1.2 Significant Figures
PRE-HEALTH PREP
Scientific Notation & Significant Figures
General Chemistry 1.2
Lesson Overview
Scientific notation provides a standardized method for expressing very large and very small numbers as a coefficient multiplied by a power of ten, eliminating ambiguity about which digits are meaningful. Significant figures follow specific rules—particularly regarding leading, trailing, and embedded zeros—and carry through mathematical operations differently depending on whether the calculation involves multiplication/division or addition/subtraction. Precision and accuracy have distinct scientific meanings: precision measures the repeatability of measurements, while accuracy measures closeness to the true value.
What Is Scientific Notation?
Why We Use Scientific Notation
Scientific notation is a convenient way of writing numbers that are either very large or very small — exactly the kinds of numbers that appear constantly in chemistry. Consider the diameter of an atom: it is a tiny fraction of a nanometer, which translates to an extraordinarily small decimal. Or consider a sample that fits in the palm of your hand: it contains so many atoms or molecules that the count runs into the trillions of trillions. Writing either extreme by hand is tedious and error-prone. Scientific notation solves both problems elegantly.
The Core Rule: A number in proper scientific notation has exactly one nonzero digit to the left of the decimal point, multiplied by a power of ten. For example, 4.72 × 10⁵ is correct; 47.2 × 10⁴ is not proper form, even though it represents the same value.
How to Convert Large Numbers to Scientific Notation
Start with the number as written and locate the decimal point (which may be implied at the right end of an integer). Move the decimal point to the left until only one nonzero digit remains to its left. The number of places you moved the decimal becomes a positive exponent on the power of ten.
Example: Converting 472,000
The decimal point is implied at the right end of 472,000. Move it five places to the left:
472,000 = 4.72 × 10⁵
The exponent is positive (+5) because the original number is much larger than 1. Notice that moving the decimal left made the coefficient smaller (from 472,000 down to 4.72), so the power of ten must be large and positive to compensate.
A useful check: if the original number is greater than 1, the exponent is positive. The more you shrink the coefficient, the bigger the exponent must be to keep the overall value the same.
How to Convert Small Numbers to Scientific Notation
For numbers smaller than 1, move the decimal point to the right until one nonzero digit sits to its left. The number of places moved becomes a negative exponent.
Example: Converting 0.007349
Move the decimal point three places to the right:
0.007349 = 7.349 × 10¯³
The exponent is negative (−3) because the original number is much smaller than 1. Moving the decimal right enlarged the coefficient (from 0.007349 up to 7.349), so the power of ten must be small (negative) to compensate.
Quick Memory Aid: Think of it as a balance. If your coefficient gets smaller, your exponent gets bigger (more positive). If your coefficient gets larger, your exponent gets smaller (more negative). The two always move in opposite directions.
What Are Significant Figures and Why Do They Matter?
Significant figures — "sig figs" for short — are a way of communicating how precisely a number is known. Many students memorize the rules for counting sig figs without ever understanding their purpose. That's a mistake, because once you understand the concept, the rules become far more intuitive.
A Practical Example: Driving Distance
Imagine someone asks how far it is from Phoenix, Arizona to Portland, Maine. Here are four possible answers, each more precise than the last:
| Distance Stated | Sig Figs | Precision (plus or minus) | What It Implies |
|---|---|---|---|
| 3,000 miles | 1 | ± 1,000 miles | A rough estimate; could be anywhere from 2,000 to 4,000 |
| 2,700 miles | 2 | ± 100 miles | Narrows it considerably — enough to matter for fuel planning |
| 2,740 miles | 3 | ± 10 miles | Fairly precise; the trailing zero is not significant here |
| 2,739.1 miles | 5 | ± 0.1 miles | Highly precise; known to the nearest tenth of a mile |
Each step reveals more information about how well the distance is actually known. In science and engineering, this matters enormously. If you are designing a bridge and your measurements are rounded too aggressively, the actual structure may be weaker than your calculations predict. Significant figures are the formal system for tracking — and communicating — that precision.
