Basic Algebra Review

12 min read

Quick Reference

The mean $\bar{x}$, the standard deviation $s$, and the standardized value $z$ all rest on substituting numbers into a formula and carrying out the operations in the correct order. A reviewer with a physical copy of the 9th edition should fill the page ranges for Sections 1.3 and 1.4 and flip pages_verified to true.

Try This First

Before any rule or formula, work this one by hand with a pencil. Take your time. There is no rush, and a wrong turn here is common and worth understanding.

You are handed this expression:

$$3 + 4 \times 2^2$$

Write down the single number you think it equals. Then read the next sentence.

Two readers often disagree here. One reader gets 196, by going strictly left to right: $3 + 4 = 7$, then $7 \times 2 = 14$, then $14^2 = 196$. Another reader gets 19. Only one of those follows the agreed order that every statistics formula assumes. Keep both numbers in mind. By the end of the first section you will know which one is correct and exactly why the other reader went wrong.

Where this shows up

Every number you will report in this course (a mean, a spread, a $z$-score) is the output of a small machine. You feed in data values, the machine applies a fixed recipe of operations, and a single number comes back out. If two people feed in the same data but apply the operations in a different order, they get different answers, and at most one of them is right. The rules in this node are the shared recipe so that everyone who computes a standard deviation from the same data gets the same standard deviation.

Think of a formula as a process: it takes inputs (the data, the sample size $n$) and returns an output (a summary number). The work of this node is making that process reliable, so that the statistics built on top of it are trustworthy.

Prerequisite Hub

This is a foundation node. Nothing in the course comes before it.

Builds on

Unlocks (each of these uses substitution and order of operations directly)

Cross-course prerequisites: none.

graph LR
    subgraph ThisSkill["This Skill"]
        C["Basic Algebra<br/>Review"]
    end

    subgraph Unlocks
        D["Sigma (Summation)<br/>Notation"]
        E["Mean and<br/>Median"]
        F["Variance<br/>and SD"]
        G["Standardizing<br/>and z-scores"]
    end

    C --> D
    C --> E
    C --> F
    C --> G

    style C fill:#d1fae5,stroke:#a565f0,stroke-width:3px

Quick Reference

Order of Operations at a Glance

A common memory aid is PEMDAS. Read it as four tiers, not six steps: multiplication and division share one tier (left to right), and so do addition and subtraction.

Key Concepts

The Order of Operations

The order of operations is the agreed sequence for carrying out the operations in an expression so that every reader gets the same value.

Definition (Evaluating an algebraic expression). Evaluating an expression means substituting given numerical values for the variables and carrying out the operations in the correct order: parentheses, exponents and roots, multiplication and division, then addition and subtraction. (Source: foundational algebra underlying the computational formulas of IPS 9th edition, Sections 1.3 and 1.4 (mean, standard deviation, and standardizing).)

Return to the opener, $3 + 4 \times 2^2$. Work it by tier:

1. Parentheses: none.
2. Exponents: $2^2 = 4$, giving $3 + 4 \times 4$.
3. Multiplication and division: $4 \times 4 = 16$, giving $3 + 16$.
4. Addition and subtraction: $3 + 16 = 19$.

The value is $19$. The reader who got $196$ added before multiplying and squared last, which reverses the agreed order. Notice that the disagreement was never about arithmetic facts. Both readers can add and multiply. The disagreement was entirely about order.

One way to see this, and another way. One way to keep the tiers straight is the PEMDAS memory aid. Another way is to picture the expression as nested boxes: the most tightly bound operations (exponents, then products) form inner boxes that must be closed before the looser outer boxes (sums) can be added up. Which picture do you prefer, and why? Both lead to the same value $19$.

Square Roots

Definition (Square root). The square root of a nonnegative number $a$, written $\sqrt{a}$, is the nonnegative number whose square is $a$. (Source: foundational algebra underlying the computational formulas of IPS 9th edition, Sections 1.3 and 1.4.)

In symbols, if $a \geq 0$, then $\sqrt{a} = b$ means $b \geq 0$ and $b^2 = a$.

Two facts follow directly from the definition and matter for the standard deviation:

• $\sqrt{a}$ is never negative. The square root of $9$ is $3$, not $-3$, even though $(-3)^2 = 9$ is also true. The definition picks the nonnegative one.
• The bar over the radical is a grouping symbol. In $\sqrt{16 + 9}$, you add first, then take the root: $\sqrt{25} = 5$. This is not the same as $\sqrt{16} + \sqrt{9} = 4 + 3 = 7$.

The standard deviation $s$ is a square root of a sum divided by a count. That square root sits at the very end of the recipe, after all the squaring, summing, and dividing are done.

Substituting Into a Formula

Substitution is the act of replacing each variable with its given numerical value and then evaluating.

The three formulas this node supports, stated as in IPS 9th edition Chapter 1, are:

• The mean: $\bar{x} = \dfrac{1}{n}\sum_{i=1}^{n} x_i$
• The (sample) standard deviation: $s = \sqrt{\dfrac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2}$
• The standardized value: $z = \dfrac{x - \bar{x}}{s}$

(Formulas as stated in IPS 9e, Chapter 1, Sections 1.3 and 1.4.)

