1. Density Curves & Normal Distributions Textbook Section 2.2

2. Remember…  When exploring quantitative data, ALWAYS  Plot the data (dot plot, stem plot, or histogram)  Look at overall shape, center, spread, look for outliers  Calculate a numerical summary to describe center and spread  Sometimes, the pattern is so regular and the number of observations so large, we can describe it as a smooth curve called a Density Curve.

3. Density Curves  A density curve is a curve that  Is always on or above the horizontal axis  Has the area of exactly 1 underneath it.  Density curves come in many shapes (like distributions).  Outliers are not included in the curve  No set of real data is exactly described by a density curve. The curve is an approximation that is easy to use and accurate enough for practical use.

4. Describing a Density Curve  The median of a density curve is the “equal areas” point – the point at which the area under the curve is equal on both sides.  The mean of a density curve is the “balance point” – the point at which the curve would balance if it were made of solid material.  In a symmetric curve, the mean and median are equal and at the center.  In a skewed curve, the mean is “pulled” towards the tail. 

5. Normal Distributions  One of the most important density curves in statistics is the Normal Curve.  All Normal curves have the same shape: symmetric, single-peaked, and bell shaped  Normal curves are specified by their mean (μ) and standard deviation (σ).  The mean is located at the center, and the standard deviation determines the spread of the curve.  Normal distributions naturally occur in many standardized tests, characteristics of a biological population (lengths of crickets, etc.), the results of chance outcomes (probability).

6. 68-95-99.7 Rule 68% of the observations fall within 1 standard deviation of the mean 95% of the observations fall within 2 standard deviations of the mean 99.7% of the observations fall within 3 standard deviations of the mean

7. Check your Understanding…  The distribution of young women ages 18-24 in approximately N(64.5, 2.5) 1. Sketch a Normal Density Curve for the distribution of young women’s heights. Label the points one, two and three standard deviations from the mean. 2. What percent of young women have heights greater than 67 inches? Show your work. 3. What percent of young women have heights between 62 and 72 inches? Show your work.

8. The STANDARD Normal Distribution  Because the mean and standard deviations of real data sets vary, we need to be able to standardize the measurements in order to compare them.  The Standard Normal Distribution has a mean of 0 and a standard deviation of 1.  We standardize the measurements by finding a z-score.  The 68-95-99.7 rule tells us that 68% of the data fall between z = -1 and z = 1, but it doesn’t help us when we have values like z = 1.25…

9. The Standard Normal Table  The Standard Normal Table gives us the area under the curve to the LEFT of any z-score calculated.  Look at Table A – Standard Normal Table. Find the value listed for z = 0.81 by finding 0.8 in the left column and 0.01 in the top column.  The area under the curve to the left of z = 0.81 is 0.7910 which also means that the z-score of 0.81 is in the 79 th percentile of data. z.00.01.02 0.7.7580.7611.7642 0.8.7881.7910.7939 0.9.8159.8186.8212

10. Check your understanding…  Remember, the value from Table A is the area to the LEFT of the z- score. If you want to find the area to the RIGHT of your critical value, you must subtract from 1 (remember: the total area under the curve is 1) 1. Find the proportion of observations from a standard Normal distribution. Sketch and shade a curve to assist you. a. z -2.15c. -0.56 < z < 1.81 2. Use Table A to find the value of z from the standard Normal distribution that corresponds the following conditions. a. the 20 th percentileb. 45% of all observations are greater than z.

11. Technology Helps  You can use the applet at www.whfreeman.com/tps5e to assist you in finding areas. Just adjust the mean to 0 and standard deviation to 1.www.whfreeman.com/tps5e  You can also use your calculator.  Press 2 nd + VARS (DISTR) and choose normalcdf  The input for this function is (low value, high value, μ, σ)  Examples:  Z > -1.78  normalcdf(-1.78,10^99,0,1)  -1.25 < z < 0.81  normalcdf (-1.25,0.81,0,1)  What is the z-score for the 90 th percentile of data?  Press 2 nd + VARS and choose InvNorm  Then input (Percentile, mean, standard deviation)  InvNorm(0.90, 0, 1 ) = 1.2815  z-score of 1.2815

12. Normal Distribution Calculations  On the driving range, Tiger Woods practices his swing with a particular club by hitting many, many golf balls. Suppose that when Tiger hits the driver, the distance the ball travels follows a Normal Distribution with mean 304 yards and standard deviation of 8 yards.  What percent of Tiger’s drives travel at least 290 yards? 1. State the distribution and values of interest. Sketch a curve and label appropriately. 2. Perform calculations – show your work! 1. Answer the question: About 96% of Tiger Woods’s drives travel at least 290 yards. OR normalcdf (290,10^99, 304,8) If you use the calculator, you MUST draw and label curve!

13. Assessing Normality  While some distributions of real data tend to be normal, it’s difficult to know if the data you have is one of those normal distributions.  First, plot the data – if the graph is clearly skewed or has multiple peaks, that’s evidence AGAINST Normality. However, if it looks Normal (or close to Normal), we need more evidence.  Make a Normal Probability Plot on your calculator (the last option in the STATPLOT menu) and check for linearity.  If the points are close to linear, the data is approximately Normal. Don’t worry too much about small deviations/wiggles.  Outliers will appear completely separate.  Beware of obvious curves (graph will look logarithmic or exponential)