Chapter 3 The Normal Distribution

Density Curves

What are density curves? A density curve is always on or above the x-axis and has an area of exactly 1 underneath it The median of a density curve divides the curve in half The mean of a density curve is known as the “balance point” of a density curve When a density curve is symmetric, the mean and median are the same in the center of the curve. When a density curve is skewed, the mean is pulled away from the median in the direction of the long tail

Breaking news! σ for standard deviation of a density curve We now finally use: σ for standard deviation of a density curve μ for the mean of a density curve

Normal Distributions

Characteristics All normal curves are symmetric, single-peaked, bell-shaped Any specific normal curve is completely described by its mean (μ) and standard deviation (σ) Mean and median are in the center, so changing μ without changing σ simply moves the graph along the x-axis Standard deviation controls the spread of a normal curve

Are these curves normal?

Empirical Rule

Warm-Up 09/13 Are these normal? Why or why not?

Warm-Up 09/14 Consider a normally distributed data set with a mean of 520 and a standard deviation of 98. What percent of values are below 422? What interval, centered on the mean, contains 68% of the values? What percent of values is between 324 and 422?

Heights Activity Questions Are these curves approximately normal? How are the graphs the same/different? Which represents a normal curve more? Find the mean and standard deviation of the data. Assuming it’s normal,estimate how many students fall within one, two and three standard deviations. Compare with the actual data. How could we make this graph more normal?

Z-Scores

Calculating a z-score

“The BeyonCE Rule” For Z-Scores I love Stats! “The BeyonCE Rule” For Z-Scores Z-Scores measure everything to the LEFT

Warm-Up 09/19 You take a sample of 100 student SAT scores and find its average to be 1140. If the population has a standard deviation of 194, what percentile is a score of 1300 in?

Warm-Up 09/20 A study selected 100,000 Facebook users and found that the individuals in the sample had an average of 232 friends. If the population standard deviation is 64, what percentage of the users sampled had between 100 and 200 friends?

Warm-up 09/21 Sally studied insects in Australia and measured their diameters. After taking a sample with population standard deviation of 0.44 cm and population mean 20 cm, she wanted to know what diameter would define the 85th percentile. Find that diameter.