Sigma (Summation) Notation
Quick Reference
| Field | Value |
|---|---|
| Textbook | Introduction to the Practice of Statistics (IPS), 9th edition. Moore, McCabe, and Craig. |
| Chapter | 0 (prerequisite review; supports IPS Chapter 1) |
| Section | Prerequisite review (not a numbered IPS section) |
| Subsection | The summation symbol and indexed data $x_1, x_2, \ldots, x_n$ |
| Pages | Page numbers pending faculty verification. |
| Course | MATH380 |
The summation symbol is not given its own numbered section in IPS, yet IPS Section 1.3 relies on it to define the sample mean and the sample variance. A student who reads those definitions without a firm grip on the sigma symbol will read past the formulas without seeing what they say.
Before You Start: Prerequisite Check
📋 Can you do these? (Click to reveal self-test)
Test yourself on these prerequisite skills:
Adding a short list: Add the numbers $4, 7, 2, 9$.
Check
$4 + 7 + 2 + 9 = 22$.
Substituting into an expression: If $x = 5$, what is $x - 3$? If $x = 2$, what is $x - 3$?
Check
When $x = 5$, $x - 3 = 2$. When $x = 2$, $x - 3 = -1$.
Reading a subscript: A data list is written $x_1, x_2, x_3, x_4$. Which number is $x_3$ if the list is $10, 20, 30, 40$?
Check
$x_3 = 30$. The subscript names the position in the list, not the value.
If you struggled:
- Review Basic Algebra Review for adding lists and substituting values.
Try This First
Before any formula, try a small case by hand.
A study records the number of hours four students slept last night: $7, 5, 8, 6$.
- Write down the total number of hours for all four students.
- Now suppose a fifth student is added who slept $4$ hours. Write the new total.
- Predict, in one sentence, what symbol a statistics book might use so that it does not have to keep rewriting “add up all the values” every time the list grows.
Show what you should have found
- $7 + 5 + 8 + 6 = 26$ hours.
- $7 + 5 + 8 + 6 + 4 = 30$ hours.
- A single symbol that means “add up every value in the list” would let the book write one short expression that works whether the list has $4$ values or $4000$. That symbol is the capital Greek letter sigma.
The point of the symbol is economy. Writing $x_1 + x_2 + \cdots + x_n$ over and over is slow and hides the pattern. One symbol carries the instruction “add all of them” no matter how long the list is.
Intuition: A Symbol That Means “Add Them All Up”
Picture a list of measurements, one per subject in a study: heights, test scores, reaction times. The list has a length, call it $n$, and the entries are named in order:
$$x_1, \; x_2, \; x_3, \; \ldots, \; x_n.$$
The subscript is an address, not a value. The symbol $x_1$ means “the first measurement”, whatever number that happens to be. The symbol $x_i$ means “the measurement at position $i$”, where $i$ is a placeholder that will walk through the positions $1, 2, 3$, and so on.
Summation notation is a process that takes this list and gives back a single number: the total. The capital sigma symbol, $\sum$, is the instruction “run the index through every position and add the results”. Three representations of the same idea sit side by side:
| Representation | What it looks like |
|---|---|
| Worded instruction | Add the first value, then the second, then the third, all the way to the last. |
| Spelled-out sum | $x_1 + x_2 + x_3 + \cdots + x_n$ |
| Sigma notation | $\displaystyle\sum_{i=1}^{n} x_i$ |
Read the sigma expression out loud as: “the sum, as $i$ goes from $1$ to $n$, of $x_i$”. The bottom of the sigma says where the index starts, the top says where it stops, and the expression to the right of the sigma says what to add at each step.
Prerequisite Hub
Builds on:
| Skill | Why it is needed |
|---|---|
basic-algebra-review |
Adding a list of numbers and substituting values into an expression are the two mechanical steps inside every sum. |
Unlocks:
| Skill | How this feeds it |
|---|---|
measures-of-center-mean-median |
The sample mean is defined as $\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$. The mean is unreadable without the sigma. |
measures-of-spread-variance-sd |
The sample variance is defined as $s^2 = \frac{1}{n-1}\sum_{i=1}^{n} (x_i - \bar{x})^2$, a sum of squared deviations. |
Cross-course prerequisites: none recorded for this node.
