Chapter 13, Part 1

STA 200 Summer I 2011

At this point…

• we have a couple of methods for graphing a data set (histogram, stem-and-leaf plot)

• we have a general idea of what to look for in a graph (shape, outliers)

• we have a couple of ways to describe a distribution numerically (mean and standard deviation, five number summary)

Density Curves

• Sometimes, the pattern of a large data set is so regular that we can describe it by a smooth curve (a density curve).

• A density curve can be obtained by drawing a curve through the tops of the bars in a histogram.

• Density curves always show percentages or proportions (rather than counts), and the total percentage under the curve is always 100%.

Density Curves and Shape

• We use the same terminology to describe the shape of a density curve as we would to describe the shape of a histogram (symmetric, left-skewed, right-skewed).

• Regarding measures of center:– if a distribution is exactly symmetric: mean = median– if a distribution is left-skewed: mean < median– if a distribution is right-skewed: mean > median

Normal Distribution

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

• A specific normal curve is completely described by its mean and standard deviation.

• The mean is located at the center of the curve.

• The standard deviation is the distance from the mean to one of the change-of-curvature points on either side.

68-95-99.7 Rule

• For a normal distribution,– approximately 68% of the data will be within 1

standard deviation of the mean– approximately 95% of the data will be within 2

standard deviations of the mean– approximately 99.7% of the data will be within 3

standard deviations of the mean

• This is also known as the Empirical Rule.

68-95-99.7 Rule (cont.)

Example

• An IQ test is normally distributed with mean 100 and standard deviation 15.– Between what two values do approximately 68%

of IQ scores lie?– Between what two values do approximately 95%

of IQ scores lie?– Between what two values do approximately 99.7%

of IQ scores lie?

Example (cont.)

• More questions:– What percentage of IQ scores are less than 115?– What percentage of IQ scores are greater than

130?– What percentage of IQ scores are less than 85?– What percentage of IQ scores are between 70 and

115%?