Normal Distribution

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The normal distribution is one of the most important ideas in statistics because many real measurements cluster around an average and become less common as you move away from it. Its graph is the bell curve, a symmetric shape centered at the mean. Understanding this curve helps students describe data, compare values, and estimate probabilities. It is used in fields such as test scoring, biology, manufacturing, and social science.

Standard deviation tells how spread out the data are, while $z$ -scores tell how far a value is from the $\text{mean}$ in units of standard deviation. Together, they let you place any observation on the bell curve and compare values from different data sets. For a normal distribution, fixed percentages of data fall within $1$ , $2$ , and $3$ standard deviations of the mean. This makes the normal model a powerful tool for prediction, interpretation, and decision making.

Key Facts

A normal distribution is symmetric about its mean $\mu$ , with $\text{mean} = \text{median} = \text{mode}$ .
Standard deviation measures spread: larger σ means a wider, flatter bell curve.
Z-score formula: z = (x - μ) / σ
About 68% of data lie within μ ± 1σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
A z-score of 0 means the value equals the mean; positive z is above the mean and negative z is below it.
The total area under a normal curve equals 1, representing 100% of the probability.

Vocabulary

Normal distribution: A symmetric probability distribution shaped like a bell, where most values cluster near the mean.
Mean: The average value of a data set, found by adding all values and dividing by the number of values.
Standard deviation: A measure of how far data values typically spread from the mean.
Z-score: A standardized value that shows how many standard deviations a data point is above or below the $\text{mean}$ .
Probability density: A way of describing how probability is distributed across values in a continuous distribution.

Common Mistakes to Avoid

Confusing standard deviation with variance, which is wrong because variance is measured in squared units while standard deviation is in the original units of the data.
Using z = (μ - x) / σ, which is wrong because the correct formula is z = (x - μ) / σ and reversing the order changes the sign.
Assuming all data sets are normally distributed, which is wrong because some data are skewed, uniform, or have outliers that do not fit a bell curve well.
Thinking 95% of data are always exactly between any two values near the center, which is wrong because the 68-95-99.7 rule applies specifically to normal distributions and to intervals based on standard deviations.

Practice Questions

1 A test score distribution has mean $\mu = 70$ and standard deviation $\sigma = 10$ . Find the z-score for a student who scored $x = 85$ .
2 The heights of a plant species are approximately normal with mean $40$ cm and standard deviation $5$ cm. Using the 68-95-99.7 rule, what percentage of plants are expected to have heights between $35$ cm and $45$ cm?
3 Two students took different exams. One scored $78$ on a test with mean $70$ and standard deviation $4$ . The other scored $88$ on a test with mean $80$ and standard deviation $10$ . Explain which student performed better relative to their class.

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