The Official Definition
Summation notation. For a list of $n$ observations written $x_1, x_2, \ldots, x_n$, the symbol $\sum x_i$ (capital Greek sigma) denotes the sum $x_1 + x_2 + \cdots + x_n$. The index $i$ runs over the observations. IPS uses exactly this notation to define the mean as $\bar{x} = \frac{1}{n}\sum x_i$ and the variance as $s^2 = \frac{1}{n-1}\sum (x_i - \bar{x})^2$.
Cite: Notation as used in IPS 9th edition Section 1.3 to define the sample mean and sample variance (Moore, McCabe, Craig, IPS 9e, Ch. 1, Sec. 1.3).
Reading Every Part of the Symbol
The fully written form names the start and the stop explicitly:
$$\sum_{i=1}^{n} x_i.$$
| Part | Name | What it says |
|---|---|---|
| $\sum$ | Sigma (capital Greek S) | Add the terms produced below. |
| $i$ | Index of summation | A counter that steps through whole numbers. |
| $i = 1$ (bottom) | Lower limit | Start the counter at $1$. |
| $n$ (top) | Upper limit | Stop the counter at $n$, the last position. |
| $x_i$ | Summand | The quantity to compute at each value of $i$. |
When the context already makes the range clear (as IPS often does when it sums over all observations in a sample), the limits are dropped and the book writes simply $\sum x_i$. Both forms mean the same total.
No key theorems are attached to this prerequisite-review node. The work here is reading and expanding the notation correctly, which the worked examples below practice directly.
Worked Examples
Example 1: Expand and Evaluate a Basic Sum
A data list is $x_1 = 3$, $x_2 = 7$, $x_3 = 2$, $x_4 = 8$. Compute $\sum_{i=1}^{4} x_i$.
Predict first. All four values are positive and none is large, so the total should land somewhere in the high teens or low twenties. Hold that estimate.
Expand. The index starts at $1$ and stops at $4$, so write one term per position:
$$\sum_{i=1}^{4} x_i = x_1 + x_2 + x_3 + x_4.$$
Substitute the values.
$$x_1 + x_2 + x_3 + x_4 = 3 + 7 + 2 + 8.$$
Add.
$$3 + 7 + 2 + 8 = 20.$$
Check against the prediction. The result, $20$, sits in the low-twenties range that was predicted. The estimate and the computation agree.
Example 2: A Summand That Is an Expression, Not Just $x_i$
Using the same list $x_1 = 3$, $x_2 = 7$, $x_3 = 2$, $x_4 = 8$, compute $\sum_{i=1}^{4} (x_i - 5)$.
Predict first. Subtracting $5$ from each of four values removes $20$ in total from the sum found in Example 1, which was $20$. So the answer should be near zero.
Expand. The summand is now $x_i - 5$, so each term subtracts $5$:
$$\sum_{i=1}^{4} (x_i - 5) = (x_1 - 5) + (x_2 - 5) + (x_3 - 5) + (x_4 - 5).$$
Substitute.
$$(3 - 5) + (7 - 5) + (2 - 5) + (8 - 5) = (-2) + (2) + (-3) + (3).$$
Add.
$$(-2) + 2 + (-3) + 3 = 0.$$
Check. The prediction said “near zero”, and the sum is exactly $0$. This previews a fact used later: the sum of the deviations of data from a well-chosen center can vanish.
Example 3: A Sum of Squares (Preview of Variance)
Using the list $x_1 = 3$, $x_2 = 7$, $x_3 = 2$, $x_4 = 8$ with mean $\bar{x} = 5$ (from Example 1, since $\frac{20}{4} = 5$), compute $\sum_{i=1}^{4} (x_i - \bar{x})^2$.
Predict first. Squaring makes every term nonnegative, so unlike Example 2 the total cannot be zero. It will be a positive number that measures how far the values spread from $5$.
Expand.
$$\sum_{i=1}^{4} (x_i - \bar{x})^2 = (x_1 - 5)^2 + (x_2 - 5)^2 + (x_3 - 5)^2 + (x_4 - 5)^2.$$
Substitute and square each term.
$$(3 - 5)^2 + (7 - 5)^2 + (2 - 5)^2 + (8 - 5)^2 = (-2)^2 + (2)^2 + (-3)^2 + (3)^2.$$
$$= 4 + 4 + 9 + 9.$$
Add.
$$4 + 4 + 9 + 9 = 26.$$
Check. The result, $26$, is positive, as predicted. Dividing by $n - 1 = 3$ would give the sample variance $s^2 = \frac{26}{3} \approx 8.67$, which is exactly the formula the variance node uses. The sum of squares is the engine inside that formula.
