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

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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
Density Curves 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.

  • A density curve is an idealized mathematical model for a set of data.

    • It ignores minor irregularities and outliers


Density curves1
Density Curves 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.

Page 79-80


Density curves2
Density Curves 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.

Page 79-80

0.303

0.293


Density curve
Density Curve 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.

  • Always on or above the horizontal axis

  • Has an area of exactly 1 underneath it


Types of density curves
Types of Density Curves 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.

  • 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 curve1
Density Curve 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.

 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 times?

20% or 0.2

10% or 0.1

Relative Frequency

1 2 3 4 5 6

Outcomes

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


In a perfect world
In a perfect world… times?

  • 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
Other Density Curves times?

  • 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 curves1
Other Density Curves times?

  • What percent of observations are between 3 and 4?

Area: (1)(.2) = .2 = 20%


Normal curve
Normal curve times?


Density curves skewed
Density Curves: Skewed times?

M

Median: the equal-areas point of the curve

Half of the area on each side


Density curves skewed1
Density Curves: Skewed times?

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


Mean and median of density curves
Mean and Median times?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
Notation times?

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


Notation1
Notation times?

  • 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
Normal Curves are always: times?

  • Described in terms of their mean (µ) and standard deviation (σ)

  • Symmetric

  • One peak and two tails


Normal curves
Normal Curves times?

Concave down

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

Inflection point

σ

Concave up

µ



The empirical rule
The Empirical Rule times?

  • The 68-95-99.7 Rule

-3 -2 -1 0 1 2 3


The empirical rule1
The Empirical Rule times?

  • 68% of the observations fall within σ of the mean µ.

68 % of data

-3 -2 -1 0 1 2 3


The empirical rule2
The Empirical Rule times?

  • 95% of the observations fall with 2σofµ.

95% of data

-3 -2 -1 0 1 2 3


The empirical rule3
The Empirical Rule times?

  • 99.7% of the observations fall within 3σof µ.

99.7% of data

-3 -2 -1 0 1 2 3


Heights of young women
Heights of Young Women times?

  • 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 women1
Heights of Young Women times?

  • 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 women2
Heights of Young Women times?

  • 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
Shorthand with Normal Dist. times?

  • 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 women3
Heights of Young Women times?

  • 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 women4
Heights of Young Women times?

  • 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 women5
Heights of Young Women times?

  • 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
Practice times?

  • For homework:

    • 2.1, 2.3, 2.4 p. 83

    • 2.6, 2.7, 2.8 p. 89


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