Understanding the 68-95-99.7 Rule

Adult male heights in a country are normally distributed with mean $\mu = 175$ cm and standard deviation $\sigma = 7$ cm. What percentage of men are between 168 cm and 182 cm tall?

Identify the boundaries in terms of standard deviations: $168 = 175 - 7 = \mu - 1\sigma$ $182 = 175 + 7 = \mu + 1\sigma$ = We are looking at $\pm 1$ standard deviation from the mean

Apply the 68-95-99.7 rule: Within 1 standard deviation of the mean: 68% Within 2 standard deviations: 95% Within 3 standard deviations: 99.7% = 68% of data falls within 1 standard deviation

State the answer: Since 168 cm and 182 cm are exactly 1 standard deviation from the mean = Approximately 68% of men are between 168 cm and 182 cm tall

Answer: Approximately 68% of adult men have heights between 168 cm and 182 cm.

Calculating Z-Scores

On a standardized test, scores are normally distributed with $\mu = 500$ and $\sigma = 100$. A student scores 650. What is their z-score, and what does it mean?

Write the z-score formula: $Z = \frac{X - \mu}{\sigma}$ = Formula to convert raw score to z-score

Substitute the values: $Z = \frac{650 - 500}{100}$ = $Z = \frac{150}{100}$

Calculate: $Z = 1.5$ = The z-score is 1.5

Interpret the result: A z-score of 1.5 means the student scored 1.5 standard deviations above the mean = This is a strong score, better than approximately 93% of test-takers

Answer: The z-score is $Z = 1.5$, meaning the student scored 1.5 standard deviations above average.

Finding Values from Z-Scores

The weights of apples from an orchard are normally distributed with $\mu = 180$ g and $\sigma = 20$ g. What weight corresponds to the 84th percentile (z-score of approximately 1)?

Use the z-score formula solved for X: $X = \mu + Z \cdot \sigma$ = Rearranged formula to find the raw value

Substitute the values: $X = 180 + 1 \cdot 20$ = $X = 180 + 20$

Calculate: $X = 200$ g = The 84th percentile weight is 200 g

Interpret: An apple weighing 200 g is heavier than approximately 84% of all apples from this orchard = Only about 16% of apples are heavier

Answer: An apple at the 84th percentile weighs 200 g.

Mistake: Confusing standard deviation with variance

Why: Variance ($\sigma^2$) is the square of standard deviation. Using variance instead of standard deviation in z-score calculations gives wrong results.

Correct: Always use standard deviation ($\sigma$) in the z-score formula: $Z = \frac{X - \mu}{\sigma}$

Mistake: Forgetting that the normal distribution is symmetric

Why: Students sometimes calculate probabilities for only one tail and forget to account for symmetry.

Correct: The probability of being more than 2 standard deviations above the mean (2.5%) equals the probability of being more than 2 standard deviations below the mean (2.5%).

Mistake: Applying the 68-95-99.7 rule to non-normal distributions

Why: This rule only applies to normal distributions. Other distributions may have very different percentages.

Correct: First verify that your data follows a normal distribution before applying these rules.

Mistake: Thinking a negative z-score is bad

Why: A negative z-score simply means the value is below the mean, not that it is wrong or invalid.

Correct: Z-scores can be positive (above mean) or negative (below mean). Context determines whether higher or lower is desirable.

Quality Control in Manufacturing

Factories use normal distributions to monitor product quality and set acceptable tolerance ranges.

A bolt factory produces bolts with mean diameter $\mu = 10.00$ mm and $\sigma = 0.02$ mm. Bolts outside $\pm 3\sigma$ (9.94 mm to 10.06 mm) are rejected as defective.

Medical Testing and Diagnosis

Doctors use normal distributions to interpret medical test results and determine what is considered healthy or abnormal.

Blood pressure readings are approximately normally distributed. A reading more than 2 standard deviations above the population mean may indicate hypertension.

Q: Why is it called a bell curve?

A: The shape of the normal distribution graph resembles a bell - wide in the middle where most values occur, and tapering off symmetrically at both ends where extreme values are rare.

Q: Can a z-score be negative?

A: Yes! A negative z-score means the value is below the mean. For example, z = -1.5 means the value is 1.5 standard deviations below average. This is not bad - it depends on context.

Q: What is the difference between normal and standard normal distribution?

A: Any normal distribution can have any mean and standard deviation. The standard normal distribution specifically has mean = 0 and standard deviation = 1. We convert to standard normal using z-scores.

Q: Does all data follow a normal distribution?

A: No. Income distributions are typically skewed (not symmetric). Counts and proportions follow different distributions. Always check if normality is a reasonable assumption for your data.