Describing Location in a Distribution Normal Distributions

Normal Distribution Described by a Normal density curve (bell-shaped curves) Specified by its mean, μ, and standard deviation σ Always symmetric with the mean at the center, but exact shape depends on μ and σ Change in curvature (point of inflection) shows where 1 standard deviation from the mean is located. Abbreviate the Normal distribution as N(μ, σ)

Normal Distribution Graphically…

Empirical Rule (68-95-99.7 Rule)

Example IQ scores are Normally distributed with a mean of 100 and a std dev of 15. What % of people have IQ scores… Between 70 and 130? Less than 85? Greater than 145? Less than 115? Between 55 and 70?

Probability Calculations If we standardize the distribution by calculating z scores, we create a standard normal distribution where N(0,1) with a mean = 0 and std dev = 1 Standard normal table- a table of areas under the standard normal curve. The table entry for each value z is the area under the curve to the left of z

Solving Problems w/ Normal Distributions State the problem in terms of the observed variable, x. Draw a picture of the distribution and shade the area of interest under the curve. Standardize and draw a picture. Standardize x to restate the problem in terms of a standard normal variable z. Draw a picture to show the area of interest under the standard normal curve. Use the table or calculator to find the required area under the curve. Remember the area under the entire curve is 1. Conclusion. Answer the question in context!

Example IQ scores are Normally distributed with a mean of 100 and a std dev of 15. What percent of people have IQ scores less than 82? Less than 121? What percent of people have IQ scores greater than 107? What percent of people have IQ scores between 88 and 104?

Finding a Value (given proportion) Work backwards to find the value Draw a normal curve with the area of interest shaded and the mean, standard deviation, and unknown boundary value Either use the table by finding the area and locating the z value or use the calculator invNorm function to find the z value. Use the formula for z value to find x If using invNorm (area to left, μ, σ) Answer the question in context!

Example A person is considered a genius if they are in the top 2% in terms of IQ. What IQ score does a person need to be considered a genius?

Calculator Usage To calculate the % of observations within a certain interval, use the z-table or the graphing calculator. 2nd Vars (Dist), choose option 2. Normalcdf (min, max, μ, σ) To calculate raw data scores from percentiles: 2nd Vars (Dist), choose option 3. invNorm(%, μ, σ)

Assessing Normality Construct a histogram or stemplot If graph is approximately bell shaped and symmetric about the mean then Normal Interpret a Normal Probability Plot If points on plot lie close to a straight line, the data is Normal Not on the AP exam! Can you your calculator https://www.youtube.com/watch?v=cZjKdm4TzDo