Density curves and normal distributions
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Density Curves and Normal Distributions. Section 2.1.

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Density Curves and Normal Distributions

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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

Page 79-80


Density Curves

Page 79-80

0.303

0.293


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.


30% or 0.3

20% or 0.2

10% or 0.1

Relative Frequency

123456

Outcomes

  • What would the results look like if we rolled a fair die 100 times?


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.


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


Density Curves: Skewed

Mean: the balance point of the curve (if it was made of solid material)


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 points – points at which this change of curvature takes place.

Inflection point

σ

Concave up

µ


Normal Curves


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 2σofµ.

95% of data

-3 -2 -1 0 1 2 3


The Empirical Rule

  • 99.7% of the observations fall within 3σof 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.

  • N(µ,σ)

    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

  • What percentile of young women are 64.5 inches or shorter?

50%

99.7% of data

5759.5 62 64.5 67 69.5 72

Height (in inches)


Heights of Young Women

  • What percentile of young women are 59.5 inches or shorter?

2.5%

99.7% of data

5759.5 62 64.5 67 69.5 72

Height (in inches)


Heights of Young Women

  • 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

5759.5 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


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