Where Sig Figs Matter Most in Your Course: Significant figures receive heavy emphasis on your first exam and are a persistent concern in the laboratory. Most instructors will deduct points for incorrect sig figs on lab reports throughout the entire semester, even after the topic fades from lecture.
3 Rules for Significant Figures: Zeros
Every nonzero digit is always significant. The only source of ambiguity is zeros — they can be either significant or not, depending on their position. Three rules cover all cases.
Rule 1: Leading Zeros Are Never Significant
Any zeros that appear at the beginning of a number — before the first nonzero digit — serve only as placeholders and carry no information about precision. They are never significant, regardless of which side of the decimal point they fall on.
Example: 0.00047
The three leading zeros are not significant. Only the 4 and the 7 count.
Significant figures: 2 (the digit 4 and the digit 7)
Rule 2: Trailing Zeros — It Depends on the Decimal Point
Zeros at the end of a number are significant only if the number contains a written decimal point. This is one of the most commonly confused rules in introductory chemistry.
Example A: 5,400 (no decimal point written)
The two trailing zeros are not significant. This number has only 2 significant figures (the 5 and the 4). It is precise only to the nearest hundred.
Example B: 2.5400 (decimal point present, zeros follow it)
All five digits are significant, including both trailing zeros. Writing those zeros explicitly asserts that the number is known all the way to the ten-thousandths place. Significant figures: 5.
Common Mistake: Students often assume that trailing zeros are always insignificant. They are only insignificant when no decimal point is written. The presence of a decimal point — even if it seems redundant — is a deliberate signal that those zeros were actually measured.
Rule 3: Zeros Between Significant Digits Are Always Significant
Any zero that sits between two nonzero significant digits is itself significant, no exceptions.
Example: 200.16
The digits 2, 0, 0, 1, and 6 are all significant. The two zeros in the hundreds and tens places are flanked by significant digits on both sides, so they are significant.
Significant figures: 5
3 Zero Rules at a Glance
| Zero Type | Example | Significant? |
|---|---|---|
| Leading zeros | 0.00047 | No — never |
| Trailing zeros, no decimal | 5,400 | No |
| Trailing zeros, decimal present | 2.5400 | Yes |
| Embedded zeros | 200.16 | Yes — always |
How Scientific Notation Removes Sig Fig Ambiguity
What if a number like 5,400 really is known to the nearest mile — all four digits intentionally significant? Written as an integer, it is ambiguous: the reader cannot tell whether those trailing zeros are meaningful or merely placeholders. Scientific notation resolves this ambiguity cleanly.
Example: Expressing 5,400 with 4 Significant Figures
Convert to scientific notation by moving the decimal three places to the left:
5,400 = 5.400 × 10³
The zeros now sit to the right of the decimal point in the coefficient, making them unambiguously significant. All four digits count.
This is one of the most practical advantages of scientific notation: because a properly formatted coefficient must begin with a nonzero digit, the number can never have leading zeros. Any zeros that appear in the coefficient are always either embedded or trailing-after-decimal — and therefore always significant. In scientific notation, every digit you write is a significant digit.
Alternative for Handwritten Work: Some textbooks indicate a significant trailing zero in an integer by placing a bar (overline) above it — for example, writing 5,400 with a bar over the second zero. This notation works but is cumbersome. Scientific notation is the cleaner, universally understood approach.
Sig Fig Rules for Multiplication and Division
Identifying the number of significant figures in a single measurement is only half the challenge. The other half is carrying that precision correctly through calculations. The rules differ depending on whether you are multiplying/dividing or adding/subtracting.
Multiplication and Division
Multiplication and division are by far the most common operations in general chemistry calculations — think unit conversions, density, molarity, and nearly every stoichiometry problem. The rule is straightforward:
Rule: The result of a multiplication or division can have no more significant figures than the measurement with the fewest significant figures. Count sig figs in each factor; your answer matches the smallest count.
