Density curves

• Like drawing a curve through the tops of the bars in a histogram and smoothing out the irregular ups and downs.

• Show the proportion of observations in any region by the area under the curve

• Choose a scale so that the total area under the curve is 1

The center and spread of a density curve

• The median is the equal-areas point—the point that divides the area under the curve in half

• The mean is the balance point, at which the curve would balance if made of solid material

• The mean and median of a symmetric curve are in the same place

• The mean of a skewed curve is pulled toward the long tail

The normal distribution

• Normal curves are symmetric, single-peaked, and bell-shaped.

• A normal curve can be described completely by its mean and standard deviation

• The standard deviation determines the shape of the curve

• Distributions of many scientific variables end up being normal

• The 68-95-99.7% rule

Standard scores

• Standard scores are scores that are expressed in standard deviations above or below the mean

• Standard score = (observation –mean) /std. dev

Percentiles

• The xth percentile of a distribtuion is a value such that x percent of the observations lie below it and the rest above it.

• Median = 50% percentile• Quartiles = ?• Standard scores translate into specific percentiles that

are not affected by the mean and standard deviation• Finding the observation that matches a percentile