Density Curves and Normal Distributions Section 2.1

Sometimes the overall pattern of a distribution can be described by a smooth curve. This histogram shows the distribution of vocab scores. We could use it to see how many students scored at each value, or what percent of students got 4’s, above 10, etc.

Density Curves A density curve is an idealized mathematical model for a set of data. It ignores minor irregularities and outliers

Density Curves   Eyes respond to the areas of the bars in a histogram. Bar areas represent proportions of the observations. Page 79-80

Density Curves Page 79-80 Now look at the curve drawn through the bars. This line is called a Density curve. The area under the curve to the left of 6.0 is shaded. The area is 0.293. Only 0.01 away from the histogram result. Good approximation. 0.303 0.293

Always on or above the horizontal axis Density Curve Always on or above the horizontal axis Has an area of exactly 1 underneath it

Types of Density Curves Normal curves Uniform density curves Later we’ll see important density curves that are skewed left/right and other curves related to the normal curve

Density Curve  Area = 1, corresponds to 100% of the data

What would the results look like if we rolled a fair die 100 times? Press STAT ENTER Choose a list: highlight the name and press ENTER. Type: MATH  PRB 5:randInt(1,6,100) ENTER Look at a histogram of the results: 2ND Y= ENTER Press WINDOW and change your settings Press GRAPH. Use TRACE button to see heights. Let them work this out. Then they sketch. Theirs will not necessarily look the same!

What would the results look like if we rolled a fair die 100 times? 1 2 3 4 5 6 Outcomes 30% or 0.3 20% or 0.2 10% or 0.1 Relative Frequency Let them work this out. Then they sketch. Theirs will not necessarily look the same!

In a perfect world… The different outcomes when you roll a die are equally likely, so the ideal distribution would look something like this: An example of a uniform density curve. Example of Uniform density curve.

Other Density Curves What percent of observations are between 0 and 2? (area between 0 and 2) Area of rectangle: 2(.2) = .4 Area of triangle: ½ (2)(.2) = .2 Total Area = .4 + .2 = .6 = 60%

Other Density Curves What percent of observations are between 3 and 4? Area: (1)(.2) = .2 = 20%

Normal curve    

Density Curves: Skewed M Median: the equal-areas point of the curve Half of the area on each side Mark where you think the Median would be.

Density Curves: Skewed Mean: the balance point of the curve (if it was made of solid material) Mark where you think the mean would be.

Mean and Median Of Density Curves Just remember: Symmetrical distribution Mean and median are in the center Skewed distribution Mean gets pulled towards the skew and away from the median.

Notation Since density curves are idealized, the mean and standard deviation of a density curve will be slightly different from the actual mean and standard deviation of the distribution (histogram) that we’re approximating, and we want a way to distinguish them

Notation For actual observations (our sample): use and s. For idealized (theoretical): use μ (mu) for mean and σ (sigma) for the standard deviation.

Normal Curves are always: Described in terms of their mean (µ) and standard deviation (σ) Symmetric One peak and two tails

Normal Curves Concave down Inflection point σ Concave up µ Inflection points – points at which this change of curvature takes place.

Normal Curves Larger std. dev. = more spread out, Smaller std. dev. = more pulled in

The Empirical Rule The 68-95-99.7 Rule -3 -2 -1 0 1 2 3

The Empirical Rule 68% of the observations fall within σ of the mean µ. 68 % of data -3 -2 -1 0 1 2 3

The Empirical Rule 95% of the observations fall with 2σ of µ. 95% of data -3 -2 -1 0 1 2 3

The Empirical Rule 99.7% of the observations fall within 3σ of µ. 99.7% of data -3 -2 -1 0 1 2 3

Heights of Young Women The distribution of heights of young women aged 18 to 24 is approximately normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches. 64.5 – 2.5 = 62 64.5 + 2.5 = 67 62 64.5 67 Height (in inches)

Heights of Young Women The distribution of heights of young women aged 18 to 24 is approximately normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches. 5 62 64.5 67 Height (in inches) 59.5 62 64.5 67 69.5 Height (in inches)

Heights of Young Women The distribution of heights of young women aged 18 to 24 is approximately normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches. 99.7% of data 62 64.5 67 Height (in inches) 59.5 62 64.5 67 69.5 Height (in inches)

Shorthand with Normal Dist. Ex: The distribution of young women’s heights is N(64.5, 2.5). What this means: Normal Distribution centered at µ = 64.5 with a standard deviation σ = 2.5.

Heights of Young Women 50% What percentile of young women are 64.5 inches or shorter? 50% 99.7% of data 57 59.5 62 64.5 67 69.5 72 Height (in inches)

Heights of Young Women 2.5% What percentile of young women are 59.5 inches or shorter? 2.5% 99.7% of data 57 59.5 62 64.5 67 69.5 72 Height (in inches)

Heights of Young Women 64.5 or less = 50% 50% – 2.5% = 47.5% What percentile of young women are between 59.5 inches and 64.5 inches? 64.5 or less = 50% 59.5 or less = 2.5% 50% – 2.5% = 47.5% 99.7% of data 57 59.5 62 64.5 67 69.5 72 Height (in inches)

Practice For homework: 2.1, 2.3, 2.4 p. 83 2.6, 2.7, 2.8 p. 89