When multiplying or dividing a long chain of numbers, you can carry extra digits through all intermediate steps and apply the rounding rule just once at the very end.
Example: 4.52 × 3.978
- 4.52 has 3 significant figures
- 3.978 has 4 significant figures
- The answer is limited to 3 significant figures (the smaller count)
Calculator result: 17.9802...
Rounded to 3 sig figs: 18.0
Note that the trailing zero in 18.0 is significant — it appears to the right of the decimal and is required to communicate the correct precision.
The underlying logic: if one of your inputs is only a rough approximation, multiplying by a highly precise number does not magically make the result more precise. Your answer can only be as exact as your least-exact input.
Practice Problem: Compute the following and express your answer in the proper number of significant figures.
(2.73) × (6.584) =
Sig Fig Rules for Addition and Subtraction
Addition and subtraction use a completely different rule from multiplication and division. This is one of the most common points of confusion — students forget the rule is different and apply the multiplication rule incorrectly.
Rule: The result rounds to the same decimal place as the least precise term in the sum or difference. You are not counting total significant figures here — you are identifying the rightmost significant column among all the numbers, then rounding the answer to that column.
The easiest way to apply this rule is to line up the numbers vertically, as you would in grade school arithmetic, and identify which number has its last significant digit furthest to the left. That is your limiting term.
Example: 1,540 + 327.4 + 0.267
- 0.267 — last significant digit is in the thousandths place (± 0.001)
- 327.4 — last significant digit is in the tenths place (± 0.1)
- 1,540 — last significant digit is the 4, in the tens place (± 10); the trailing zero is not significant
The least precise term is 1,540, with its last significant digit in the tens place. The answer must therefore round to the tens place.
Sum: 1,540 + 327.4 + 0.267 = 1,867.667...
Rounded to the tens place: 1,870
The answer happens to have three significant figures, but that is a consequence of where rounding occurred — not a result of counting sig figs in the inputs.
Mixed Operations: Applying Both Sig Fig Rules in 1 Problem
Problems that combine both types of operations require you to propagate significant figures at each step, not just at the end. Follow the order of operations, apply the appropriate sig fig rule to each intermediate result, then carry that rounded result into the next operation.
Example: (4.23 + 7.6) ÷ 0.50
Step 1 — Addition (apply the addition rule):
- 4.23: last significant digit in hundredths place
- 7.6: last significant digit in tenths place
- Least precise: tenths place → round sum to tenths place
4.23 + 7.6 = 11.83 → rounded to tenths: 11.8 (3 sig figs)
Step 2 — Division (apply the multiplication/division rule):
- 11.8 has 3 significant figures
- 0.50 has 2 significant figures
- Answer limited to 2 significant figures
11.8 ÷ 0.50 = 23.6 → rounded to 2 sig figs: 24
Mixed-Operation Problems Are Rare — But Not Impossible: This type of problem appears infrequently, usually only in the first chapter or on a comprehensive final exam. Do not let it catch you off guard. When in doubt, identify the operation type at each step and apply the correct rule before moving to the next.
Sig Figs for the Rest of the Semester
Significant figures receive intense focus during this first unit and its exam. After that, the emphasis shifts — most general chemistry exams present multiple-choice answers with different values rather than different levels of precision, so the sig fig rules matter less for earning points on those tests. That said, the laboratory remains a different story. Lab reports almost always require proper sig figs, and most instructors will deduct points throughout the entire course for careless rounding. Nail the rules now, and keep them in the back of your mind for lab all semester.
What Is the Difference Between Precision and Accuracy?
These two words are used interchangeably in everyday speech, but in science and engineering they have distinct, technical meanings. Confusing them is a guaranteed way to lose points on an exam.
Defining the 2 Terms
Accuracy describes how close a measurement is to the true value. A highly accurate measurement is one that hits the mark.
Precision describes how repeatable a measurement is — how close multiple independent measurements are to each other, regardless of whether any of them are close to the true value. Precision is about consistency, not correctness.
