What We Know So Far…

1. Data plots or graphs

2. Look for the overall pattern and identify deviations and outliers

3. Numerical summary to briefly describe center and spread

Density Curves

The key idea is that we can use a mathematical model (a density curve) as an approximation to the overall pattern of data.If the pattern is sufficiently regular, approximate it with a smooth curve.

Density Curves Software can replace the separate bars of a histogram with a smooth curve that represents the overall shape of a distribution.

ExampleHere is a histogram of vocabulary scores of 947 seventh graders. The smooth curve drawn over the histogram is a mathematical “idealization” for the distribution.

Example

The areas of the shaded bars in this histogram represent the proportion ofscores in the observed data that are less than or equal to 6.0. This proportion is equal to 0.303.

ExampleNow the area under the smooth curve is shaded. Its proportion to the total area isnow equal to 0.293 (not 0.303). This is what the proportion on the previous slide would equal to if we had LOTS of data.

Density Curves: Definition• Always on or above the horizontal axis • Have area exactly 1 underneath curve • Area under the curve indicates the

“theoretical” proportion of values in that range.• Remember the density is only an

approximation, but it simplifies analysis and is generally accurate enough for practical use.

• Come in a variety of shapes, bell-shaped density is commonly used.

Mean

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

Median The median of a density curve is the equal-areas point, the point that divides the area under the curve in half.

Mode The mode is the peak point of the curve.

Mode

The median and the mean are always equal on a symmetric density curve

Mean > median for a right-skeweddistribution

Mean < median for a left-skewed distribution

Median

ExampleDetermine which letter corresponds to the median, mode, and the mean of the following density curves.

ExampleCopy this picture into your notes.Guess where you think the Median and the Mean are by drawing an arrow pointing to themGuess what values they actually are.

ExamplesCopy this histogram into your notes. Sketch a smooth curve that describes the distribution well. Mark your best guess with an arrow for the mean and median of the distribution.

ExamplesSketch a smooth curve that describes the distribution well. Mark your best guess with an arrow for the mean and median of the distribution.

Uniform DistributionThe figure below shows the density curve of a uniform distribution. The curve takes the constant value. Sketch the curve.

ExampleThe curve takes the constant value 1 over the interval from 0 to 1. This means that data described by this distribution take values that are uniformly spread between 0 and 1.

Questions for Previous Picture(a) Why is the total area under this curve equal to 1? The area under the curve is a rectangle with height 1 and width 1. Thus the total area under the curve = 1 ´ 1 = 1 (b) What percent of the observations lies above 0.8? 20%. (The region is a rectangle with height 1 and base width 0.2; hence the area is 0.2.)(c) What percent of the observations lie below 0.6? 60%.(d) What percent of the observations lie between 0.25 and 0.75? 50%.(e) What is the mean of this distribution? 0.5, the “balance point” of the density curve

Summary• The mode is the peak point of the curve• The median of a density curve is the equal-

areas point• The mean of a density curve is the balance

point• The median and the mean are always equal

on a symmetric density curve• Mean < median for a left-skewed distribution • Mean > median for a right-skewed

distribution • Uniform distribution has a constant curve.

HOMEWORK

Density Curve Worksheet Due Wednesday