Presentation on theme: "Density Curves and Normal Distributions"— Presentation transcript:
1 Density Curves and Normal Distributions Section 2.1
2 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.
3 Density CurvesA density curve is an idealized mathematical model for a set of data.It ignores minor irregularities and outliers
4 Density CurvesEyes respond to the areas of the bars in a histogram. Bar areas represent proportions of the observations.Page 79-80
5 Density CurvesPage 79-80Now 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 Only 0.01 away from the histogram result. Good approximation.0.3030.293
6 Always on or above the horizontal axis Density CurveAlways on or above the horizontal axisHas an area of exactly 1 underneath it
7 Types of Density Curves Normal curvesUniform density curvesLater we’ll see important density curves that are skewed left/right and other curves related to the normal curve
8 Density Curve Area = 1, corresponds to 100% of the data
9 What would the results look like if we rolled a fair die 100 times? Press STAT ENTERChoose a list: highlight the name and press ENTER.Type: MATH PRB 5:randInt(1,6,100) ENTERLook at a histogram of the results: ND Y= ENTERPress WINDOW and change your settingsPress GRAPH. Use TRACE button to see heights.Let them work this out. Then they sketch. Theirs will not necessarily look the same!
10 What would the results look like if we rolled a fair die 100 times? Outcomes30% or 0.320% or 0.210% or 0.1Relative FrequencyLet them work this out. Then they sketch. Theirs will not necessarily look the same!
11 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.
12 Other Density CurvesWhat percent of observations are between 0 and 2? (area between 0 and 2)Area of rectangle: 2(.2) = .4Area of triangle: ½ (2)(.2) = .2Total Area = = .6 = 60%
13 Other Density Curves What percent of observations are between 3 and 4? Area: (1)(.2) = .2 = 20%
15 Density Curves: Skewed MMedian: the equal-areas point of the curveHalf of the area on each sideMark where you think the Median would be.
16 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.
17 Mean and Median Of Density Curves Just remember:Symmetrical distributionMean and median are in the centerSkewed distributionMean gets pulled towards the skew and away from the median.
18 NotationSince 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
19 Notation For actual observations (our sample): use and s. For idealized (theoretical): use μ (mu) for mean and σ (sigma) for the standard deviation.
20 Normal Curves are always: Described in terms of their mean (µ) and standard deviation (σ)SymmetricOne peak and two tails
21 Normal CurvesConcave downInflection pointσConcave upInflection points – points at which this change of curvature takes place.
22 Normal CurvesLarger std. dev. = more spread out, Smaller std. dev. = more pulled in
24 The Empirical Rule68% of the observations fall within σ of the mean µ.68 % of data
25 The Empirical Rule 95% of the observations fall with 2σ of µ. 95% of data
26 The Empirical Rule 99.7% of the observations fall within 3σ of µ. 99.7% of data
27 Heights of Young WomenThe 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= 67Height (in inches)
28 Heights of Young WomenThe 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.5Height (in inches)Height (in inches)
29 Heights of Young WomenThe 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 dataHeight (in inches)Height (in inches)
30 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.
31 Heights of Young Women 50% What percentile of young women are 64.5 inches or shorter?50%99.7% of dataHeight (in inches)
32 Heights of Young Women 2.5% What percentile of young women are 59.5 inches or shorter?2.5%99.7% of dataHeight (in inches)
33 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 dataHeight (in inches)
34 PracticeFor homework:2.1, 2.3, p. 832.6, 2.7, p. 89