The Archery Bullseye Analogy
The classic illustration is a target with a bullseye. Imagine shooting four arrows under each of three scenarios:
| Scenario | Grouping | Near Bullseye? | Verdict |
|---|---|---|---|
| All four arrows cluster in the bullseye | Tight | Yes | Both precise and accurate |
| All four arrows cluster tightly, but far from center | Tight | No | Precise but not accurate (e.g., miscalibrated sight) |
| Arrows scattered randomly across the target | Scattered | No | Neither precise nor accurate |
Chemistry Context
A balance that consistently reads 0.50 g too high is precise (every measurement is repeatable) but not accurate (every measurement is wrong by the same amount). This is the signature of a systematic error — a miscalibrated instrument, for instance. Random errors, by contrast, scatter measurements in unpredictable directions and destroy both precision and accuracy.
The Key Distinction: Accuracy = how close to the truth. Precision = how close to each other. A set of measurements can be precise without being accurate, but it is very difficult to be consistently accurate without also being precise.
Key Takeaways
- Scientific notation expresses any number as a coefficient (one nonzero digit left of the decimal) multiplied by a power of ten. Numbers greater than 1 have positive exponents; numbers less than 1 have negative exponents.
- Every digit in a properly written scientific notation expression is significant, making it the unambiguous way to convey precision for large or small numbers.
- Significant figures communicate how precisely a number is known. Nonzero digits are always significant; zeros follow 3 rules based on their position (leading, trailing, or embedded).
- Leading zeros are never significant. Trailing zeros are significant only when a decimal point is explicitly written. Zeros embedded between significant digits are always significant.
- For multiplication and division, the result carries the same number of significant figures as the input with the fewest sig figs.
- For addition and subtraction, the result rounds to the same decimal place as the least precise input term — this is a column-based rule, not a count-based rule.
- When a calculation mixes both operation types, apply the appropriate sig fig rule at each intermediate step before moving on to the next operation.
- Accuracy measures closeness to the true value; precision measures the repeatability of measurements. A result can be precise without being accurate, but not the reverse.
- Significant figures are most heavily tested on your first exam and in the laboratory throughout the semester. Keep the rules sharp for lab reports even after they fade from lecture.
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Scientific Notation & Significant Figures
General Chemistry 1.2
Lesson Overview
Scientific notation provides a standardized method for expressing very large and very small numbers as a coefficient multiplied by a power of ten, eliminating ambiguity about which digits are meaningful. Significant figures follow specific rules—particularly regarding leading, trailing, and embedded zeros—and carry through mathematical operations differently depending on whether the calculation involves multiplication/division or addition/subtraction. Precision and accuracy have distinct scientific meanings: precision measures the repeatability of measurements, while accuracy measures closeness to the true value.
What Is Scientific Notation?
Why We Use Scientific Notation
Scientific notation is a convenient way of writing numbers that are either very large or very small — exactly the kinds of numbers that appear constantly in chemistry. Consider the diameter of an atom: it is a tiny fraction of a nanometer, which translates to an extraordinarily small decimal. Or consider a sample that fits in the palm of your hand: it contains so many atoms or molecules that the count runs into the trillions of trillions. Writing either extreme by hand is tedious and error-prone. Scientific notation solves both problems elegantly.
The Core Rule: A number in proper scientific notation has exactly one nonzero digit to the left of the decimal point, multiplied by a power of ten. For example, 4.72 × 10⁵ is correct; 47.2 × 10⁴ is not proper form, even though it represents the same value.
How to Convert Large Numbers to Scientific Notation
Start with the number as written and locate the decimal point (which may be implied at the right end of an integer). Move the decimal point to the left until only one nonzero digit remains to its left. The number of places you moved the decimal becomes a positive exponent on the power of ten.
Example: Converting 472,000
The decimal point is implied at the right end of 472,000. Move it five places to the left:
472,000 = 4.72 × 10⁵
The exponent is positive (+5) because the original number is much larger than 1. Notice that moving the decimal left made the coefficient smaller (from 472,000 down to 4.72), so the power of ten must be large and positive to compensate.
