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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 4’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 we’ll see important density curves that are skewed left/right and other curves related to the normal 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?

- 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: ½ (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 we’re approximating, and we want a way to distinguish them

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

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

Concave down

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

Inflection point

σ

Concave up

µ

- The 68-95-99.7 Rule

-3 -2 -1 0 1 2 3

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

68 % of data

-3 -2 -1 0 1 2 3

- 95% of the observations fall with 2σofµ.

95% of data

-3 -2 -1 0 1 2 3

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

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 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)

- 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)

- 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)

- 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.

- 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% – 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