1. January 24, 2012

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

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

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

5.  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.

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

7.  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)

8.  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.

9. A curve will only balance if its median & mean are the same.

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

11.  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…

12. Day 2

13.  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.

14.  In a normal distribution, approximately… ◦ 68% of all data fall within 1 standard deviation of the mean. ◦ 95% of all data fall within 2 standard deviations of the mean. ◦ 99.7% of all data fall within 3 standard deviations of the mean. ◦ Remember, the amount of data will have an impact on how close these numbers are to reality.

15. 34 — 34 13.5 13.5 2.35 2.35 0.15 0.15

16. 1.How much of the data is greater than 500? Answer: 50% 2.How much of the data falls between 400 and 500? Answer: 34% 3.How much of the data falls between 200 and 400? Answer: 50-34-0.15=15.85% 4.How much of the data falls between 300 and 700? Answer: 95% 5.How much of the data is less than 300? Answer: 100-50-34-13.5=2.5%

17.  What kind of curve is being used if the area under the curve is defined by a proportion (a value between 0 and 1). A DENSITY CURVE  To use the 68-95-99.7 Rule, it also must be a normal curve.  Symmetric  Bell shaped  Single peak  Tails fall off  No outliers  Mean and median are same number

18.  You can use Chebychev’s Theorem if the distribution is not bell shaped, or if the shape is not known.  The portion of any data set lying within “k” standard deviations (k > 1) of the mean is at least:

19.  If you wanted to know how much data fell within 3 standard deviations of a data set and it wasn’t a bell shaped curve or you didn’t know the shape:

20.  This is used to standardize numbers which may be on different scales.

21.  Answer can be between -3.49 and +3.49.  Z-scores represent the number of standard deviations above or below the mean the number is. ◦ If z=1, the value is 1 standard deviation above the mean. ◦ If z=-2, the value is 2 standard deviations below the mean.

22.  During the 2003 regular season, the Kansas City Chiefs (NFL) scored 63 touchdowns. During the 2003 regular season the Tampa Bay Storm (Arena Football) scored 119 touchdowns. The mean number of touchdowns for Kansas City is 37.4 with a standard deviation of 9.3. The mean number of touchdowns for Tampa Bay is 111.7 with a standard deviation of 17.3.  Find the z-score for each.  KC = 2.75TB = 0.42  Kansas City had a better record of touchdowns for the season (much higher above the mean).

24.  C th percentile of a distribution is a value such that C percent of the observations lie below it and the rest lie above it. ◦ 80 th percentile =  80% below, 20% above = Top 20% ◦ 90 th percentile =  90% below, 10% above = Top 10% ◦ 99 th percentile =  99% below, 1% above = Top 1%

25.  Please note, you cannot use this identical activity for your project.  Everyone in class is to measure their height in centimeters (nearest whole number) and record the result on the white board.  Arrange the values from smallest to largest.  Find the mean and standard deviation (both to the nearest tenth)… on your calculator it’s x bar and s(x).  Calculate the borders for the 68-95-99.7 regions. How many students fall into each category? How many should, based on the 68- 95-99.7 rule?

27. ◦ 1.Find the mean and standard deviation, to the nearest tenth. ◦ 2.Construct a 68-95-99.7 curve to determine the number of presidents who fall into each category and determine 0 if the curve was accurate. ◦ 3.What is the minimum age a person can be to be president? ◦ 4.Who will win the Super Bowl? ◦ Data: 5761575758576154 6851496450486552 5646544951475555 5442515655515451 6061435556615269 64465448