A useful check: if the original number is greater than 1, the exponent is positive. The more you shrink the coefficient, the bigger the exponent must be to keep the overall value the same.
How to Convert Small Numbers to Scientific Notation
For numbers smaller than 1, move the decimal point to the right until one nonzero digit sits to its left. The number of places moved becomes a negative exponent.
Example: Converting 0.007349
Move the decimal point three places to the right:
0.007349 = 7.349 × 10⁻³
The exponent is negative (−3) because the original number is much smaller than 1. Moving the decimal right enlarged the coefficient (from 0.007349 up to 7.349), so the power of ten must be small (negative) to compensate.
Quick Memory Aid: Think of it as a balance. If your coefficient gets smaller, your exponent gets bigger (more positive). If your coefficient gets larger, your exponent gets smaller (more negative). The two always move in opposite directions.
What Are Significant Figures and Why Do They Matter?
Significant figures — "sig figs" for short — are a way of communicating how precisely a number is known. Many students memorize the rules for counting sig figs without ever understanding their purpose. That's a mistake, because once you understand the concept, the rules become far more intuitive.
A Practical Example: Driving Distance
Imagine someone asks how far it is from Phoenix, Arizona to Portland, Maine. Here are four possible answers, each more precise than the last:
| Distance Stated | Sig Figs | Precision (plus or minus) | What It Implies |
|---|---|---|---|
| 3,000 miles | 1 | ± 1,000 miles | A rough estimate; could be anywhere from 2,000 to 4,000 |
| 2,700 miles | 2 | ± 100 miles | Narrows it considerably — enough to matter for fuel planning |
| 2,740 miles | 3 | ± 10 miles | Fairly precise; the trailing zero is not significant here |
| 2,739.1 miles | 5 | ± 0.1 miles | Highly precise; known to the nearest tenth of a mile |
Each step reveals more information about how well the distance is actually known. In science and engineering, this matters enormously. If you are designing a bridge and your measurements are rounded too aggressively, the actual structure may be weaker than your calculations predict. Significant figures are the formal system for tracking — and communicating — that precision.
3 Rules for Significant Figures: Zeros
Every nonzero digit is always significant. The only source of ambiguity is zeros — they can be either significant or not, depending on their position. Three rules cover all cases.
Rule 1: Leading Zeros Are Never Significant
Any zeros that appear at the beginning of a number — before the first nonzero digit — serve only as placeholders and carry no information about precision. They are never significant, regardless of which side of the decimal point they fall on.
Example: 0.00047
The three leading zeros are not significant. Only the 4 and the 7 count.
Significant figures: 2 (the digit 4 and the digit 7)
Rule 2: Trailing Zeros — It Depends on the Decimal Point
Zeros at the end of a number are significant only if the number contains a written decimal point. This is one of the most commonly confused rules in introductory chemistry.
Example A: 5,400 (no decimal point written)
The two trailing zeros are not significant. This number has only 2 significant figures (the 5 and the 4). It is precise only to the nearest hundred.
Example B: 2.5400 (decimal point present, zeros follow it)
All five digits are significant, including both trailing zeros. Writing those zeros explicitly asserts that the number is known all the way to the ten-thousandths place. Significant figures: 5.
Common Mistake: Students often assume that trailing zeros are always insignificant. They are only insignificant when no decimal point is written. The presence of a decimal point — even if it seems redundant — is a deliberate signal that those zeros were actually measured.
Rule 3: Zeros Between Significant Digits Are Always Significant
Any zero that sits between two nonzero significant digits is itself significant, no exceptions.
Example: 200.16
The digits 2, 0, 0, 1, and 6 are all significant. The two zeros in the hundreds and tens places are flanked by significant digits on both sides, so they are significant.
