Section 2.1 Density Curves & the Normal Distributions Honors Statistics September 15, 2014 Mr. Calise

AP Statistics, Section 2.1, Part 1 Density Curves A density curve is similar to a histogram, but there are several important distinctions. 1. A smooth curve is used to represent data rather than bars. However, a density curve describes the proportions of the observations that fall in each range rather than the actual number of observations. 2. The scale should be adjusted so that the total area under the curve is exactly 1. This represents the proportion 1 (or 100%). AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 Density Curve “Mathematical Models” The area under the curve is related to the distribution of values “Idealized description” AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 Density Curves Characteristics Always above the x-axis Area always equal to 1 The area under the curve and above any range of values is the proportion of all observations that fall in that range. AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 Median and Mean Since the area represents portions of the population, the Median is the spot where the area to the left is the same as the area to the right. AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 Median and Mean The Mean represents the “balance point”. Imagine a “see-saw” AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 Median and Mean The Mean is always pulled towards the tail in a skewed distribution AP Statistics, Section 2.1, Part 1

“Idealized Distributions” We use different notation for density curve (which represent entire populations) as compared to data sets (which represent samples) AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 Normal Distribution A special, bell shaped, symmetric, single-peaked distribution AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 Normal Distribution Because of the symmetry, the mean and median are the same and at the line symmetry AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 Normal Curves: μ and σ The "control factors" are the mean μ and the standard deviation σ. Changing only μ will move the curve along the horizontal axis. The standard deviation σ controls the spread of the distribution. Remember that a large σ implies that the data is spread out. AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 Normal Distribution The inflection points (where the curve starts to flatten out) represent the width of the standard deviation μ-σ μ μ+σ AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 68-95-99.7 Rule AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 Heights of Young Women The distribution of heights of young women aged 18 to 24 is approximately normally distributed with mean  = 64.5 inches and standard deviation  = 2.5 inches. AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 68-95-99.7 Rule with N(64.5,2.5) AP Statistics, Section 2.1, Part 1

Where do the middle 95% of heights fall? What percent of the heights are above 69.5 inches? A height of 62 inches is what percentile? What percent of the heights are between 62 and 67 inches? What percent of heights are less than 57 in.?

Why Use the Normal Distribution??? 1. They are frequently in large data sets (all SAT scores), repeated measurements of the same quantity, and in biological populations (lengths of roaches). 2. They are often good approximations to chance outcomes (like coin flipping). 3. We can apply things we learn in studying normal distributions to other distributions. AP Statistics, Section 2.1, Part 1

AP Statistics, Section 2.1, Part 1 Assignment Exercises 2.1 – 2.9, The Practice of Statistics, for Monday. AP Statistics, Section 2.1, Part 1