Apply properties of the normal distribution

A normal distribution is a continuous, symmetric, bell-shaped model determined by its mean and standard deviation, with mean, median, and mode equal at the center. Learners interpret standardized scores $z=(x-\mu)/\sigma$ , use the standard normal curve or the 68–95–99.7 rule to estimate probabilities and proportions within specified intervals, and understand that area represents probability rather than the probability of an individual exact value. More advanced topics, such as calculus-based derivations, parameter estimation, and formal hypothesis testing, are not included.

Detailed Explanation: Apply properties of the normal distribution

A normal distribution is a symmetric, bell-shaped model. Its center is the mean, $\mu$ , which is also the median and mode. The standard deviation, $\sigma$ , tells how spread out the data are.

To locate a value relative to the mean, use its standardized score, or $z$ -score:

z=\frac{x-\mu}{\sigma}

A $z$ -score tells how many standard deviations a value is above or below the mean.

Worked example

The heights of students at a school are approximately normally distributed with mean $170$ cm and standard deviation $6$ cm. Estimate the proportion of students whose heights are between $164$ cm and $176$ cm.

Step 1: Identify the mean and standard deviation.

\mu=170,\qquad \sigma=6

Step 2: Find the $z$ -score for each boundary.

For $164$ cm:

z=\frac{164-170}{6}=\frac{-6}{6}=-1

For $176$ cm:

z=\frac{176-170}{6}=\frac{6}{6}=1

So the interval from $164$ cm to $176$ cm is from $1$ standard deviation below the mean to $1$ standard deviation above the mean.

Step 3: Use the 68–95–99.7 rule.

In a normal distribution, approximately $68\%$ of the data lie within $1$ standard deviation of the mean.

Therefore,

P(164\leq X\leq176)\approx 0.68

So, approximately 68% of the students are between $164$ cm and $176$ cm tall.

Because the normal distribution is symmetric, about half of this $68\%$ lies below the mean and half lies above it:

about $34\%$ are between $164$ cm and $170$ cm;
about $34\%$ are between $170$ cm and $176$ cm.

Remember that for a continuous distribution, the probability of one exact value, such as being exactly $170$ cm, is $0$ . Probabilities describe areas over intervals, such as being between $164$ cm and $176$ cm.

Learn by doing: Apply properties of the normal distribution

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Statistics - Standard Deviation - Mean, SD and Range to Percent in Three Sections

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