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.
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 68-95-99.7 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 68-95-99.7 Rule, z-scores, and percentiles… We will be doing group activities. Please bring your calculators and books!!! Homework: None…