Each one is a process. Read $z = \frac{x - \bar{x}}{s}$ left to right as a recipe: subtract the mean from the value (numerator, grouped by the fraction bar), then divide by the standard deviation. The fraction bar groups the whole numerator, so the subtraction happens before the division even though no parentheses are printed.

Worked Examples

Worked Example 1: Order of operations end to end

Evaluate $20 - 2 \times (3 + 1)^2 \div 4$.

Predict first. Before computing, predict whether the answer is closer to $20$ or closer to $0$. The expression starts at $20$ and subtracts a positive quantity, so a value somewhere below $20$ is reasonable. Hold that prediction.

Now compute, by tier.

1. Parentheses: $3 + 1 = 4$, giving $20 - 2 \times 4^2 \div 4$.
2. Exponents: $4^2 = 16$, giving $20 - 2 \times 16 \div 4$.
3. Multiplication and division, left to right: $2 \times 16 = 32$, then $32 \div 4 = 8$, giving $20 - 8$.
4. Addition and subtraction: $20 - 8 = 12$.

Compare to the prediction. The result $12$ is below $20$, matching the prediction. A reader who divided before multiplying would compute $16 \div 4 = 4$ first, then $2 \times 4 = 8$, and land on the same $8$ here only by coincidence of these numbers. The reliable rule is left to right within the multiply-divide tier.

Worked Example 2: A square root over a sum

Evaluate $s = \sqrt{\dfrac{36 + 4 + 0 + 16 + 4}{5 - 1}}$.

This is the exact shape of a standard deviation: a square root of a sum divided by a count.

Predict first. The five numbers on top sum to something around $60$, divided by $4$ is around $15$, and the square root of $15$ is between $3$ and $4$. Predict an answer near $4$.

Now compute, inside out.

1. The fraction bar groups numerator and denominator. Numerator first: $36 + 4 + 0 + 16 + 4 = 60$.
2. Denominator: $5 - 1 = 4$.
3. Divide: $\frac{60}{4} = 15$.
4. Square root last: $\sqrt{15} \approx 3.873$.

Compare to the prediction. The result $\approx 3.87$ sits between $3$ and $4$, matching the prediction. Taking the square root before the division (computing $\sqrt{60}$ and $\sqrt{4}$ separately) would be a different and wrong value, because the bar over the radical groups the entire fraction underneath it.

Worked Example 3: Substituting into the z-score formula

A data value is $x = 78$. The mean is $\bar{x} = 70$ and the standard deviation is $s = 4$. Compute the standardized value $z = \dfrac{x - \bar{x}}{s}$.

Predict first. The value $78$ is above the mean $70$, so $z$ should be positive. The gap is $8$, which is two times the standard deviation $4$, so predict $z$ near $2$.

Now substitute and evaluate.

1. Replace each variable: $z = \dfrac{78 - 70}{4}$.
2. Numerator (grouped by the bar): $78 - 70 = 8$.
3. Divide: $\frac{8}{4} = 2$.

So $z = 2$. Compare to the prediction. The value is positive and equals $2$, exactly the prediction. In words, $78$ is two standard deviations above the mean. That meaning, not just the number, is what the standardized value reports back into the situation.

Common Misconceptions

Practice Problems

Ask Why

How would you convince a classmate that $\sqrt{16 + 9} \neq \sqrt{16} + \sqrt{9}$ without a calculator? One short argument: $\sqrt{16} + \sqrt{9} = 4 + 3 = 7$, and $7^2 = 49$, not $25$. So $7$ cannot be the square root of $25$. The grouping bar changes the answer, and squaring the candidate is a quick way to test a square root claim. Try writing that argument in your own words.

Mastery Checklist

Novice (Level 1-2):

• I can name the four tiers of the order of operations and explain why multiply-divide share one tier
• I can evaluate an expression with parentheses, exponents, and the four basic operations
• I can state the definition of a square root and explain why it is never negative

Competent (Level 3-4):

• I can evaluate a square root taken over a sum divided by a count
• I can substitute given values into the mean formula and get the correct number
• I can flatten a fraction onto one line with the correct parentheses

Proficient (Level 5):

• I can compute a full sample standard deviation by hand, in the correct order
• I can spot when a one-line formula is missing parentheses and predict the wrong answer it would give
• I can explain in words what a standardized value reports back into the situation

Connections

Looking ahead:

• Sigma (Summation) Notation packages the repeated addition in the mean and standard deviation into the symbol $\sum$
• Measures of Center: Mean and Median builds the mean $\bar{x}$ directly on substitution and division
• Measures of Spread: Variance and SD builds $s$ on squaring, summing, dividing, and a final square root
• Standardizing and z-scores builds $z$ on subtract-then-divide

Real-world connection: every reported statistic, from a class average to a national unemployment rate, is the output of substituting data into a formula and evaluating in a fixed order. Shared rules are what let two analysts reproduce the same numbers from the same data.

Resources

• Textbook (where these operations are applied): Moore, McCabe, and Craig, Introduction to the Practice of Statistics, 9th edition, Chapter 1, Sections 1.3 (Describing Distributions with Numbers) and 1.4 (Density Curves and Normal Distributions). Page numbers pending faculty verification.
• OpenStax (open companion): Introductory Statistics 2e, Chapter 2 Formula Review, where the mean and standard deviation operations are applied to data.

Last updated: 2026-06-16