Standard Deviation and Normal Distribution

6.1 The Standard Normal Distribution

Introduction

The normal probability distribution plays a key role not only in statistical theory but also in many real data phenomena. For example, after measuring an infant, a pediatrician predicts the infant's height as an adult. This prediction relies on the fact that height is normally distributed. Unlike discrete distributions that model specific counts, the normal distribution models a great number of continuous measurements in medicine, science, education, etc.

A distribution with a continuous random variable that is symmetric and bell-shaped.
Properties:

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Definition

​Inequalities and Symbols

The region under the curve.

The total area under the curve is equal to 1.

Area

The distance along the horizontal scale of the standard normal distribution (corresponding to the number of standard deviations above or below the mean)

Z score

Terminology

Finding z-scores from known Areas.

Procedure

1. Draw a bell-shaped curve and identify the region under the curve that corresponds to the given probability.

2. Use STATCRUNCH by selecting: STAT-- Calculators--Normal from the submenu

-Type in the designated area in the probability box and press enter.

Percentiles

Example:

Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the bone density test score corresponding to the given information. Round to two decimal places.

a. Find the 99^th percentile. This is the bone density score separating the bottom 99% from the top 1 %.

b. Find the 10^th percentile. This is the bone density score separating the bottom 10% from the top 90 %.

c. Find the bone density scores that could be used as cutoff values separating the lowest 3% and highest 3%.

Critical Values

Definition

​Examples