Density Curves Sections - PDF Free Download

Density Curves Sections 3.1-3.2 Lecture 8 Robb T. Koether Hampden-Sydney College Wed, Jan 27, 2016 Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 1 / 18

Outline 1 Density Curves 2 Creating Density Curves 3 Describing Density Curves 4 Mean and Standard Deviation 5 Assignment Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 2 / 18

Density Curves Definition (Density Curve) A density curve is similar to a histogram, with the following differences. The vertical scale is adjusted so that the total area is 1. The density curve may use rectangles, but it will more likely be a smooth curve. The smooth curve would be the result of using so many rectangles that the eye could no longer distinguish them. That would require an enormous amount of data. Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 4 / 18

Creating Density Curves Find the total area by computing the area of each rectangle and then adding up the area. Divide the numbers on the vertical scale by the total area. The effect is to change the scale so that the total area is now 1. Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 6 / 18

Example Example (August Rainfall) No. of months 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 August rainfall What is the total area of the rectangles in the histogram? Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 7 / 18

Example Example (August Rainfall) Multiply width (2) by the height of each rectangle and add: Area = (2 3) + (2 12) + (2 8) + (2 5) + (2 1) + (2 1) = 2 (3 + 12 + 8 + 5 + 1 + 1) = 60. Divide the numbers on the vertical scale by 60. Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 8 / 18

Example Example (August Rainfall) 12/60 Density 10/60 8/60 6/60 4/60 2/60 0 0 2 4 6 8 10 12 14 16 18 August rainfall Now the total area is 1. Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 9 / 18

Example Example (August Rainfall) 0.20 Density 0.15 0.10 0.05 0 0 2 4 6 8 10 12 14 16 18 August rainfall Relabel the vertical scale in more standard units. Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 10 / 18

Example Example (August Rainfall) No. of months 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 August rainfall Suppose had used a class width of 1 instead of 2. Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 11 / 18

Example Example (August Rainfall) Density 0.25 0.20 0.15 0.10 0.05 0 0 2 4 6 8 10 12 14 16 18 August rainfall The histogram is suggestive of a smooth curve representing the true distribution. Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 13 / 18

Describing Density Curves We can describe the density curve as Symmetric Skewed right Skewed left Uniform Flat, no peak Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 14 / 18

For density curves, The symbol µ (the Greek letter mu) is used for the mean. The symbol σ (the Greek letter sigma) is used for the standard deviation. For the different density-curve descriptions, where would be place µ? Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 16 / 18

For density curves, The symbol µ (the Greek letter mu) is used for the mean. The symbol σ (the Greek letter sigma) is used for the standard deviation. For the different density-curve descriptions, where would be place µ? How does σ relate to the graph? Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 16 / 18

Assignment Assignment Read Sections 3.1, 3.2. Apply Your Knowledge: 3.1, 3.2, 3.3, 3.4. Robb T. Koether (Hampden-Sydney College) Density CurvesSections 3.1-3.2 Wed, Jan 27, 2016 18 / 18