(Day 1)

 So far, we have used histograms to represent the overall shape of a distribution. Now smooth curves can be used:

 If the curve is symmetric, single peaked, and bell-shaped, it is called a normal curve.

 Plot the data: usually a histogram or a stem plot.  Look for overall pattern ◦ Shape ◦ Center ◦ Spread ◦ Outliers

 Choose either 5 number summary or “Mean and Standard Deviation” to describe center and spread of numbers ◦ 5 number summary used when there are outliers and graph is skewed; center is the median. ◦ Mean and Standard Deviation used when there are no outliers and graph is symmetric; center is the mean  Now, if the overall pattern of a large number of observations is so regular, it can be described by a normal curve.

 The tails of normal curves fall off quickly.  There are no outlier  There are no outliers.  The mean and median are the same number, located at the center (peak) of graph.

 Most histograms show the “counts” of observations in each class by the heights of their bars and therefore by the area of the bars. ◦ (12 = Type A)  Curves show the “proportion” of observations in each region by the area under the curve. The scale of the area under the curve equals 1. This is called a density curve. ◦ (0.45 = Type A)

 Median: “Equal-areas” point – half area is to the right, half area is to the left.  Mean: The balance point at which the curve would balance if made of a solid material (see next slide).  Area: ¼ of area under curve is to the left of Quartile 1, ¾ of area under curve is to the left of Quartile 3. (Density curves use areas “to the left”).  Symmetric: Confirms that mean and median are equal.  Skewed: See next slide.

 The mean of a skewed distribution is pulled along the long tail (away from the median).

 Uniform Distributions (height = 1)

 If the curve is a normal curve, the standard deviation can be seen by sight. It is the point at which the slope changes on the curve.  A small standard deviation shows a graph which is less spread out, more sharply peaked…

 Carl Gauss used standard deviations to describe small errors by astronomers and surveyors in repeated careful measurements. A normal curve showing the standard deviations was once referred to as an “error curve”.  The Rule shows the area under the curve which shows 1, 2, and 3 standard deviations to the right and the left of the center of the curve…more accurate than by sight.

 More about Rule, z-scores, and percentiles…  We will be doing group activities. Please bring your calculators and books!!!  Homework: None… 