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Density Curves and Normal Distributions. Section 2.1.

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 4â€™s, above 10, etc.

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

Page 79-80

Page 79-80

0.303

0.293

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

- Normal curves
- Uniform density curves
- Later weâ€™ll see important density curves that are skewed left/right and other curves related to the normal curve

ïƒŸ 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 ïƒ 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?

- 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.

- What percent of observations are between 0 and 2? (area between 0 and 2)

Area of rectangle: 2(.2) = .4

Area of triangle: Â½ (2)(.2) = .2

Total Area = .4 + .2 = .6 = 60%

- What percent of observations are between 3 and 4?

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

M

Median: the equal-areas point of the curve

Half of the area on each side

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

Just remember:

- Symmetrical distribution
- Mean and median are in the center

- Skewed distribution
- Mean gets pulled towards the skew and away from the median.

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 weâ€™re approximating, and we want a way to distinguish them

- For actual observations (our sample): use and s.
- For idealized (theoretical): use Î¼ (mu) for mean and Ïƒ (sigma) for the standard deviation.

- Described in terms of their mean (Âµ) and standard deviation (Ïƒ)
- Symmetric
- One peak and two tails

Concave down

Inflection points â€“ points at which this change of curvature takes place.

Inflection point

Ïƒ

Concave up

Âµ

- The 68-95-99.7 Rule

-3 -2 -1 0 1 2 3

- 68% of the observations fall within Ïƒ of the mean Âµ.

68 % of data

-3 -2 -1 0 1 2 3

- 95% of the observations fall with 2ÏƒofÂµ.

95% of data

-3 -2 -1 0 1 2 3

- 99.7% of the observations fall within 3Ïƒof Âµ.

99.7% of data

-3 -2 -1 0 1 2 3

- The distribution of heights of young women aged 18 to 24 is approximately normal with mean Âµ = 64.5 inches and standard deviation Ïƒ = 2.5 inches.

64.5 â€“ 2.5 = 62

64.5 + 2.5 = 67

62 64.5 67 Height (in inches)

- The distribution of heights of young women aged 18 to 24 is approximately normal with mean Âµ = 64.5 inches and standard deviation Ïƒ = 2.5 inches.

5

62 64.5 67 Height (in inches)

59.5 62 64.5 67 69.5

Height (in inches)

- The distribution of heights of young women aged 18 to 24 is approximately normal with mean Âµ = 64.5 inches and standard deviation Ïƒ = 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)

- N(Âµ,Ïƒ)
Ex: The distribution of young womenâ€™s heights is N(64.5, 2.5).

What this means:

Normal Distribution centered at Âµ = 64.5 with a standard deviation Ïƒ = 2.5.

- 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)

- 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)

- 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% â€“ 2.5% = 47.5%

99.7% of data

5759.5 62 64.5 67 69.5 72

Height (in inches)

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