum is 33.
Does the index have to start at 1?
No. The lower limit can be any integer, such as 0, 3 or −2. Σ4k=0 2k = 1 + 2 + 4 + 8 + 16 = 31 starts at 0. Changing the starting value changes which terms are included, so always read both limits before you expand the sum, and count n − m + 1 terms.
What does the sigma symbol mean in statistics?
Capital Σ still means “add up”. The formula for the mean, x̄ = Σ xn, says to add all the data values and divide by how many there are. When no limits are shown, it means sum over every data value. Lowercase σ is different: it stands for the population standard deviation.
Updated November 2026·Reviewed by the Effortless Math team·Standards: n/a
Cite this entry
Effortless Math Team. “Sigma Notation.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/sigma-notation/
Flashcard
TermSigma NotationΣnk=1ak
DefinitionShorthand for a sum: plug each index value from the bottom limit to the top into the rule and add. Σ4k=1k = 10.
SeriesA series is the sum of the terms of a sequence. The sequence 3, 7, 11, 15 is a list; the series 3 + 7 + 11 + 15 = 36 is what you get by adding it. A finite series stops after n terms, written Sn. An infinite series has a sum only if its partial sums approach a fixed number.Functions · Algebra 2, Precalculus, CalculusNumberA 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 2MeanThe 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, StatisticsEvaluateEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Algebra · Grade 6, Grade 7, Algebra 1DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3PopulationA 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 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 6SumA sum is the result you get when you add two or more numbers. In 8 + 5 = 13, the numbers 8 and 5 are the addends and 13 is the sum. The order does not change it: 5 + 8 is also 13. Word problems ask for a sum with words like total, in all and altogether.Operations · Grade 1, Grade 2, Grade 3SequenceA sequence is an ordered list of numbers that follows a rule, such as 3, 7, 11, 15, …. Each number is a term, and its position is 1st, 2nd, 3rd and so on. You can describe a sequence with a formula for the nth term, an, or by how each term comes from the one before it.Functions · Grade 8, Algebra 1, Algebra 2VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8IntegralAn integral adds up infinitely many tiny pieces to find a total, such as the area under a curve. The definite integral ∫baf(x) dx is the signed area between the graph of f and the x-axis from x = a to x = b. An indefinite integral is an antiderivative: ∫ 2xdx = x2 + C.Calculus · CalculusArithmetic SequenceAn arithmetic sequence is a list of numbers in which you add the same amount each time to get the next term. That fixed amount is the common difference, d. In 4, 7, 10, 13, … the common difference is 3. The nth term is an = a1 + (n − 1)d.Functions · Grade 8, Algebra 1, Algebra 2Geometric SequenceA geometric sequence is a list of numbers where each term is the term before it times the same number, called the common ratio r. In 3, 6, 12, 24, … the ratio is 2. The nth term is an = a1 · rn − 1. The terms grow when |r| > 1 and shrink when |r| < 1.Functions · Algebra 1, Algebra 2, Precalculus