# 5.2 The Normal Distribution, Revisited

#exploringdata

#sampling

#experimentation

#anticipatingpatterns

#statisticalinference

written by josh argo

brianna bukowski harrison burnside

(editor)

### Definition

A normal model is unimodal (one mound, bell-shaped) and symmetric. The median and mean are equal to each other.

Furthermore, a model (such as a sampling distribution) is approximately normal if the following conditions are met for the following types of data:

 Categorical (Proportions) Quantitative (Means) The number of successes and failures is at least 10 (Large Counts) The sample size is at least 30 (Central Limit Theorem which you’ll learn about in the next post)ORPopulation is normally distributed

## Empirical Rule Review

The area underneath any density curve (including the normal curve) is equal to 1, or 100% of the data. There is 50% of the data on each size of the mean.

• 68% of the data is within 1 standard deviation of the mean. (34% on each side)

• 95% of the data is within 2 standard deviations of the mean (an additional 13.5% on each side)

• 99.7% of the data is within 3 standard deviations of the mean (an additional 2.35% on each side) Source: Medium

🎥Watch: AP Stats - Normal Distributions

A z-score indicates how many standard deviations above or below the mean a piece of data is. If a z-score is -2, it is 2 standard deviations below the mean. If a z-score is 1, it is 1 standard deviation above the mean.  When asked to interpret a z-score, it is imperative that you indicate the direction (positive or negative). The magnitude of the z-scores is also important, especially when making inferences.

Z-scores are often used to compare scores regarding two sets of data, like SAT and ACT scores. The z-scores are resistant to units. Therefore the units do not matter and the z-scores remain the same if we convert the units.  Think of z-scores as a way of comparing apples 🍎 to oranges 🍊.