Example 4: The Index Does Not Have to Start at 1
Compute $\sum_{i=2}^{4} x_i$ for the list $x_1 = 3$, $x_2 = 7$, $x_3 = 2$, $x_4 = 8$.
Predict first. This sum skips $x_1 = 3$, so the answer should be $3$ less than the full sum of $20$ from Example 1, near $17$.
Expand. The lower limit is $2$, so the first term is $x_2$, and the index stops at $4$:
$$\sum_{i=2}^{4} x_i = x_2 + x_3 + x_4 = 7 + 2 + 8 = 17.$$
Check. The result, $17$, equals $20 - 3$, matching the prediction. The lower limit controls where the addition begins.
Common Misconceptions
the index $i$ is a data value rather than a position counter. Some readers see $\sum_{i=1}^{n} x_i$ and try to plug the number $1$ into $x$, computing “$x$ times $1$” or similar. The symbol $i$ is an address that walks through positions $1, 2, 3, \ldots, n$; at each position it selects the value stored there. For the list $x_1 = 3, x_2 = 7$, the term at $i = 1$ is the value $3$, not the number $1$. Reading $i$ as a value confuses the input (the position) with the output (the measurement at that position).
the bottom and top numbers tell you what to add, instead of where to start and stop. A reader might see $\sum_{i=1}^{4} x_i$ and add $1 + 4 = 5$. The $1$ and the $4$ are the range of positions, not terms in the sum. The terms come from the summand $x_i$ evaluated at each position. To check yourself, always expand the sum into the spelled-out form $x_1 + x_2 + x_3 + x_4$ before computing.
$\sum (x_i - \bar{x})$ and $\sum (x_i - \bar{x})^2$ are the same up to a sign. Example 2 gives a sum of deviations equal to $0$, while Example 3 gives a sum of squared deviations equal to $26$. Squaring before summing changes the result entirely, because squaring removes the cancellation between positive and negative deviations. The square is the whole reason the variance measures spread instead of collapsing to zero.
Practice Problems
For the list $x_1 = 10$, $x_2 = 20$, $x_3 = 30$, write $\sum_{i=1}^{3} x_i$ in spelled-out form and evaluate it.
For the list $x_1 = 5$, $x_2 = 1$, $x_3 = 9$, $x_4 = 4$, $x_5 = 6$, compute $\sum_{i=3}^{5} x_i$.
For the list $x_1 = 2$, $x_2 = 4$, $x_3 = 6$, compute $\sum_{i=1}^{3} 2x_i$. Then compute $2\sum_{i=1}^{3} x_i$. What do you notice?
A list is $x_1 = 8$, $x_2 = 12$, $x_3 = 10$ with mean $\bar{x} = 10$. Predict the value of $\sum_{i=1}^{3} (x_i - \bar{x})$ before computing, then compute it.
A sample of reaction times (in milliseconds) is $x_1 = 220$, $x_2 = 260$, $x_3 = 240$, $x_4 = 280$. Use the IPS definition $\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$ to compute the mean, then compute the sum of squared deviations $\sum_{i=1}^{4} (x_i - \bar{x})^2$.
Mastery Checklist
Novice (Level 1-2):
Competent (Level 3-4):
Proficient (Level 5):
Connections
Looking back:
- Basic Algebra Review supplies the addition and substitution that every sum is built from.
Looking ahead:
- Measures of Center (Mean, Median): the sample mean $\bar{x} = \frac{1}{n}\sum x_i$ is a sum divided by the count.
- Measures of Spread (Variance, SD): the sample variance $s^2 = \frac{1}{n-1}\sum (x_i - \bar{x})^2$ is a sum of squared deviations.
Real-world connections:
- Any total computed from a data table (total sales, total rainfall, total votes) is a sum written compactly with sigma.
- Spreadsheet functions such as a column total carry out exactly the operation that $\sum x_i$ describes.
Resources
| Resource | What it offers |
|---|---|
| Introduction to the Practice of Statistics (IPS) 9e, Section 1.3 (Moore, McCabe, Craig) | The mean and variance defined with sigma notation; the primary text for this node. |
| OpenStax Introductory Statistics 2e, Ch. 2 Formula Review | Open-access formula review listing the sample mean and standard deviation formulas in sigma notation. |
| Previous | Up | Next |
|---|---|---|
| Basic Algebra Review | Skills Index | Measures of Center (Mean, Median) |
Last updated: 2026-06-16