Significant figures: 5
3 Zero Rules at a Glance
| Zero Type | Example | Significant? |
|---|---|---|
| Leading zeros | 0.00047 | No — never |
| Trailing zeros, no decimal | 5,400 | No |
| Trailing zeros, decimal present | 2.5400 | Yes |
| Embedded zeros | 200.16 | Yes — always |
How Scientific Notation Removes Sig Fig Ambiguity
What if a number like 5,400 really is known to the nearest mile — all four digits intentionally significant? Written as an integer, it is ambiguous: the reader cannot tell whether those trailing zeros are meaningful or merely placeholders. Scientific notation resolves this ambiguity cleanly.
Example: Expressing 5,400 with 4 Significant Figures
Convert to scientific notation by moving the decimal three places to the left:
5,400 = 5.400 × 10³
The zeros now sit to the right of the decimal point in the coefficient, making them unambiguously significant. All four digits count.
This is one of the most practical advantages of scientific notation: because a properly formatted coefficient must begin with a nonzero digit, the number can never have leading zeros. Any zeros that appear in the coefficient are always either embedded or trailing-after-decimal — and therefore always significant. In scientific notation, every digit you write is a significant digit.
Alternative for Handwritten Work: Some textbooks indicate a significant trailing zero in an integer by placing a bar (overline) above it — for example, writing 5,400 with a bar over the second zero. This notation works but is cumbersome. Scientific notation is the cleaner, universally understood approach.
Sig Fig Rules for Multiplication and Division
Identifying the number of significant figures in a single measurement is only half the challenge. The other half is carrying that precision correctly through calculations. The rules differ depending on whether you are multiplying/dividing or adding/subtracting.
Multiplication and Division
Multiplication and division are by far the most common operations in general chemistry calculations — think unit conversions, density, molarity, and nearly every stoichiometry problem. The rule is straightforward:
Rule: The result of a multiplication or division can have no more significant figures than the measurement with the fewest significant figures. Count sig figs in each factor; your answer matches the smallest count.
When multiplying or dividing a long chain of numbers, you can carry extra digits through all intermediate steps and apply the rounding rule just once at the very end.
Example: 4.52 × 3.978
- 4.52 has 3 significant figures
- 3.978 has 4 significant figures
- The answer is limited to 3 significant figures (the smaller count)
Calculator result: 17.9802...
Rounded to 3 sig figs: 18.0
Note that the trailing zero in 18.0 is significant — it appears to the right of the decimal and is required to communicate the correct precision.
The underlying logic: if one of your inputs is only a rough approximation, multiplying by a highly precise number does not magically make the result more precise. Your answer can only be as exact as your least-exact input.
Practice Problem: Compute the following and express your answer in the proper number of significant figures.
(2.73) × (6.584) =
Sig Fig Rules for Addition and Subtraction
Addition and subtraction use a completely different rule from multiplication and division. This is one of the most common points of confusion — students forget the rule is different and apply the multiplication rule incorrectly.
Rule: The result rounds to the same decimal place as the least precise term in the sum or difference. You are not counting total significant figures here — you are identifying the rightmost significant column among all the numbers, then rounding the answer to that column.
The easiest way to apply this rule is to line up the numbers vertically, as you would in grade school arithmetic, and identify which number has its last significant digit furthest to the left. That is your limiting term.
Example: 1,540 + 327.4 + 0.267
- 0.267 — last significant digit is in the thousandths place (± 0.001)
- 327.4 — last significant digit is in the tenths place (± 0.1)
- 1,540 — last significant digit is the 4, in the tens place (± 10); the trailing zero is not significant
The least precise term is 1,540, with its last significant digit in the tens place. The answer must therefore round to the tens place.
Sum: 1,540 + 327.4 + 0.267 = 1,867.667...
Rounded to the tens place: 1,870
The answer happens to have three significant figures, but that is a consequence of where rounding occurred — not a result of counting sig figs in the inputs.
Mixed Operations: Applying Both Sig Fig Rules in 1 Problem
Problems that combine both types of operations require you to propagate significant figures at each step, not just at the end. Follow the order of operations, apply the appropriate sig fig rule to each intermediate result, then carry that rounded result into the next operation.
Example: (4.23 + 7.6) ÷ 0.50
Step 1 — Addition (apply the addition rule):
- 4.23: last significant digit in hundredths place
- 7.6: last significant digit in tenths place
- Least precise: tenths place → round sum to tenths place
4.23 + 7.6 = 11.83 → rounded to tenths: 11.8 (3 sig figs)
Step 2 — Division (apply the multiplication/division rule):
- 11.8 has 3 significant figures
- 0.50 has 2 significant figures
- Answer limited to 2 significant figures
11.8 ÷ 0.50 = 23.6 → rounded to 2 sig figs: 24
Mixed-Operation Problems Are Rare — But Not Impossible: This type of problem appears infrequently, usually only in the first chapter or on a comprehensive final exam. Do not let it catch you off guard. When in doubt, identify the operation type at each step and apply the correct rule before moving to the next.
Sig Figs for the Rest of the Semester
Significant figures receive intense focus during this first unit and its exam. After that, the emphasis shifts — most general chemistry exams present multiple-choice answers with different values rather than different levels of precision, so the sig fig rules matter less for earning points on those tests. That said, the laboratory remains a different story. Lab reports almost always require proper sig figs, and most instructors will deduct points throughout the entire course for careless rounding. Nail the rules now, and keep them in the back of your mind for lab all semester.
What Is the Difference Between Precision and Accuracy?
These two words are used interchangeably in everyday speech, but in science and engineering they have distinct, technical meanings. Confusing them is a guaranteed way to lose points on an exam.
Defining the 2 Terms
Accuracy describes how close a measurement is to the true value. A highly accurate measurement is one that hits the mark.
Precision describes how repeatable a measurement is — how close multiple independent measurements are to each other, regardless of whether any of them are close to the true value. Precision is about consistency, not correctness.
The Archery Bullseye Analogy
The classic illustration is a target with a bullseye. Imagine shooting four arrows under each of three scenarios:
| Scenario | Grouping | Near Bullseye? | Verdict |
|---|---|---|---|
| All four arrows cluster in the bullseye | Tight | Yes | Both precise and accurate |
| All four arrows cluster tightly, but far from center | Tight | No | Precise but not accurate (e.g., miscalibrated sight) |
| Arrows scattered randomly across the target | Scattered | No | Neither precise nor accurate |
Chemistry Context
A balance that consistently reads 0.50 g too high is precise (every measurement is repeatable) but not accurate (every measurement is wrong by the same amount). This is the signature of a systematic error — a miscalibrated instrument, for instance. Random errors, by contrast, scatter measurements in unpredictable directions and destroy both precision and accuracy.
The Key Distinction: Accuracy = how close to the truth. Precision = how close to each other. A set of measurements can be precise without being accurate, but it is very difficult to be consistently accurate without also being precise.
Key Takeaways
- Scientific notation expresses any number as a coefficient (one nonzero digit left of the decimal) multiplied by a power of ten. Numbers greater than 1 have positive exponents; numbers less than 1 have negative exponents.
- Every digit in a properly written scientific notation expression is significant, making it the unambiguous way to convey precision for large or small numbers.
- Significant figures communicate how precisely a number is known. Nonzero digits are always significant; zeros follow 3 rules based on their position (leading, trailing, or embedded).
- Leading zeros are never significant. Trailing zeros are significant only when a decimal point is explicitly written. Zeros embedded between significant digits are always significant.
- For multiplication and division, the result carries the same number of significant figures as the input with the fewest sig figs.
- For addition and subtraction, the result rounds to the same decimal place as the least precise input term — this is a column-based rule, not a count-based rule.
- When a calculation mixes both operation types, apply the appropriate sig fig rule at each intermediate step before moving on to the next operation.
- Accuracy measures closeness to the true value; precision measures the repeatability of measurements. A result can be precise without being accurate, but not the reverse.
- Significant figures are most heavily tested on your first exam and in the laboratory throughout the semester. Keep the rules sharp for lab reports even after they fade from lecture.